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                               **Ray Tracing: The Rest of Your Life**
                   [Peter Shirley][], [Trevor David Black][], [Steve Hollasch][]
                                                <br>
                                     Version 4.0.0-alpha.1, 2023-08-06
                                                <br>
                      Copyright 2018-2023 Peter Shirley. All rights reserved.



Overview
====================================================================================================
In _Ray Tracing in One Weekend_ and _Ray Tracing: the Next Week_, you built a “real” ray tracer.

If you are motivated, you can take the source and information contained in those books to implement
any visual effect you want. The source provides a meaningful and robust foundation upon which to
build out a raytracer for a small hobby project. Most of the visual effects found in commercial ray
tracers rely on the techniques described in these first two books. However, your capacity to add
increasingly complicated visual effects like subsurface scattering or nested dielectrics will be
severely limited by a missing mathematical foundation. In this volume, I assume that you are either
a highly interested student, or are someone who is pursuing a career related to ray tracing. We will
be diving into the math of creating a very serious ray tracer. When you are done, you should be
well equipped to use and modify the various commercial ray tracers found in many popular domains,
such as the movie, television, product design, and architecture industries.

There are many many things I do not cover in this short volume. For example, there are many ways of
writing Monte Carlo rendering programs--I dive into only one of them. I don’t cover shadow rays
(deciding instead to make rays more likely to go toward lights), nor do I cover bidirectional
methods, Metropolis methods, or photon mapping. You'll find many of these techniques in the
so-called "serious ray tracers", but they are not covered here because it is more important to cover
the concepts, math, and terms of the field. I think of this book as a deep exposure that should be
your first of many, and it will equip you with some of the concepts, math, and terms that you'll
need in order to study these and other interesting techniques.

I hope that you find the math as fascinating as I do.

As before, https://in1weekend.blogspot.com/ will have further readings and references.

Thanks to everyone who lent a hand on this project. You can find them in the acknowledgments section
at the end of this book.



A Simple Monte Carlo Program
====================================================================================================
Let’s start with one of the simplest Monte Carlo programs. If you're not familiar with Monte Carlo
programs, then it'll be good to pause and catch you up. There are two kinds of randomized
algorithms: Monte Carlo and Las Vegas. Randomized algorithms can be found everywhere in computer
graphics, so getting a decent foundation isn't a bad idea. A randomized algorithm uses some amount
of randomness in its computation. A Las Vegas (LV) random algorithm always produces the correct
result, whereas a Monte Carlo (MC) algorithm _may_ produce a correct result--and frequently gets it
wrong! But for especially complicated problems such as ray tracing, we may not place as huge a
priority on being perfectly exact as on getting an answer in a reasonable amount of time. LV
algorithms will eventually arrive at the correct result, but we can't make too many guarantees on
how long it will take to get there. The classic example of an LV algorithm is the _quicksort_
sorting algorithm. The quicksort algorithm will always complete with a fully sorted list, but, the
time it takes to complete is random. Another good example of an LV algorithm is the code that we use
to pick a random point in a unit sphere:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    inline vec3 random_in_unit_sphere() {
        while (true) {
            auto p = vec3::random(-1,1);
            if (p.length_squared() < 1)
                return p;
        }
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [las-vegas-algo]: <kbd>[vec3.h]</kbd> A Las Vegas algorithm]

This code will always eventually arrive at a random point in the unit sphere, but we can't say
beforehand how long it'll take. It may take only 1 iteration, it may take 2, 3, 4, or even longer.
Whereas, an MC program will give a statistical estimate of an answer, and this estimate will get
more and more accurate the longer you run it. Which means that at a certain point, we can just
decide that the answer is accurate _enough_ and call it quits. This basic characteristic of simple
programs producing noisy but ever-better answers is what MC is all about, and is especially good for
applications like graphics where great accuracy is not needed.


Estimating Pi
--------------
The canonical example of a Monte Carlo algorithm is estimating $\pi$, so let's do that. There are
many ways to estimate $\pi$, with the Buffon Needle problem being a classic case study. We’ll do a
variation inspired by this method. Suppose you have a circle inscribed inside a square:

  ![Figure [circ-square]: Estimating π with a circle inside a square
  ](../images/fig-3.01-circ-square.jpg)

Now, suppose you pick random points inside the square. The fraction of those random points that end
up inside the circle should be proportional to the area of the circle. The exact fraction should in
fact be the ratio of the circle area to the square area:

  $$ \frac{\pi r^2}{(2r)^2} = \frac{\pi}{4} $$

<div class='together'>
Since the $r$ cancels out, we can pick whatever is computationally convenient. Let’s go with $r=1$,
centered at the origin:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    #include "rtweekend.h"

    #include <iostream>
    #include <iomanip>
    #include <math.h>
    #include <stdlib.h>

    int main() {
        int N = 100000;
        int inside_circle = 0;
        for (int i = 0; i < N; i++) {
            auto x = random_double(-1,1);
            auto y = random_double(-1,1);
            if (x*x + y*y < 1)
                inside_circle++;
        }
        std::cout << std::fixed << std::setprecision(12);
        std::cout << "Estimate of Pi = " << (4.0 * inside_circle) / N << '\n';
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [estpi-1]: <kbd>[pi.cc]</kbd> Estimating π]

The answer of $\pi$ found will vary from computer to computer based on the initial random seed.
On my computer, this gives me the answer `Estimate of Pi = 3.143760000000`.

</div>


Showing Convergence
--------------------
If we change the program to run forever and just print out a running estimate:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    #include "rtweekend.h"

    #include <iostream>
    #include <iomanip>
    #include <math.h>
    #include <stdlib.h>

    int main() {
        int inside_circle = 0;
        int runs = 0;
        std::cout << std::fixed << std::setprecision(12);
        while (true) {
            runs++;
            auto x = random_double(-1,1);
            auto y = random_double(-1,1);
            if (x*x + y*y < 1)
                inside_circle++;

            if (runs % 100000 == 0)
                std::cout << "Estimate of Pi = "
                          << (4.0 * inside_circle) / runs
                          << '\n';
        }
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [estpi-2]: <kbd>[pi.cc]</kbd> Estimating π, v2]


Stratified Samples (Jittering)
-------------------------------
We get very quickly near $\pi$, and then more slowly zero in on it. This is an example of the _Law
of Diminishing Returns_, where each sample helps less than the last. This is the worst part of Monte
Carlo. We can mitigate this diminishing return by _stratifying_ the samples (often called
_jittering_), where instead of taking random samples, we take a grid and take one sample within
each:

  ![Figure [jitter]: Sampling areas with jittered points](../images/fig-3.02-jitter.jpg)

<div class='together'>
This changes the sample generation, but we need to know how many samples we are taking in advance
because we need to know the grid. Let’s take a million and try it both ways:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    #include "rtweekend.h"

    #include <iostream>
    #include <iomanip>

    int main() {
        int inside_circle = 0;
        int inside_circle_stratified = 0;
        int sqrt_N = 1000;
        for (int i = 0; i < sqrt_N; i++) {
            for (int j = 0; j < sqrt_N; j++) {
                auto x = random_double(-1,1);
                auto y = random_double(-1,1);
                if (x*x + y*y < 1)
                    inside_circle++;
                x = 2*((i + random_double()) / sqrt_N) - 1;
                y = 2*((j + random_double()) / sqrt_N) - 1;
                if (x*x + y*y < 1)
                    inside_circle_stratified++;
            }
        }

        std::cout << std::fixed << std::setprecision(12);
        std::cout
            << "Regular    Estimate of Pi = "
            << (4.0 * inside_circle) / (sqrt_N*sqrt_N) << '\n'
            << "Stratified Estimate of Pi = "
            << (4.0 * inside_circle_stratified) / (sqrt_N*sqrt_N) << '\n';
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [estpi-3]: <kbd>[pi.cc]</kbd> Estimating π, v3]

</div>

<div class='together'>
On my computer, I get:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Regular    Estimate of Pi = 3.141184000000
    Stratified Estimate of Pi = 3.141460000000
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Where the first 12 decimal places of pi are:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    3.141592653589
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

</div>

Interestingly, the stratified method is not only better, it converges with a better asymptotic rate!
Unfortunately, this advantage decreases with the dimension of the problem (so for example, with the
3D sphere volume version the gap would be less). This is called the _Curse of Dimensionality_. Ray
tracing is a very high-dimensional algorithm, where each reflection adds two new dimensions:
$\phi_o$ and $\theta_o$. We won't be stratifying the output reflection angle in this book, simply
because it is a little bit too complicated to cover, but there is a lot of interesting research
currently happening in this space.

As an intermediary, we'll be stratifying the locations of the sampling positions around each pixel
location.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    #include "rtweekend.h"

    #include "camera.h"
    #include "color.h"
    #include "hittable_list.h"
    #include "material.h"
    #include "quad.h"
    #include "sphere.h"

    int main() {
        hittable_list world;

        auto red   = make_shared<lambertian>(color(.65, .05, .05));
        auto white = make_shared<lambertian>(color(.73, .73, .73));
        auto green = make_shared<lambertian>(color(.12, .45, .15));
        auto light = make_shared<diffuse_light>(color(15, 15, 15));

        // Cornell box sides
        world.add(make_shared<quad>(point3(555,0,0), vec3(0,0,555), vec3(0,555,0), green));
        world.add(make_shared<quad>(point3(0,0,555), vec3(0,0,-555), vec3(0,555,0), red));
        world.add(make_shared<quad>(point3(0,555,0), vec3(555,0,0), vec3(0,0,555), white));
        world.add(make_shared<quad>(point3(0,0,555), vec3(555,0,0), vec3(0,0,-555), white));
        world.add(make_shared<quad>(point3(555,0,555), vec3(-555,0,0), vec3(0,555,0), white));

        // Light
        world.add(make_shared<quad>(point3(213,554,227), vec3(130,0,0), vec3(0,0,105), light));

        // Box 1
        shared_ptr<hittable> box1 = box(point3(0,0,0), point3(165,330,165), white);
        box1 = make_shared<rotate_y>(box1, 15);
        box1 = make_shared<translate>(box1, vec3(265,0,295));
        world.add(box1);

        // Box 2
        shared_ptr<hittable> box2 = box(point3(0,0,0), point3(165,165,165), white);
        box2 = make_shared<rotate_y>(box2, -18);
        box2 = make_shared<translate>(box2, vec3(130,0,65));
        world.add(box2);

        camera cam;

        cam.aspect_ratio      = 1.0;
        cam.image_width       = 600;
        cam.samples_per_pixel = 64;
        cam.max_depth         = 50;
        cam.background        = color(0,0,0);

        cam.vfov     = 40;
        cam.lookfrom = point3(278, 278, -800);
        cam.lookat   = point3(278, 278, 0);
        cam.vup      = vec3(0, 1, 0);

        cam.defocus_angle = 0;

        cam.render(world, lights);
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [estpi-3]: <kbd>[main.cc]</kbd> Stratifying the samples inside pixels]


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class camera {
      public:
        ...
        void render(const hittable& world) {
            initialize();

            std::cout << "P3\n" << image_width << ' ' << image_height << "\n255\n";

            for (int j = 0; j < image_height; ++j) {
                std::clog << "\rScanlines remaining: " << (image_height - j) << ' ' << std::flush;
                for (int i = 0; i < image_width; ++i) {
                    color pixel_color(0,0,0);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
                    for (int s_j = 0; s_j < sqrt_spp; ++s_j) {
                        for (int s_i = 0; s_i < sqrt_spp; ++s_i) {
                            ray r = get_ray(i, j, s_i, s_j);
                            pixel_color += ray_color(r, max_depth, world);
                        }
                    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
                    write_color(std::cout, pixel_color, samples_per_pixel);
                }
            }

            std::clog << "\rDone.                 \n";
        }
        ...
      private:
        ...
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        ray get_ray(int i, int j, int s_i, int s_j) const {
            // Get a randomly-sampled camera ray for the pixel at location i,j, originating from
            // the camera defocus disk, and randomly sampled around the pixel location.
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            auto pixel_center = pixel00_loc + (i * pixel_delta_u) + (j * pixel_delta_v);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            auto pixel_sample = pixel_center + pixel_sample_square(s_i, s_j);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            auto ray_origin = (defocus_angle <= 0) ? center : defocus_disk_sample();
            auto ray_direction = pixel_sample - ray_origin;
            auto ray_time = random_double();

            return ray(ray_origin, ray_direction, ray_time);
        }


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        vec3 pixel_sample_square(int s_i, int s_j) const {
            // Returns a random point in the square surrounding a pixel at the origin, given
            // the two subpixel indices.
            auto px = -0.5 + recip_sqrt_spp * (s_i + random_double());
            auto py = -0.5 + recip_sqrt_spp * (s_j + random_double());
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
            return (px * pixel_delta_u) + (py * pixel_delta_v);
        }

        ...
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [render-estpi-3]: <kbd>[camera.h]</kbd> Stratifying the samples inside pixels (render)]

<div class='together'>
If we compare the results from without stratification:

  ![<span class='num'>Image 1:</span> Cornell box, no stratification
  ](../images/img-3.01-cornell-no-strat.png class='pixel')

</div>

<div class='together'>
To after, with stratification:

  ![<span class='num'>Image 2:</span> Cornell box, with stratification
  ](../images/img-3.02-cornell-strat.png class='pixel')

You should, if you squint, be able to see sharper contrast at the edges of planes and at the edges
of boxes. The effect will be more pronounced at locations that have a higher frequency of change.
High frequency change can also be thought of as high information density. For our cornell box scene,
all of our materials are matte, with a soft area light overhead, so the only locations of high
information density are at the edges of objects. The effect will be more obvious with textures and
reflective materials.

If you are ever doing single-reflection or shadowing or some strictly 2D problem, you definitely
want to stratify.

</div>



One Dimensional Monte Carlo Integration
====================================================================================================
Our Buffon Needle example is a way of calculating $\pi$ by solving for the ratio of the area of the
circle and the area of the inscribing square:

    $$ \frac{\operatorname{area}(\mathit{circle})}{\operatorname{area}(\mathit{square})}
       = \frac{\pi}{4}
    $$

We picked a bunch of random points in the inscribing square and counted the fraction of them that
were also in the unit circle. This fraction was an estimate that tended toward $\frac{\pi}{4}$ as
more points were added. If we didn't know the area of a circle, we could still solve for it using
the above ratio. We know that the ratio of areas of the unit circle and the inscribing square is
$\frac{\pi}{4}$, and we know that the area of a inscribing square is $4r^2$, so we could then use
those two quantities to get the area of a circle:

    $$ \frac{\operatorname{area}(\mathit{circle})}{\operatorname{area}(\mathit{square})}
       = \frac{\pi}{4}
    $$

    $$ \frac{\operatorname{area}(\mathit{circle})}{(2r)^2} = \frac{\pi}{4} $$

    $$ \operatorname{area}(\mathit{circle}) = \frac{\pi}{4} 4r^2 $$

    $$ \operatorname{area}(\mathit{circle}) = \pi r^2 $$

We choose a circle with radius $r = 1$ and get:

    $$ \operatorname{area}(\mathit{circle}) = \pi $$

<div class='together'>
Our work above is equally valid as a means to solve for $pi$ as it is a means to solve for the area
of a circle:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    #include "rtweekend.h"

    #include <iostream>
    #include <iomanip>
    #include <math.h>
    #include <stdlib.h>

    int main() {
        int N = 100000;
        int inside_circle = 0;
        for (int i = 0; i < N; i++) {
            auto x = random_double(-1,1);
            auto y = random_double(-1,1);
            if (x*x + y*y < 1)
                inside_circle++;
        }
        std::cout << std::fixed << std::setprecision(12);
        std::cout << "Estimated area of unit circle = " << (4.0 * inside_circle) / N << '\n';
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [estunitcircle]: <kbd>[pi.cc]</kbd> Estimating area of unit circle]

</div>


Expected Value
--------------
Let's take a step back and think about our Monte Carlo algorithm a little bit more generally.

If we assume that we have all of the following:

1. A list of values $X$ that contains members $x_i$:

    $$ X = (x_0, x_1, ..., x_{N-1})  $$

2. A continuous function $f(x)$ that takes members from the list:

    $$ y_i = f(x_i) $$

3. A function $F(X)$ that takes the list $X$ as input and produces the list $Y$ as output:

    $$ Y = F(X) $$

4. Where output list $Y$ has members $y_i$:

    $$ Y = (y_0, y_1, ..., y_{N-1}) = (f(x_0), f(x_1), ..., f(x_{N-1})) $$

If we assume all of the above, then we could solve for the arithmetic mean--the average--of the
list $Y$ with the following:

    $$ \operatorname{average}(Y) = E[Y] = \frac{1}{N} \sum_{i=0}^{N-1} y_i $$
    $$ = \frac{1}{N} \sum_{i=0}^{N-1} f(x_i) $$
    $$ = E[F(X)] $$

Where $E[Y]$ is referred to as the _expected value of_ $Y$. If the values of $x_i$ are chosen
randomly from a continuous interval $[a,b]$ such that $ a \leq x_i \leq b $ for all values of $i$,
then $E[F(X)]$ will approximate the average of the continuous function $f(x')$ over the the same
interval $ a \leq x' \leq b $.

    $$ E[f(x') | a \leq x' \leq b] \approx E[F(X) | X =
        \{\small x_i | a \leq x_i \leq b \normalsize \} ] $$
    $$ \approx E[Y = \{\small y_i = f(x_i) | a \leq x_i \leq b \normalsize \} ] $$

    $$ \approx \frac{1}{N} \sum_{i=0}^{N-1} f(x_i) $$

If we take the number of samples $N$ and take the limit as $N$ goes to $\infty$, then we get the
following:

    $$ E[f(x') | a \leq x' \leq b]  = \lim_{N \to \infty} \frac{1}{N} \sum_{i=0}^{N-1} f(x_i) $$

Within the continuous interval $[a,b]$, the expected value of continuous function $f(x')$ can be
perfectly represented by summing an infinite number of random points within the interval. As this
number of points approaches $\infty$ the average of the outputs tends to the exact answer. This is a
Monte Carlo algorithm.

Sampling random points isn't our only way to solve for the expected value over an interval. We can
also choose where we place our sampling points. If we had $N$ samples over an interval $[a,b]$ then
we could choose to equally space points throughout:

    $$ x_i = a + i \Delta x $$
    $$ \Delta x = \frac{b - a}{N} $$

Then solving for their expected value:

    $$ E[f(x') | a \leq x' \leq b] \approx \frac{1}{N} \sum_{i=0}^{N-1} f(x_i)
        \Big|_{x_i = a + i \Delta x} $$
    $$ E[f(x') | a \leq x' \leq b] \approx \frac{\Delta x}{b - a} \sum_{i=0}^{N-1} f(x_i)
        \Big|_{x_i = a + i \Delta x} $$
    $$ E[f(x') | a \leq x' \leq b] \approx \frac{1}{b - a} \sum_{i=0}^{N-1} f(x_i) \Delta x
        \Big|_{x_i = a + i \Delta x} $$

Take the limit as $N$ approaches $\infty$

    $$ E[f(x') | a \leq x' \leq b] = \lim_{N \to \infty} \frac{1}{b - a} \sum_{i=0}^{N-1}
        f(x_i) \Delta x \Big|_{x_i = a + i \Delta x} $$

This is, of course, just a regular integral:

    $$ E[f(x') | a \leq x' \leq b] = \frac{1}{b - a} \int_{a}^{b} f(x) dx $$

If you recall your introductory calculus class, the integral of a function is the area under the
curve over that interval:

    $$ \operatorname{area}(f(x), a, b) = \int_{a}^{b} f(x) dx$$

Therefore, the average over an interval is intrinsically linked with the area under the curve in
that interval.

    $$  E[f(x) | a \leq x \leq b] = \frac{1}{b - a} \cdot \operatorname{area}(f(x), a, b) $$

Both the integral of a function and a Monte Carlo sampling of that function can be used to solve for
the average over a specific interval. While integration solves for the average with the sum of
infinitely many infinitesimally small slices of the interval, a Monte Carlo algorithm will
approximate the same average by solving the sum of ever increasing random sample points within the
interval. Counting the number of points that fall inside of an object isn't the only way to measure
its average or area. Integration is also a common mathematical tool for this purpose. If a closed
form exists for a problem, integration is frequently the most natural and clean way to formulate
things.

I think a couple of examples will help.


Integrating x²
---------------
Let’s look at a classic integral:

    $$ I = \int_{0}^{2} x^2 dx $$

We could solve this using integration:

    $$ I = \frac{1}{3} x^3 \Big|_{0}^{2} $$
    $$ I = \frac{1}{3} (2^3 - 0^3) $$
    $$ I = \frac{8}{3} $$

Or, we could solve the integral using a Monte Carlo approach. In computer sciency notation, we might
write this as:

    $$ I = \operatorname{area}( x^2, 0, 2 ) $$

We could also write it as:

    $$  E[f(x) | a \leq x \leq b] = \frac{1}{b - a} \cdot \operatorname{area}(f(x), a, b) $$
    $$ \operatorname{average}(x^2, 0, 2) = \frac{1}{2 - 0} \cdot \operatorname{area}( x^2, 0, 2 ) $$
    $$ \operatorname{average}(x^2, 0, 2) = \frac{1}{2 - 0} \cdot I $$
    $$ I = 2 \cdot \operatorname{average}(x^2, 0, 2) $$

<div class='together'>
The Monte Carlo approach:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    #include "rtweekend.h"

    #include <iostream>
    #include <iomanip>
    #include <math.h>
    #include <stdlib.h>

    int main() {
        int a = 0;
        int b = 2;
        int N = 1000000;
        auto sum = 0.0;
        for (int i = 0; i < N; i++) {
            auto x = random_double(a, b);
            sum += x*x;
        }
        std::cout << std::fixed << std::setprecision(12);
        std::cout << "I = " << (b - a) * (sum / N) << '\n';
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [integ-xsq-1]: <kbd>[integrate_x_sq.cc]</kbd> Integrating x^2]

</div>

<div class='together'>
This, as expected, produces approximately the exact answer we get with integration, _i.e._
$I = 8/3$. You could rightly point to this example and say that the integration is actually a lot
less work than the Monte Carlo. That might be true in the case where the function is $f(x) = x^2$,
but there exist many functions where it might be simpler to solve for the Monte Carlo than for the
integration, like $f(x) = sin^5(x)$.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
        for (int i = 0; i < N; i++) {
            auto x = random_double(a, b);
            sum += pow(sin(x), 5.0);
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [integ-sin5]: Integrating sin^5]

</div>

<div class='together'>
We could also use the Monte Carlo algorithm for functions where an analytical integration does not
exist, like $f(x) = \ln(\sin(x))$.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    for (int i = 0; i < N; i++) {
        auto x = random_double(a, b);
        sum += log(sin(x));
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [integ-ln-sin]: Integrating ln(sin)]

</div>

In graphics, we often have functions that we can write down explicitly but that have a complicated
analytic integration, or, just as often, we have functions that _can_ be evaluated but that _can't_
be written down explicitly, and we will frequently find ourselves with a function that can _only_ be
evaluated probabilistically. The function `ray_color` from the first two books is an example of a
function that can only be determined probabilistically. We can’t know what color can be seen from
any given place in all directions, but we can statistically estimate which color can be seen from
one particular place, for a single particular direction.


Density Functions
------------------
The `ray_color` function that we wrote in the first two books, while elegant in its simplicity, has
a fairly _major_ problem. Small light sources create too much noise. This is because our uniform
sampling doesn’t sample these light sources often enough. Light sources are only sampled if a ray
scatters toward them, but this can be unlikely for a small light, or a light that is far away. If
the background color is black, then the only real sources of light in the scene are from the lights
that are actually placed about the scene. There might be two rays that intersect at nearby points on
a surface, one that is randomly reflected toward the light and one that is not. The ray that is
reflected toward the light will appear a very bright color. The ray that is reflected to somewhere
else will appear a very dark color. The two intensities should really be somewhere in the middle.
We could lessen this problem if we steered both of these rays toward the light, but this would cause
the scene to be inaccurately bright.

For any given ray, we usually trace from the camera, through the scene, and terminate at a light.
But imagine if we traced this same ray from the light source, through the scene, and terminated at
the camera. This ray would start with a bright intensity and would lose energy with each
successive bounce around the scene. It would ultimately arrive at the camera, having been dimmed and
colored by its reflections off various surfaces. Now, imagine if this ray was forced to bounce
toward the camera as soon as it could. It would appear inaccurately bright because it hadn't been
dimmed by successive bounces. This is analogous to sending more random samples toward the light. It
would go a long way toward solving our problem of having a bright pixel next to a dark pixel, but it
would then just make _all_ of our pixels bright.

We can remove this inaccuracy by downweighting those samples to adjust for the over-sampling. How do
we do this adjustment? Well, we'll first need to understand the concept of a
_probability density function_. But to understand the concept of a _probability density function_,
we'll first need to know what a _density function_ is.

A _density function_ is just the continuous version of a histogram. Here’s an example of a histogram
from the histogram Wikipedia page:

  ![Figure [histogram]: Histogram example](../images/fig-3.03-histogram.jpg)

If we had more items in our data source, the number of bins would stay the same, but each bin would
have a higher frequency of each item. If we divided the data into more bins, we'd have more bins,
but each bin would have a lower frequency of each item. If we took the number of bins and raised it
to infinity, we'd have an infinite number of zero-frequency bins. To solve for this, we'll replace
our histogram, which is a _discrete function_, with a _discrete density function_. A
_discrete density function_ differs from a _discrete function_ in that it normalizes the y-axis to a
fraction or percentage of the total, _i.e_ its density, instead of a total count for each bin.
Converting from a _discrete function_ to a _discrete density function_ is trivial:

    $$ \text{Density of Bin i} = \frac{\text{Number of items in Bin i}}
                                      {\text{Number of items total}} $$

Once we have a _discrete density function_, we can then convert it into a _density function_ by
changing our discrete values into continuous values.

    $$ \text{Bin Density} = \frac{(\text{Fraction of trees between height }H\text{ and }H’)}
                            {(H-H’)} $$

So a _density function_ is a continuous histogram where all of the values are normalized against a
total. If we had a specific tree we wanted to know the height of, we could create a
_probability function_ that would tell us how likely it is for our tree to fall within a specific
bin.

    $$ \text{Probability of Bin i} = \frac{\text{Number of items in Bin i}}
                                          {\text{Number of items total}} $$

If we combined our _probability function_ and our (continuous) _density function_, we could
interpret that as a statistical predictor of a tree’s height:

    $$ \text{Probability a random tree is between } H \text{ and } H’ =
        \text{Bin Density}\cdot(H-H’)$$

Indeed, with this continuous probability function, we can now say the likelihood that any given tree
has a height that places it within any arbitrary span of multiple bins. This is a
_probability density function_ (henceforth _PDF_). In short, a PDF is a continuous function that
can be  integrated over to determine how likely a result is over an integral.


Constructing a PDF
-------------------
Let’s make a PDF and play around with it to build up an intuition. We'll use the following function:

  ![Figure [linear-pdf]: A linear PDF](../images/fig-3.04-linear-pdf.jpg)

What does this function do? Well, we know that a PDF is just a continuous function that defines the
likelihood of an arbitrary range of values. This function $p(r)$ is constrained between $0$ and $2$
and linearly increases along that interval. So, if we used this function as a PDF to generate a
random number then the _probability_ of getting a number near zero would be less than the
probability of getting a number near two.

The PDF $p(r)$ is a linear function that starts with $0$ at $r=0$ and monotonically increases to its
highest point at $p(2)$ for $r=2$. What is the value of $p(2)$? What is the value of $p(r)$? Maybe
$p(2)$ is 2? The PDF increases linearly from 0 to 2, so guessing that the value of $p(2)$ is 2
seems reasonable. At least it looks like it can't be 0.

Remember that the PDF is a probability function. We are constraining the PDF so that it lies in the
range [0,2]. The PDF represents the continuous density function for a probabilistic list. If we know
that everything in that list is contained within 0 and 2, we can say that the probability of getting
a value between 0 and 2 is 100%. Therefore, the area under the curve must sum to 1:

    $$ \operatorname{area}(p(r), 0, 2) = 1 $$

All linear functions can be represented as a constant term multiplied by a variable.

    $$ p(r) = C \cdot r $$

We need to solve for the value of $C$. We can use integration to work backwards.

    $$ 1 = \operatorname{area}(p(r), 0, 2) $$
    $$ = \int_{0}^{2} C \cdot r dr $$
    $$ = C \cdot \int_{0}^{2} r dr $$
    $$ = C \cdot \frac{r^2}{2} \Big|_{0}^{2} $$
    $$ = C ( \frac{2^2}{2} - \frac{0}{2} ) $$
    $$ C = \frac{1}{2} $$

That gives us the PDF of $p(r) = r/2$. Just as with histograms we can sum up (integrate) the
region to figure out the probability that $r$ is in some interval $[x_0,x_1]$:

    $$ \operatorname{Probability} (r | x_0 \leq r \leq x_1 )
       = \operatorname{area}(p(r), x_0, x_1)
    $$

    $$ \operatorname{Probability} (r | x_0 \leq r \leq x_1 ) = \int_{x_0}^{x_1}  \frac{r}{2} dr $$

To confirm your understanding, you should integrate over the region $r=0$ to $r=2$, you should get a
probability of 1.

After spending enough time with PDFs you might start referring to a PDF as the probability that a
variable $r$ is value $x$, _i.e._ $p(r=x)$. Don't do this. For a continuous function, the
probability that a variable is a specific value is always zero. A PDF can only tell you the
probability that a variable will fall within a given interval. If the interval you're checking
against is a single value, then the PDF will always return a zero probability because its "bin" is
infinitely thin (has zero width). Here's a simple mathematical proof of this fact:

    $$ \operatorname{Probability} (r = x) = \int_{x}^{x}  p(r) dr $$
    $$ = P(r) \Big|_{x}^{x} $$
    $$ = P(x) - P(x) $$
    $$ = 0 $$

Finding the probability of a region surrounding x may not be zero:

    $$ \operatorname{Probability} (r | x - \Delta x < r < x + \Delta x ) =
         \operatorname{area}(p(r), x - \Delta x, x + \Delta x) $$
    $$ = P(x + \Delta x) - P(x - \Delta x) $$


Choosing our Samples
--------------------
If we have a PDF for the function that we care about, then we have the probability that the function
will return a value within an arbitrary interval. We can use this to determine where we should
sample. Remember that this started as a quest to determine the best way to sample a scene so that we
wouldn't get very bright pixels next to very dark pixels. If we have a PDF for the scene, then we
can probabilistically steer our samples toward the light without making the image inaccurately
bright. We already said that if we steer our samples toward the light then we _will_ make the image
inaccurately bright. We need to figure out how to steer our samples without introducing this
inaccuracy, this will be explained a little bit later, but for now we'll focus on generating samples
if we have a PDF. How do we generate a random number with a PDF? For that we will need some more
machinery. Don’t worry -- this doesn’t go on forever!

<div class='together'>
Our random number generator `random_double()` produces a random double between 0 and 1. The number
generator is uniform between 0 and 1, so any number between 0 and 1 has equal likelihood. If our PDF
is uniform over a domain, say $[0,10]$, then we can trivially produce perfect samples for this
uniform PDF with

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    10.0 * random_double()
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

</div>

That's an easy case, but the vast majority of cases that we're going to care about are nonuniform.
We need to figure out a way to convert a uniform random number generator into a nonuniform random
number generator, where the distribution is defined by the PDF. We'll just _assume_ that there
exists a function $f(d)$ that takes uniform input and produces a nonuniform distribution weighted by
PDF. We just need to figure out a way to solve for $f(d)$.

For the PDF given above, where $p(r) = \frac{r}{2}$, the probability of a random sample is higher
toward 2 than it is toward 0. There is a greater probability of getting a number between 1.8 and
2.0 than between 0.0 and 0.2. If we put aside our mathematics hat for a second and put on our
computer science hat, maybe we can figure out a smart way of partitioning the PDF. We know that
there is a higher probability near 2 than near 0, but what is the value that splits the probability
in half? What is the value that a random number has a 50% chance of being higher than and a 50%
chance of being lower than? What is the $x$ that solves:

    $$ 50\% = \int_{0}^{x}  \frac{r}{2} dr  = \int_{x}^{2}  \frac{r}{2} dr $$

Solving gives us:

    $$ 0.5 = \frac{r^2}{4} \Big|_{0}^{x} $$
    $$ 0.5 = \frac{x^2}{4} $$
    $$ x^2 = 2$$
    $$ x = \sqrt{2}$$

As a crude approximation we could create a function `f(d)` that takes as input
`double d = random_double()`. If `d` is less than (or equal to) 0.5, it produces a uniform number
in $[0,\sqrt{2}]$, if `d` is greater than 0.5, it produces a uniform number in $[\sqrt{2}, 2]$.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    double f(double d)
    {
        if (d <= 0.5)
            return sqrt(2.0) * random_double();
        else
            return sqrt(2.0) + (2 - sqrt(2.0)) * random_double();
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [crude-approx]: A crude, first-order approximation to nonuniform PDF]

<div class='together'>
While our initial random number generator was uniform from 0 to 1:

   ![Figure [uniform-dist]: A uniform distribution](../images/fig-3.05-uniform-dist.jpg)

</div>

<div class='together'>
Our, new, crude approximation for $\frac{r}{2}$ is nonuniform (but only just):

    ![Figure [approx-f]: A nonuniform distribution for r/2](../images/fig-3.06-nonuniform-dist.jpg)

</div>

We had the analytical solution to the integration above, so we could very easily solve for the 50%
value. But we could also solve for this 50% value experimentally. There will be functions that we
either can't or don't want to solve for the integration. In these cases, we can get an experimental
result close to the real value. Let's take the function:

    $$ p(x) = e^{\frac{-x}{2 \pi}} sin^2(x) $$

<div class='together'>
Which looks a little something like this:

    ![Figure [exp-sin2]: A function that we don't want to solve analytically
    ](../images/fig-3.07-exp-sin2.jpg)

</div>

<div class='together'>
At this point you should be familiar with how to experimentally solve for the area under a curve.
We'll take our existing code and modify it slightly to get an estimate for the 50% value. We want to
solve for the $x$ value that gives us half of the total area under the curve. As we go along and
solve for the rolling sum over N samples, we're also going to store each individual sample alongside
its `p(x)` value. After we solve for the total sum, we'll sort our samples and add them up until we
have an area that is half of the total. From $0$ to $2\pi$ for example:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    #include "rtweekend.h"

    #include <algorithm>
    #include <vector>
    #include <iostream>
    #include <iomanip>
    #include <math.h>
    #include <cmath>
    #include <stdlib.h>

    struct sample {
        double x;
        double p_x;
    };

    bool compare_by_x(const sample& a, const sample& b) {
        return a.x < b.x;
    }

    int main() {
        int N = 10000;
        double sum = 0.0;

        // iterate through all of our samples
        std::vector<sample> samples;
        for (int i = 0; i < N; i++) {
            // Get the area under the curve
            auto x = random_double(0, 2*pi);
            auto sin_x = sin(x);
            auto p_x = exp(-x / (2*pi)) * sin_x * sin_x;
            sum += p_x;
            // store this sample
            sample this_sample = {x, p_x};
            samples.push_back(this_sample);
        }

        // Sort the samples by x
        std::sort(samples.begin(), samples.end(), compare_by_x);

        // Find out the sample at which we have half of our area
        double half_sum = sum / 2.0;
        double halfway_point = 0.0;
        double accum = 0.0;
        for (int i = 0; i < N; i++){
            accum += samples[i].p_x;
            if (accum >= half_sum){
                halfway_point = samples[i].x;
                break;
            }
        }

        std::cout << std::fixed << std::setprecision(12);
        std::cout << "Average = " << sum / N << '\n';
        std::cout << "Area under curve = " << 2 * pi * sum / N << '\n';
        std::cout << "Halfway = " << halfway_point << '\n';
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [est-halfway]: <kbd>[estimate_halfway.cc]</kbd> Estimating the 50% point of a function]

</div>

<div class='together'>
This code snippet isn't too different from what we had before. We're still solving for the sum
over an interval (0 to $2\pi$). Only this time, we're also storing and sorting all of our samples by
their input and output. We use this to determine the point at which they subtotal half of the sum
across the entire interval. Once we know that our first $j$ samples sum up to half of the total sum,
we know that the $j\text{th}$ $x$ roughly corresponds to our halfway point:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Average = 0.314686555791
    Area under curve = 1.977233943713
    Halfway = 2.016002314977
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

</div>

If you solve for the integral from $0$ to $2.016$ and from $2.016$ to $2\pi$ you should get almost
exactly the same result for both.

We have a method of solving for the halfway point that splits a PDF in half. If we wanted to, we
could use this to create a nested binary partition of the PDF:

  1. Solve for halfway point of a PDF
  2. Recurse into lower half, repeat step 1
  3. Recurse into upper half, repeat step 1

Stopping at a reasonable depth, say 6–10. As you can imagine, this could be quite computationally
expensive. The computational bottleneck for the code above is probably sorting the samples. A naive
sorting algorithm can have an algorithmic complexity of $\mathcal{O}(\mathbf{n^2})$ time, which is
tremendously expensive. Fortunately, the sorting algorithm included in the standard library is
usually much closer to $\mathcal{O}(\mathbf{n\log{}n})$ time, but this can still be quite expensive,
especially for millions or billions of samples. But this will produce decent nonuniform
distributions of nonuniform numbers. This divide and conquer method of producing nonuniform
distributions is used somewhat commonly in practice, although there are much more efficient means of
doing so than a simple binary partition. If you have an arbitrary function that you wish to use as
the PDF for a distribution, you'll want to research the _Metropolis-Hastings Algorithm_.


Approximating Distributions
---------------------------
This was a lot of math and work to build up a couple of notions. Let's return to our initial PDF.
For the intervals without an explicit probability, we assume the PDF to be zero. So for our example
from the beginning of the chapter, $p(r) = 0$, for $r \notin [0,2]$. We can rewrite our $p(r)$ in
piecewise fashion:

    $$ p(r)=\begin{cases}
            0           & r < 0           \\
            \frac{r}{2} & 0 \leq r \leq 2 \\
            0           & 2 < r           \\
       \end{cases}
    $$

If you consider what we were trying to do in the previous section, a lot of math revolved around the
_accumulated_ area (or _accumulated_ probability) from zero. In the case of the function

    $$ f(x) = e^{\frac{-x}{2 \pi}} sin^2(x)  $$

we cared about the accumulated probability from $0$ to $2\pi$ (100%) and the accumulated probability
from $0$ to $2.016$ (50%). We can generalize this to an important term, the
_Cumulative Distribution Function_ $P(x)$ is defined as:

    $$ P(x) =  \int_{-\infty}^{x}  p(x') dx' $$

Or,

    $$ P(x) = \operatorname{area}(p(x'), -\infty, x) $$

Which is the amount of _cumulative_ probability from $-\infty$. We rewrote the integral in terms of
$x'$ instead of $x$ because of calculus rules, if you're not sure what it means, don't worry about
it, you can just treat it like it's the same. If we take the integration outlined above, we get the
piecewise $P(r)$:

    $$ P(r)=\begin{cases}
           0             & r < 0           \\
           \frac{r^2}{4} & 0 \leq r \leq 2 \\
           1             & 2 < r           \\
       \end{cases}
    $$

The _Probability Density Function_ (PDF) is the probability function that explains how likely an
interval of numbers is to be chosen. The _Cumulative Distribution Function_ (CDF) is the
distribution function that explains how likely all numbers smaller than its input is to be chosen.
To go from the PDF to the CDF, you need to integrate from $-\infty$ to $x$, but to go from the CDF
to the PDF, all you need to do is take the derivative:

    $$ p(x) = \frac{d}{dx}P(x) $$

If we evaluate the CDF, $P(r)$, at $r = 1.0$, we get:

    $$ P(1.0) = \frac{1}{4} $$

This says _a random variable plucked from our PDF has a 25% chance of being 1 or lower_. We want a
function $f(d)$ that takes a uniform distribution between 0 and 1 (_i.e_ `f(random_double())`), and
returns a random value according to a distribution that has the CDF $P(x) = \frac{x^2}{4}$. We don’t
know yet know what the function $f(d)$ is analytically, but we do know that 25% of what it returns
should be less than 1.0, and 75% should be above 1.0. Likewise, we know that 50% of what it returns
should be less than $\sqrt{2}$, and 50% should be above $\sqrt{2}$. If $f(d)$ monotonically
increases, then we would expect $f(0.25) = 1.0$ and $f(0.5) = \sqrt{2}$. This can be generalized to
figure out $f(d)$ for every possible input:

    $$ f(P(x)) = x $$

Let's take some more samples:

    $$ P(0.0) = 0 $$
    $$ P(0.5) = \frac{1}{16} $$
    $$ P(1.0) = \frac{1}{4} $$
    $$ P(1.5) = \frac{9}{16} $$
    $$ P(2.0) = 1 $$

so, the function $f()$ has values

    $$ f(P(0.0)) = f(0) = 0 $$
    $$ f(P(0.5)) = f(\frac{1}{16}) = 0.5 $$
    $$ f(P(1.0)) = f(\frac{1}{4}) = 1.0 $$
    $$ f(P(1.5)) = f(\frac{9}{16}) = 1.5 $$
    $$ f(P(2.0)) = f(1) = 2.0 $$

We could use these intermediate values and interpolate between them to approximate $f(d)$:

    ![Figure [approx f]: Approximating the nonuniform f()](../images/fig-3.08-approx-f.jpg)

If you can't solve for the PDF analytically, then you can't solve for the CDF analytically. After
all, the CDF is just the integral of the PDF. However, you can still create a distribution that
approximates the PDF. If you take a bunch of samples from the random function you want the PDF from,
you can approximate the PDF by getting a histogram of the samples and then converting to a PDF.
Alternatively, you can do as we did above and sort all of your samples.

Looking closer at the equality:

    $$ f(P(x)) = x $$

That just means that $f()$ just undoes whatever $P()$ does. So, $f()$ is the inverse function:

    $$ f(d) = P^{-1}(x) $$

For our purposes, if we have PDF $p()$ and cumulative distribution function $P()$, we can use this
"inverse function" with a random number to get what we want:

    $$ f(d) = P^{-1} (\operatorname{random_double}()) $$

For our PDF $p(r) = r/2$, and corresponding $P(r)$, we need to compute the inverse of $P(r)$. If we
have

    $$ y = \frac{r^2}{4} $$

we get the inverse by solving for $r$ in terms of $y$:

    $$ r = \sqrt{4y} $$

Which means the inverse of our CDF is defined as

    $$ P^{-1}(r) = \sqrt{4y} $$

Thus our random number generator with density $p(r)$ can be created with:

    $$ f(d) = \sqrt{4\cdot\operatorname{random_double}()} $$

Note that this ranges from 0 to 2 as we hoped, and if we check our work, we replace
`random_double()` with $1/4$ to get 1, and also replace with $1/2$ to get $\sqrt{2}$, just as
expected.


Importance Sampling
--------------------
You should now have a decent understanding of how to take an analytical PDF and generate a function
that produces random numbers with that distribution. We return to our original integral and try it
with a few different PDFs to get a better understanding:

    $$ I = \int_{0}^{2} x^2 dx $$

<div class='together'>
The last time that we tried to solve for the integral we used a Monte Carlo approach, uniformly
sampling from the interval $[0, 2]$. We didn't know it at the time, but we were implicitly using a
uniform PDF between 0 and 2. This means that we're using a PDF = $1/2$ over the range $[0,2]$, which
means the CDF is $P(x) = x/2$, so $f(d) = 2d$. Knowing this, we can make this uniform PDF explicit:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    #include "rtweekend.h"

    #include <iostream>
    #include <iomanip>
    #include <math.h>
    #include <stdlib.h>


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
    double f(double d) {
        return 2.0 * d;
    }

    double pdf(double x) {
        return 0.5;
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

    int main() {
        int N = 1000000;
        auto sum = 0.0;
        for (int i = 0; i < N; i++) {
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            auto x = f(random_double());
            sum += x*x / pdf(x);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
        }
        std::cout << std::fixed << std::setprecision(12);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        std::cout << "I = " << sum / N << '\n';
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [integ-xsq-2]: <kbd>[integrate_x_sq.cc]</kbd> Explicit uniform PDF for $x^2$]

</div>

There are a couple of important things to emphasize. Every value of $x$ represents one sample of the
function $x^2$ within the distribution $[0, 2]$. We use a function $f$ to randomly select samples
from within this distribution. We were previously multiplying the average over the interval
(`sum / N`) times the length of the interval (`b - a`) to arrive at the final answer. Here, we
don't need to multiply by the interval length--that is, we no longer need to multiply the average
by $2$.

We need to account for the nonuniformity of the PDF of $x$. Failing to account for this
nonuniformity will introduce bias in our scene. Indeed, this bias is the source of our inaccurately
bright image--if we account for nonuniformity, we will get accurate results. The PDF will "steer"
samples toward specific parts of the distribution, which will cause us to converge faster, but at
the cost of introducing bias. To remove this bias, we need to down-weight where we sample more
frequently, and to up-weight where we sample less frequently. For our new nonuniform random number
generator, the PDF defines how much or how little we sample a specific portion.
So the weighting function should be proportional to $1/\mathit{pdf}$.
In fact it is _exactly_ $1/\mathit{pdf}$.
This is why we divide `x*x` by `pdf(x)`.

We can try to solve for the integral using the linear PDF $p(r) = \frac{r}{2}$, for which we were
able to solve for the CDF and its inverse. To do that, all we need to do is replace the functions
$f = \sqrt{4d}$ and $pdf = x/2$.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
    double f(double d) {
        return sqrt(4.0 * d);
    }

    double pdf(double x) {
        return x / 2.0;
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

    int main() {
        int N = 1000000;
        auto sum = 0.0;
        for (int i = 0; i < N; i++) {
            auto x = f(random_double());
            sum += x*x / pdf(x);
        }
        std::cout << std::fixed << std::setprecision(12);
        std::cout << "I = " << sum / N << '\n';
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [integ-xsq-3]: <kbd>[integrate_x_sq.cc]</kbd> Integrating $x^2$ with linear PDF]

If you compared the runs from the uniform PDF and the linear PDF, you would have probably found that
the linear PDF converged faster. If you think about it, a linear PDF is probably a better
approximation for a quadratic function than a uniform PDF, so you would expect it to converge
faster. If that's the case, then we should just try to make the PDF match the integrand by turning
the PDF into a quadratic function:

    $$ p(r)=\begin{cases}
            0           & r < 0           \\
            C \cdot r^2 & 0 \leq r \leq 2 \\
            0           & 2 < r           \\
       \end{cases}
    $$

Like the linear PDF, we'll solve for the constant $C$ by integrating to 1 over the interval:

    $$ 1 = \int_{0}^{2} C \cdot r^2 dr $$
    $$ = C \cdot \int_{0}^{2} r^2 dr $$
    $$ = C \cdot \frac{r^3}{3} \Big|_{0}^{2} $$
    $$ = C ( \frac{2^3}{3} - \frac{0}{3} ) $$
    $$ C = \frac{3}{8} $$

Which gives us:

    $$ p(r)=\begin{cases}
            0           & r < 0           \\
            \frac{3}{8} r^2 & 0 \leq r \leq 2 \\
            0           & 2 < r           \\
       \end{cases}
    $$

And we get the corresponding CDF:

    $$ P(r) = \frac{r^3}{8} $$

and

    $$ P^{-1}(x) = f(d) = 8d^\frac{1}{3} $$

<div class='together'>
For just one sample we get:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
    double f(double d) {
        return 8.0 * pow(d, 1.0/3.0);
    }

    double pdf(double x) {
        return (3.0/8.0) * x*x;
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

    int main() {
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        int N = 1;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
        auto sum = 0.0;
        for (int i = 0; i < N; i++) {
            auto x = f(random_double()));
            sum += x*x / pdf(x);
        }
        std::cout << std::fixed << std::setprecision(12);
        std::cout << "I = " << sum / N << '\n';
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [integ-xsq-5]: <kbd>[integrate_x_sq.cc]</kbd> Integrating $x^2$, final version]

This always returns the exact answer. Which, honestly, feels a bit like magic.

</div>

A nonuniform PDF "steers" more samples to where the PDF is big, and fewer samples to where the PDF
is small. By this sampling, we would expect less noise in the places where the PDF is big and more
noise where the PDF is small. If we choose a PDF that is higher in the parts of the scene that have
higher noise, and is smaller in the parts of the scene that have lower noise, we'll be able to
reduce the total noise of the scene with fewer samples. This means that we will be able to converge
to the correct scene _faster_ than with a uniform PDF. In effect, we are steering our samples toward
the parts of the distribution that are more _important_. This is why using a carefully chosen
nonuniform PDF is usually called _importance sampling_.

In all of the examples given, we always converged to the correct answer of $8/3$. We got the same
answer when we used both a uniform PDF and the "correct" PDF ($i.e. f(d)=8d^{\frac{1}{3}}$). While
they both converged to the same answer, the uniform PDF took much longer. After all, we only needed
a single sample from the PDF that perfectly matched the integral. This should make sense, as we were
choosing to sample the important parts of the distribution more often, whereas the uniform PDF just
sampled the whole distribution equally, without taking importance into account.

Indeed, this is the case for any PDF that you create--they will all converge eventually. This is
just another part of the power of the Monte Carlo algorithm. Even the naive PDF where we solved for
the 50% value and split the distribution into two halves: $[0, \sqrt{2}]$ and $[\sqrt{2}, 2]$. That
PDF will converge. Hopefully you should have an intuition as to why that PDF will converge faster
than a pure uniform PDF, but slower than the linear PDF ($i.e. f(d) = \sqrt{4d}$).

The perfect importance sampling is only possible when we already know the answer (we got $P$ by
integrating $p$ analytically), but it’s a good exercise to make sure our code works.

Let's review the main concepts that underlie Monte Carlo ray tracers:

  1. You have an integral of $f(x)$ over some domain $[a,b]$
  2. You pick a PDF $p$ that is non-zero and non-negative over $[a,b]$
  3. You average a whole ton of $\frac{f(r)}{p(r)}$ where $r$ is a random number with PDF $p$.

Any choice of PDF $p$ will always converge to the right answer, but the closer that $p$
approximates $f$, the faster that it will converge.



Monte Carlo Integration on the Sphere of Directions
====================================================================================================
In chapter One Dimensional Monte Carlo Integration we started with uniform random numbers and
slowly, over the course of a chapter, built up more and more complicated ways of producing random
numbers, before ultimately arriving at the intuition of PDFs, and how to use them to generate random
numbers of arbitrary distribution.

All of the concepts covered in that chapter continue to work as we extend beyond a single dimension.
Moving forward, we might need to be able to select a point from a two, three, or even higher
dimensional space and then weight that selection by an arbitrary PDF. An important case of this--at
least for ray tracing--is producing a random direction. In the first two books we generated a random
direction by creating a random vector and rejecting it if it fell outside of the unit sphere. We
repeated this process until we found a random vector that fell inside the unit sphere. Normalizing
this vector produced points that lay exactly on the unit sphere and thereby represent a random
direction. This process of generating samples and rejecting them if they are not inside a desired
space is called _the rejection method_, and is found all over the literature. The method covered
last chapter is referred to as _the inversion method_ because we invert a PDF.

Every direction in 3D space has an associated point on the unit sphere and can be generated by
solving for the vector that travels from the origin to that associated point. You can think of
choosing a random direction as choosing a random point in a constrained two dimensional plane: the
plane created by mapping the unit sphere to Cartesian coordinates. The same methodology as before
applies, but now we might have a PDF defined over two dimensions. Suppose we want to integrate this
function over the surface of the unit sphere:

  $$ f(\theta, \phi) = cos^2(\theta) $$

Using Monte Carlo integration, we should just be able to sample $\cos^2(\theta) / p(r)$, where the
$p(r)$ is now just $p(direction)$. But what is _direction_ in that context? We could make it based
on polar coordinates, so $p$ would be in terms of $\theta$ and $\phi$ for $p(\theta, \phi)$. It
doesn't matter which coordinate system you choose to use. Although, however you choose to do it,
remember that a PDF must integrate to one over the whole surface and that the PDF represents the
_relative probability_ of that direction being sampled. Recall that we have a `vec3` function to
generate uniform random samples on the unit sphere (`random_unit_vector()`). What is the PDF of
these uniform samples?
As a uniform density on the unit sphere, it is $1/\mathit{area}$ of the sphere, which is $1/(4\pi)$.
If the integrand is $\cos^2(\theta)$, and $\theta$ is the angle with the $z$ axis:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    double f(const vec3& d) {
        auto cosine_squared = d.z()*d.z();
        return cosine_squared;
    }

    double pdf(const vec3& d) {
        return 1 / (4*pi);
    }

    int main() {
        int N = 1000000;
        auto sum = 0.0;
        for (int i = 0; i < N; i++) {
            vec3 d = random_unit_vector();
            auto f_d = f(d);
            sum += f_d / pdf(d);
        }
        std::cout << std::fixed << std::setprecision(12);
        std::cout << "I = " << sum / N << '\n';
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [main-sphereimp]: <kbd>[sphere_importance.cc]</kbd>
    Generating importance-sampled points on the unit sphere]

The analytic answer is $\frac{4}{3} \pi$ -- if you remember enough advanced calc, check me! And the
code above produces that. The key point here is that all of the integrals and the probability and
everything else is over the unit sphere. The way to represent a single direction in 3D is its
associated point on the unit sphere. The way to represent a range of directions in 3D is the amount
of area on the unit sphere that those directions travel through. Call it direction, area, or
_solid angle_ -- it’s all the same thing. Solid angle is the term that you'll usually find in the
literature. You have radians (r) in $\theta$ over one dimension, and you have _steradians_ (sr) in
$\theta$ _and_ $\phi$ over two dimensions (the unit sphere is a three dimensional object, but its
surface is only two dimensional). Solid Angle is just the two dimensional extension of angles. If
you are comfortable with a two dimensional angle, great! If not, do what I do and imagine the area
on the unit sphere that a set of directions goes through. The solid angle $\omega$ and the projected
area $A$ on the unit sphere are the same thing.

  ![Figure [solid-angle]: Solid angle / projected area of a sphere
  ](../images/fig-3.09-solid-angle.jpg)

Now let’s go on to the light transport equation we are solving.



Light Scattering
====================================================================================================
In this chapter we won't actually program anything. We'll just be setting up for a big lighting
change in the next chapter. Our ray tracing program from the first two books scatters a ray when it
interacts with a surface or a volume. Ray scattering is the most commonly used model for simulating
light propagation through a scene. This can naturally be modeled probabilistically. There are many
things to consider when modeling the probabilistic scattering of rays.


Albedo
-------
First, is the light absorbed?

Probability of light being scattered: $A$

Probability of light being absorbed: $1-A$

Where here $A$ stands for _albedo_, which is latin for _whiteness_. Albedo is a precise technical
term in some disciplines, but in all cases it is used to define some form of
_fractional reflectance_. This _fractional reflectance_ (or albedo) will vary with color and (as we
implemented for our glass material) can also vary with incident direction (the direction of the
incoming ray). It can help to stop and remember that when we simulate light propagation, all we're
doing is simulating the movement of photons through a space. If you remember your high school
Physics then you should recall that every photon has a unique energy and wavelength associated by
the Planck constant:

    $$ E = \frac{hc}{\lambda} $$

Each individual photon has a _tiny_ amount of energy, but when you add enough of them up you get all
of the illumination in your rendering. The absorption or scattering of a photon with a surface or a
volume (or really anything that a photon can interact with) is probabilistically determined by the
albedo of the object. Albedo can depend on color because some objects are more likely to absorb
some wavelengths.

In most physically based renderers, we would use a predefined set of specific wavelengths for the
light color rather than RGB. As an example, we would replace our _tristimulus_ RGB renderer with
something that specifically samples at 300nm, 350nm, 400nm, ..., 700nm. We can extend our intuition
by thinking of R, G, and B as specific algebraic mixtures of wavelengths where R is _mostly_ red
wavelengths, G is _mostly_ green wavelengths, and B is _mostly_ blue wavelengths. This is an
approximation of the human visual system which has 3 unique sets of color receptors, called _cones_,
that are each sensitive to different algebraic mixtures of wavelengths, roughly RGB, but are
referred to as long, medium, and short cones (the names are in reference to the wavelengths that
each cone is sensitive to, not the length of the cone). Just as colors can be represented by their
strength in the RGB color space, colors can also be represented by how excited each set of cones is
in the _LMS color space_ (long, medium, short).

Scattering
-----------
If the light does scatter, it will have a directional distribution that we can describe as a PDF
over solid angle.
I will refer to this as its _scattering PDF_: $\operatorname{pScatter}()$.
The scattering PDF will vary with outgoing direction: $\operatorname{pScatter}(\omega_o)$.
The scattering PDF can also vary with _incident direction_:
  $\operatorname{pScatter}(\omega_i, \omega_o)$.
You can see this varying with incident direction when you look at reflections off a road -- they
  become mirror-like as your viewing angle (incident angle) approaches grazing.
The scattering PDF can vary with the wavelength of the light:
  $\operatorname{pScatter}(\omega_i, \omega_o, \lambda)$.
A good example of this is a prism refracting white light into a rainbow.
Lastly, the scattering PDF can also depend on the scattering position:
  $\operatorname{pScatter}(\mathbf{x}, \omega_i, \omega_o, \lambda)$.
The $\mathbf{x}$ is just math notation for the scattering position:
  $\mathbf{x} = (x, y, z)$.
The albedo of an object can also depend on these quantities:
  $A(\mathbf{x}, \omega_i, \omega_o, \lambda)$.

The color of a surface is found by integrating these terms over the unit hemisphere by the incident
direction:

    $$ \operatorname{Color}_o(\mathbf{x}, \omega_o, \lambda) = \int_{\omega_i}
        A(\mathbf{x}, \omega_i, \omega_o, \lambda) \cdot
        \operatorname{pScatter}(\mathbf{x}, \omega_i, \omega_o, \lambda) \cdot
        \operatorname{Color}_i(\mathbf{x}, \omega_i, \lambda) $$

We've added a $\operatorname{Color}_i$ term.
The scattering PDF and the albedo at the surface of an object are acting as filters to the light
  that is shining on that point.
So we need to solve for the light that is shining on that point.
This is a recursive algorithm, and is the reason our `ray_color` function returns the color of the
  current object multiplied by the color of the next ray.


The Scattering PDF
-------------------
If we apply the Monte Carlo basic formula we get the following statistical estimate:

    $$ \operatorname{Color}_o(\mathbf{x}, \omega_o, \lambda) \approx \sum
        \frac{A(\, \ldots \,) \cdot
        \operatorname{pScatter}(\, \ldots \,) \cdot
        \operatorname{Color}_i(\, \ldots \,)}
        {p(\mathbf{x}, \omega_i, \omega_o, \lambda)} $$

where $p(\mathbf{x}, \omega_i, \omega_o, \lambda)$ is the PDF of whatever outgoing direction we
randomly generate.

For a Lambertian surface we already implicitly implemented this formula for the special case where
  $pScatter(\, \ldots \,)$ is a cosine density.
The $\operatorname{pScatter}(\, \ldots \,)$ of a Lambertian surface is proportional to
  $\cos(\theta_o)$, where $\theta_o$ is the angle relative to the surface normal.
Let's solve for $C$ once more:

    $$ \operatorname{pScatter}(\mathbf{x}, \omega_i, \omega_o, \lambda) = C \cdot \cos(\theta_o) $$

All two dimensional PDFs need to integrate to one over the whole surface (remember that
  $\operatorname{pScatter}$ is a PDF).
We set $\operatorname{pScatter}(\theta_o < 0) = 0$ so that we don't scatter below the horizon.

    $$ 1 = \int_{0}^{2 \pi} \int_{0}^{\pi / 2} C \cdot cos(\theta) dA $$

To integrate over the hemisphere, remember that in spherical coordinates:

    $$ dA = \sin(\theta) d\theta d\phi $$

So:

    $$ 1 = C \cdot \int_{0}^{2 \pi} \int_{0}^{\pi / 2} cos(\theta) sin(\theta) d\theta d\phi $$
    $$ 1 = C \cdot 2 \pi \frac{1}{2} $$
    $$ 1 = C \cdot \pi $$
    $$ C = \frac{1}{\pi} $$

The integral of $\cos(\theta_o)$ over the hemisphere is $\pi$, so we need to we need to normalize
  by $\frac{1}{\pi}$.
The PDF $\operatorname{pScatter}$ is only dependent on outgoing direction ($\omega_o$), so we'll
  simplify its representation to just $\operatorname{pScatter}(\omega_o)$.
Put all of this together and you get the scattering PDF for a Lambertian surface:

    $$ \operatorname{pScatter}(\omega_o) = \frac{\cos(\theta_o)}{\pi} $$

We'll assume that the $p(\mathbf{x}, \omega_i, \omega_o, \lambda)$ is equal to the scattering PDF:

    $$ p(\omega_o) = \operatorname{pScatter}(\omega_o) = \frac{\cos(\theta_o)}{\pi} $$

The numerator and denominator cancel out, and we get:

    $$ \operatorname{Color}_o(\mathbf{x}, \omega_o, \lambda) \approx \sum
        A(\, \ldots \,) \cdot
        \operatorname{Color}_i(\, \ldots \,) $$

<div class='together'>
This is exactly what we had in our original `ray_color()` function!

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
        return attenuation * ray_color(scattered, depth-1, world);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

</div>

The treatment above is slightly non-standard because I want the same math to work for surfaces and
volumes. If you read the literature, you’ll see reflection defined by the
_Bidirectional Reflectance Distribution Function_ (BRDF). It relates pretty simply to our terms:

    $$ BRDF(\omega_i, \omega_o, \lambda) = \frac{A(\mathbf{x}, \omega_i, \omega_o, \lambda) \cdot
        \operatorname{pScatter}(\mathbf{x}, \omega_i, \omega_o, \lambda)}{\cos(\theta_o)} $$

So for a Lambertian surface for example, $BRDF = A / \pi$. Translation between our terms and BRDF is
easy. For participating media (volumes), our albedo is usually called the _scattering albedo_, and
our scattering PDF is usually called the _phase function_.

All that we've done here is outline the PDF for the Lambertian scattering of a material. However,
we'll need to generalize so that we can send extra rays in important directions, such as toward the
lights.



Playing with Importance Sampling
====================================================================================================
Our goal over the next several chapters is to instrument our program to send a bunch of extra rays
  toward light sources so that our picture is less noisy.
Let’s assume we can send a bunch of rays toward the light source using a PDF
  $\operatorname{pLight}(\omega_o)$.
Let’s also assume we have a PDF related to $operatorname{pScatter}$, and let’s call that
  $operatorname{pSurface}(\omega_o)$.
A great thing about PDFs is that you can just use linear mixtures of them to form mixture densities
  that are also PDFs.
For example, the simplest would be:

  $$ p(\omega_o) = \frac{1}{2} \operatorname{pSurface}(\omega_o) +  \frac{1}{2}
      \operatorname{pLight}(\omega_o)$$

As long as the weights are positive and add up to one, any such mixture of PDFs is a PDF. Remember,
we can use any PDF: _all PDFs eventually converge to the correct answer_. So, the game is to figure
out how to make the PDF larger where the product

    $$ \operatorname{pScatter}(\mathbf{x}, \omega_i, \omega_o) \cdot
        \operatorname{Color}_i(\mathbf{x}, \omega_i) $$

is largest.
For diffuse surfaces, this is mainly a matter of guessing where
  $\operatorname{Color}_i(\mathbf{x}, \omega_i)$ is largest.
Which is equivalent to guessing where the most light is coming from.

For a mirror, $\operatorname{pScatter}()$ is huge only near one direction, so
  $\operatorname{pScatter}()$ matters a lot more.
In fact, most renderers just make mirrors a special case, and make the
  $\operatorname{pScatter}()/p()$ implicit -- our code currently does that.


Returning to the Cornell Box
-----------------------------
Let’s adjust some parameters for the Cornell box:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    int main() {
        ...
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        cam.samples_per_pixel = 100;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
        ...
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [cornell-box]: <kbd>[main.cc]</kbd> Cornell box, refactored]

<div class='together'>
At 600×600 my code produces this image in 15min on 1 core of my Macbook:

  ![<span class='num'>Image 3:</span> Cornell box, refactored
  ](../images/img-3.03-cornell-refactor1.jpg class='pixel')

</div>

Reducing that noise is our goal. We’ll do that by constructing a PDF that sends more rays to the
light.

First, let’s instrument the code so that it explicitly samples some PDF and then normalizes for
that. Remember Monte Carlo basics: $\int f(x) \approx \sum f(r)/p(r)$. For the Lambertian material,
let’s sample like we do now: $p(\omega_o) = \cos(\theta_o) / \pi$.

<div class='together'>
We modify the base-class `material` to enable this importance sampling:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class material {
      public:
        ...


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        virtual double scattering_pdf(const ray& r_in, const hit_record& rec, const ray& scattered)
        const {
            return 0;
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [class-material]: <kbd>[material.h]</kbd>
    The material class, adding importance sampling]

</div>

<div class='together'>
And the `lambertian` material becomes:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class lambertian : public material {
      public:
        lambertian(const color& a) : albedo(make_shared<solid_color>(a)) {}
        lambertian(shared_ptr<texture> a) : albedo(a) {}

        bool scatter(const ray& r_in, const hit_record& rec, color& attenuation, ray& scattered)
        const override {
            auto scatter_direction = rec.normal + random_unit_vector();

            // Catch degenerate scatter direction
            if (scatter_direction.near_zero())
                scatter_direction = rec.normal;

            scattered = ray(rec.p, scatter_direction, r_in.time());
            attenuation = albedo->value(rec.u, rec.v, rec.p);
            return true;
        }


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        double scattering_pdf(const ray& r_in, const hit_record& rec, const ray& scattered) const {
            auto cos_theta = dot(rec.normal, unit_vector(scattered.direction()));
            return cos_theta < 0 ? 0 : cos_theta/pi;
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

      private:
        shared_ptr<texture> albedo;
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [class-lambertian-impsample]: <kbd>[material.h]</kbd>
    Lambertian material, modified for importance sampling]

</div>

<div class='together'>
And the `camera::ray_color` function gets a minor modification:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class camera {
      ...
      private:
        ...
        color ray_color(const ray& r, int depth, const hittable& world) const {
            hit_record rec;

            // If we've exceeded the ray bounce limit, no more light is gathered.
            if (depth <= 0)
                return color(0,0,0);

            // If the ray hits nothing, return the background color.
            if (!world.hit(r, interval(0.001, infinity), rec))
                return background;

            ray scattered;
            color attenuation;
            color color_from_emission = rec.mat->emitted(rec.u, rec.v, rec.p);

            if (!rec.mat->scatter(r, rec, attenuation, scattered))
                return color_from_emission;


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            double scattering_pdf = rec.mat->scattering_pdf(r, rec, scattered);
            double pdf = scattering_pdf;

            color color_from_scatter =
                (attenuation * scattering_pdf * ray_color(scattered, depth-1, world)) / pdf;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            return color_from_emission + color_from_scatter;
        }
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [ray-color-impsample]: <kbd>[camera.h]</kbd>
    The ray_color function, modified for importance sampling]

</div>

You should get exactly the same picture. Which _should make sense_, as the scattered part of
`ray_color` is getting multiplied by `scattering_pdf / pdf`, and as `pdf` is equal to
`scattering_pdf` is just the same as multiplying by one.


Using a Uniform PDF Instead of a Perfect Match
----------------------------------------------
Now, just for the experience, let's try using a different sampling PDF. We'll continue to have our
reflected rays weighted by Lambertian, so $\cos(\theta_o)$, and we'll keep the scattering PDF as is,
but we'll use a different PDF in the denominator. We will sample using a uniform PDF about the
hemisphere, so we'll set the denominator to $1/2\pi$. This will still converge on the correct
answer, as all we've done is change the PDF, but since the PDF is now less of a perfect match for
the real distribution, it will take longer to converge. Which, for the same number of samples means
a noisier image:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class camera {
      ...
      private:
        ...
        color ray_color(const ray& r, int depth, const hittable& world) const {
            hit_record rec;

            // If we've exceeded the ray bounce limit, no more light is gathered.
            if (depth <= 0)
                return color(0,0,0);

            // If the ray hits nothing, return the background color.
            if (!world.hit(r, interval(0.001, infinity), rec))
                return background;

            ray scattered;
            color attenuation;
            color color_from_emission = rec.mat->emitted(rec.u, rec.v, rec.p);

            if (!rec.mat->scatter(r, rec, attenuation, scattered))
                return color_from_emission;

            double scattering_pdf = rec.mat->scattering_pdf(r, rec, scattered);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            double pdf = 1 / (2*pi);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            color color_from_scatter =
                (attenuation * scattering_pdf * ray_color(scattered, depth-1, world)) / pdf;

            return color_from_emission + color_from_scatter;
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [ray-color-uniform]: <kbd>[camera.h]</kbd>
    The ray_color function, now with a uniform PDF in the denominator]

You should get a very similar result to before, only with slightly more noise, it may be hard to
see.

  ![<span class='num'>Image 4:</span> Cornell box, with imperfect PDF
  ](../images/img-3.04-cornell-refactor2.jpg class='pixel')

Make sure to return the PDF to the scattering PDF.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
            ...

            double scattering_pdf = rec.mat->scattering_pdf(r, rec, scattered);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            double pdf = scattering_pdf;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            ...
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [ray-color-return]: <kbd>[camera.h]</kbd> Return the PDF to the same as scattering PDF]


Random Hemispherical Sampling
---------------------------
To confirm our understanding, let's try a different scattering distribution. For this one, we'll
attempt to repeat the uniform hemispherical scattering from the first book. There's nothing wrong
with this technique, but we are no longer treating our objects as Lambertian. Lambertian is a
specific type of diffuse material that requires a $\cos(\theta_o)$ scattering distribution.
Uniform hemispherical scattering is a different diffuse material. If we keep the material the same
but change the PDF, as we did in last section, we will still converge on the same answer, but our
convergence may take more or less samples. However, if we change the material, we will have
fundamentally changed the render and the algorithm will converge on a different answer. So when we
replace Lambertian diffuse with uniform hemispherical diffuse we should expect the outcome of our
render to be _materially_ different. We're going to adjust our scattering direction and scattering
PDF:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class lambertian : public material {
      public:
        lambertian(const color& a) : albedo(make_shared<solid_color>(a)) {}
        lambertian(shared_ptr<texture> a) : albedo(a) {}

        bool scatter(const ray& r_in, const hit_record& rec, color& attenuation, ray& scattered)
        const override {
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            auto scatter_direction = random_in_hemisphere(rec.normal);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            // Catch degenerate scatter direction
            if (scatter_direction.near_zero())
                scatter_direction = rec.normal;

            scattered = ray(rec.p, scatter_direction, r_in.time());
            attenuation = albedo->value(rec.u, rec.v, rec.p);
            return true;
        }


        double scattering_pdf(const ray& r_in, const hit_record& rec, const ray& scattered) const {
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            return 1 / (2*pi);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
        }

        ...
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [scatter-mod]: <kbd>[material.h]</kbd> Modified PDF and scatter function]

This new diffuse material is actually just $p(\omega_o) = \frac{1}{2\pi}$ for the scattering PDF. So
our uniform PDF that was an imperfect match for Lambertian diffuse is actually a perfect match for
our uniform hemispherical diffuse. When rendering, we should get a slightly different image.

  ![<span class='num'>Image 5:</span> Cornell box, with uniform hemispherical sampling
  ](../images/img-3.04-cornell-refactor2.jpg class='pixel')

It’s pretty close to our old picture, but there are differences that are not just noise. The front
of the tall box is much more uniform in color. If you aren't sure what the best sampling pattern for
your material is, it's pretty reasonable to just go ahead and assume a uniform PDF, and while that
might converge slowly, it's not going to ruin your render. That said, if you're not sure what the
correct sampling pattern for your material is, your choice of PDF is not going to be your biggest
concern, as incorrectly choosing your scattering function _will_ ruin your render. At the very
least it will produce an incorrect result. You may find yourself with the most difficult kind of bug
to find in a Monte Carlo program -- a bug that produces a reasonable looking image! You won’t know
if the bug is in the first version of the program, or the second, or both!

Let’s build some infrastructure to address this.



Generating Random Directions
====================================================================================================
In this and the next two chapters, we'll harden our understanding and our tools.


Random Directions Relative to the Z Axis
-----------------------------------------
Let’s first figure out how to generate random directions. We already have a method to generate
random directions using the rejection method, so let's create one using the inversion method. To
simplify things, assume the $z$ axis is the surface normal, and $\theta$ is the angle from the
normal. We'll set everything up in terms of the $z$ axis this chapter. Next chapter we’ll get them
oriented to the surface normal vector. We will only deal with distributions that are rotationally
symmetric about $z$. So $p(\omega) = f(\theta)$.

Given a directional PDF on the sphere (where $p(\omega) = f(\theta)$), the one dimensional PDFs on
$\theta$ and $\phi$ are:

    $$ a(\phi) = \frac{1}{2\pi} $$
    $$ b(\theta) = 2\pi f(\theta)\sin(\theta) $$

For uniform random numbers $r_1$ and $r_2$, we solve for the CDF of $\theta$ and $\phi$ so that we
can invert the CDF to derive the random number generator.

    $$ r_1 = \int_{0}^{\phi} a(\phi') d\phi' $$
    $$ = \int_{0}^{\phi} \frac{1}{2\pi} d\phi' $$
    $$ = \frac{\phi}{2\pi} $$

Invert to solve for $\phi$:

    $$ \phi = 2 \pi \cdot r_1 $$

This should match with your intuition. To solve for a random $\phi$ you can take a uniform random
number in the interval [0,1] and multiply by $2\pi$ to cover the full range of all possible $\phi$
values, which is just [0,$2\pi$]. You may not have a fully formed intuition for how to solve for a
random value of $\theta$, so let's walk through the math to help you get set up. We rewrite $\phi$
as $\phi'$ and $\theta$ as $\theta'$ just like before, as a formality. For $\theta$ we have:

    $$ r_2 = \int_{0}^{\theta} b(\theta') d\theta' $$
    $$ = \int_{0}^{\theta} 2 \pi f(\theta') \sin(\theta') d\theta' $$

Let’s try some different functions for $f()$. Let’s first try a uniform density on the sphere. The
area of the unit sphere is $4\pi$, so a uniform $p(\omega) = \frac{1}{4\pi}$ on the unit sphere.

    $$ r_2 = \int_{0}^{\theta} 2 \pi \frac{1}{4\pi} \sin(\theta') d\theta' $$
    $$ = \int_{0}^{\theta} \frac{1}{2} \sin(\theta') d\theta' $$
    $$ = \frac{-\cos(\theta)}{2} - \frac{-\cos(0)}{2} $$
    $$ = \frac{1 - \cos(\theta)}{2} $$

Solving for $\cos(\theta)$ gives:

    $$ \cos(\theta) = 1 - 2 r_2 $$

We don’t solve for theta because we probably only need to know $\cos(\theta)$ anyway, and don’t want
needless $\arccos()$ calls running around.

To generate a unit vector direction toward $(\theta,\phi)$ we convert to Cartesian coordinates:

    $$ x = \cos(\phi) \cdot \sin(\theta) $$
    $$ y = \sin(\phi) \cdot \sin(\theta) $$
    $$ z = \cos(\theta) $$

And using the identity $\cos^2 + \sin^2 = 1$, we get the following in terms of random $(r_1,r_2)$:

    $$ x = \cos(2\pi \cdot r_1)\sqrt{1 - (1-2 r_2)^2} $$
    $$ y = \sin(2\pi \cdot r_1)\sqrt{1 - (1-2 r_2)^2} $$
    $$ z = 1 - 2  r_2 $$

Simplifying a little, $(1 - 2 r_2)^2 = 1 - 4r_2 + 4r_2^2$, so:

    $$ x = \cos(2 \pi r_1) \cdot 2 \sqrt{r_2(1 - r_2)} $$
    $$ y = \sin(2 \pi r_1) \cdot 2 \sqrt{r_2(1 - r_2)} $$
    $$ z = 1 - 2 r_2 $$

<div class='together'>
We can output some of these:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    #include "rtweekend.h"

    #include <iostream>
    #include <math.h>

    int main() {
        for (int i = 0; i < 200; i++) {
            auto r1 = random_double();
            auto r2 = random_double();
            auto x = cos(2*pi*r1)*2*sqrt(r2*(1-r2));
            auto y = sin(2*pi*r1)*2*sqrt(r2*(1-r2));
            auto z = 1 - 2*r2;
            std::cout << x << " " << y << " " << z << '\n';
        }
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [rand-unit-sphere-plot]: <kbd>[sphere_plot.cc]</kbd> Random points on the unit sphere]

</div>

<div class='together'>
And plot them for free on plot.ly (a great site with 3D scatterplot support):

  ![Figure [rand-pts-sphere]: Random points on the unit sphere
  ](../images/fig-3.10-rand-pts-sphere.jpg)

On the plot.ly website you can rotate that around and see that it appears uniform.

</div>


Uniform Sampling a Hemisphere
------------------------------
Now let’s derive uniform on the hemisphere. The density being uniform on the hemisphere means
$p(\omega) = f(\theta) = \frac{1}{2\pi}$. Just changing the constant in the theta equations yields:

    $$ r_2 = \int_{0}^{\theta} b(\theta') d\theta' $$
    $$ = \int_{0}^{\theta} 2 \pi f(\theta') \sin(\theta') d\theta' $$
    $$ = \int_{0}^{\theta} 2 \pi \frac{1}{2\pi} \sin(\theta') d\theta' $$
    $$ \ldots $$
    $$ \cos(\theta) = 1 - r_2 $$

This means that $\cos(\theta)$ will vary from 1 to 0, so $\theta$ will vary from 0 to $\pi/2$, which
means that nothing will go below the horizon. Rather than plot it, we'll solve for a 2D integral
with a known solution. Let’s integrate cosine cubed over the hemisphere (just picking something
arbitrary with a known solution). First we'll solve the integral by hand:

    $$ \int_\omega \cos^3(\theta) dA $$
    $$ = \int_{0}^{2 \pi} \int_{0}^{\pi /2} \cos^3(\theta) \sin(\theta) d\theta d\phi $$
    $$ = 2 \pi \int_{0}^{\pi/2} \cos^3(\theta) \sin(\theta) d\theta = \frac{\pi}{2} $$

<div class='together'>
Now for integration with importance sampling. $p(\omega) = \frac{1}{2\pi}$, so we average
$f()/p() = \cos^3(\theta) / \frac{1}{2\pi}$, and we can test this:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    #include "rtweekend.h"

    #include <iostream>
    #include <iomanip>
    #include <math.h>

    double f(double r1, double r2) {
        // auto x = cos(2*pi*r1)*2*sqrt(r2*(1-r2));
        // auto y = sin(2*pi*r1)*2*sqrt(r2*(1-r2));
        auto z = 1 - r2;
        double cos_theta = z;
        return cos_theta*cos_theta*cos_theta;
    }

    double pdf(double r1, double r2) {
        return 1.0 / (2.0*pi);
    }

    int main() {
        int N = 1000000;

        auto sum = 0.0;
        for (int i = 0; i < N; i++) {
            auto r1 = random_double();
            auto r2 = random_double();
            sum += f(r1, r2) / pdf(r1, r2);
        }

        std::cout << std::fixed << std::setprecision(12);
        std::cout << "PI/2 = " << pi / 2.0 << '\n';
        std::cout << "Estimate = " << sum / N << '\n';
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [cos-cubed]: <kbd>[cos_cubed.cc]</kbd> Integration using $cos^3(x)$]

</div>


Cosine Sampling a Hemisphere
------------------------------
We'll now continue trying to solve for cosine cubed over the horizon, but we'll change our PDF to
generate directions with $p(\omega) =  f(\theta) = \cos(\theta) / \pi$.

    $$ r_2 = \int_{0}^{\theta} b(\theta') d\theta' $$
    $$ = \int_{0}^{\theta} 2 \pi f(\theta') \sin(\theta') d\theta' $$
    $$ = \int_{0}^{\theta} 2 \pi \frac{\cos(\theta')}{\pi} \sin(\theta') d\theta' $$
    $$ = 1 - \cos^2(\theta) $$

So,

    $$ \cos(\theta) = \sqrt{1 - r_2} $$

We can save a little algebra on specific cases by noting

    $$ z = \cos(\theta) = \sqrt{1 - r_2} $$
    $$ x = \cos(\phi) \sin(\theta) = \cos(2 \pi r_1) \sqrt{1 - z^2} = \cos(2 \pi r_1) \sqrt{r_2} $$
    $$ y = \sin(\phi) \sin(\theta) = \sin(2 \pi r_1) \sqrt{1 - z^2} = \sin(2 \pi r_1) \sqrt{r_2} $$

<div class='together'>
Here's a function that generates random vectors weighted by this PDF:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    inline vec3 random_cosine_direction() {
        auto r1 = random_double();
        auto r2 = random_double();

        auto phi = 2*pi*r1;
        auto x = cos(phi)*sqrt(r2);
        auto y = sin(phi)*sqrt(r2);
        auto z = sqrt(1-r2);

        return vec3(x, y, z);
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [random-cosine-direction]: <kbd>[vec3.h]</kbd>
    Random cosine direction utility function]

</div>

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    #include "rtweekend.h"

    #include <iostream>
    #include <iomanip>
    #include <math.h>

    double f(const vec3& d) {
        auto cos_theta = d.z();
        return cos_theta*cos_theta*cos_theta;
    }

    double pdf(const vec3& d) {
        return d.z() / pi;
    }

    int main() {
        int N = 1000000;

        auto sum = 0.0;
        for (int i = 0; i < N; i++) {
            vec3 d = random_cosine_direction();
            sum += f(d) / pdf(d);
        }

        std::cout << std::fixed << std::setprecision(12);
        std::cout << "PI/2 = " << pi / 2.0 << '\n';
        std::cout << "Estimate = " << sum / N << '\n';
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [cos-density]: <kbd>[cos_density.cc]</kbd> Integration with cosine density function]

We can generate other densities later as we need them. This `random_cosine_direction()` function
produces a random direction weighted by $\cos(\theta)$ where $\theta$ is the angle from the $z$
axis.



Orthonormal Bases
====================================================================================================
In the last chapter we developed methods to generate random directions relative to the $z$ axis. If
we want to be able to produce reflections off of any surface, we are going to need to make this more
general: Not all normals are going to be perfectly aligned with the $z$ axis. So in this chapter
we are going to generalize our methods so that they support arbitrary surface normal vectors.


Relative Coordinates
---------------------
An _orthonormal basis_ (ONB) is a collection of three mutually orthogonal unit vectors. It is a
strict subtype of coordinate system. The Cartesian $xyz$ axes are one example of an orthonormal
basis. All of our renders are the result of the relative positions and orientations of the objects
in a scene projected onto the image plane of the camera. The camera and objects must be described in
the same coordinate system, so that the projection onto the image plane is logically defined,
otherwise the camera has no definitive means of correctly rendering the objects. Either the camera
must be redefined in the objects' coordinate system, or the objects must be redefined in the
camera's coordinate system. It's best to start with both in the same coordinate system, so no
redefinition is necessary. So long as the camera and scene are described in the same coordinate
system, all is well. The orthonormal basis defines how distances and orientations are represented in
the space, but an orthonormal basis alone is not enough. The objects and the camera need to
described by their displacement from a mutually defined location. This is just the origin
$\mathbf{O}$ of the scene; it represents the center of the universe for everything to displace from.

Suppose we have an origin $\mathbf{O}$ and Cartesian unit vectors $\mathbf{x}$, $\mathbf{y}$, and
$\mathbf{z}$. When we say a location is (3,-2,7), we really are saying:

    $$ \text{Location is } \mathbf{O} + 3\mathbf{x} - 2\mathbf{y} + 7\mathbf{z} $$

If we want to measure coordinates in another coordinate system with origin $\mathbf{O}'$ and basis
vectors $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$, we can just find the numbers $(u,v,w)$ such
that:

    $$ \text{Location is } \mathbf{O}' + u\mathbf{u} + v\mathbf{v} + w\mathbf{w} $$


Generating an Orthonormal Basis
--------------------------------
If you take an intro to graphics course, there will be a lot of time spent on coordinate systems and
4×4 coordinate transformation matrices. Pay attention, it’s really important stuff! But we won’t be
needing it for this book and we'll make do without it. What we do need is to generate random
directions with a set distribution relative to the surface normal vector $\mathbf{n}$. We won’t be
needing an origin for this because a direction is relative and has no specific origin. To start off
with, we need two cotangent vectors that are each perpendicular to $\mathbf{n}$ and that are also
perpendicular to each other.

Some 3D object models will come with one or more cotangent vectors for each vertex. If our model has
only one cotangent vector, then the process of making an ONB is a nontrivial one. Suppose we have
any vector $\mathbf{a}$ that is of nonzero length and nonparallel with $\mathbf{n}$. We can get
vectors $\mathbf{s}$ and $\mathbf{t}$ perpendicular to $\mathbf{n}$ by using the property of the
cross product that $\mathbf{n} \times \mathbf{a}$ is perpendicular to both $\mathbf{n}$ and
$\mathbf{a}$:

    $$ \mathbf{s} = \operatorname{unit_vector}(\mathbf{n} \times \mathbf{a}) $$

    $$ \mathbf{t} = \mathbf{n} \times \mathbf{s} $$

<div class='together'>
This is all well and good, but the catch is that we may not be given an $\mathbf{a}$ when we load a
model, and our current program doesn't have a way to generate one. If we went ahead and picked an
arbitrary $\mathbf{a}$ to use as an initial vector we may get an $\mathbf{a}$ that is parallel to
$\mathbf{n}$. So a common method is to pick an arbitrary axis and check to see if it's parallel to
$\mathbf{n}$ (which we assume to be of unit length), if it is, just use another axis:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    if (fabs(n.x()) > 0.9)
        a = vec3(0, 1, 0)
    else
        a = vec3(1, 0, 0)
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

</div>

<div class='together'>
We then take the cross product to get $\mathbf{s}$ and $\mathbf{t}$

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    vec3 s = unit_vector(cross(n, a));
    vec3 t = cross(n, s);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

</div>

Note that we don't need to take the unit vector for $\mathbf{t}$. Since $\mathbf{n}$ and
$\mathbf{s}$ are both unit vectors, their cross product $\mathbf{t}$ will be also. Once we have an
ONB of $\mathbf{s}$, $\mathbf{t}$, and $\mathbf{n}$, and we have a random $(x,y,z)$ relative to the
$z$ axis, we can get the vector relative to $\mathbf{n}$ with:

    $$ \mathit{Random vector} = x \mathbf{s} + y \mathbf{t} + z \mathbf{n} $$

If you remember, we used similar math to produce rays from a camera. You can think of that as a
change to the camera’s natural coordinate system.


The ONB Class
--------------
Should we make a class for ONBs, or are utility functions enough? I’m not sure, but let’s make a
class because it won't really be more complicated than utility functions:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    #ifndef ONB_H
    #define ONB_H

    #include "rtweekend.h"

    class onb {
      public:
        onb() {}

        vec3 operator[](int i) const { return axis[i]; }
        vec3& operator[](int i) { return axis[i]; }

        vec3 u() const { return axis[0]; }
        vec3 v() const { return axis[1]; }
        vec3 w() const { return axis[2]; }

        vec3 local(double a, double b, double c) const {
            return a*u() + b*v() + c*w();
        }

        vec3 local(const vec3& a) const {
            return a.x()*u() + a.y()*v() + a.z()*w();
        }

        void build_from_w(const vec3& w) {
            vec3 unit_w = unit_vector(w);
            vec3 a = (fabs(unit_w.x()) > 0.9) ? vec3(0,1,0) : vec3(1,0,0);
            vec3 v = unit_vector(cross(unit_w, a));
            vec3 u = cross(unit_w, v);
            axis[0] = u;
            axis[1] = v;
            axis[2] = unit_w;
        }

      public:
        vec3 axis[3];
    };


    #endif
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [class-onb]: <kbd>[onb.h]</kbd> Orthonormal basis class]

<div class='together'>
We can rewrite our Lambertian material using this to get:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class lambertian : public material {
      public:
        ...

        bool scatter(
            const ray& r_in, const hit_record& rec, color& alb, ray& scattered, double& pdf
        ) const override {
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            onb uvw;
            uvw.build_from_w(rec.normal);
            auto scatter_direction = uvw.local(random_cosine_direction());
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
            scattered = ray(rec.p, unit_vector(scatter_direction), r_in.time());
            alb = albedo->value(rec.u, rec.v, rec.p);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            pdf = dot(uvw.w(), scattered.direction()) / pi;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
            return true;
        }

        ...
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [scatter-onb]: <kbd>[material.h]</kbd> Scatter function, with orthonormal basis]

</div>

<div class='together'>
Which produces:

  ![<span class='num'>Image 6:</span> Cornell box, with orthonormal basis scatter function
  ](../images/img-3.06-cornell-ortho.jpg class='pixel')

</div>

<div class='together'>
Let’s get rid of some of that noise.

But first, let's quickly update the `isotropic` material:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class isotropic : public material {
      public:
        isotropic(color c) : albedo(make_shared<solid_color>(c)) {}
        isotropic(shared_ptr<texture> a) : albedo(a) {}


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        bool scatter(
            const ray& r_in, const hit_record& rec, color& alb, ray& scattered, double& pdf
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
        ) const override {
            scattered = ray(rec.p, random_unit_vector(), r_in.time());
            attenuation = albedo->value(rec.u, rec.v, rec.p);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            pdf = 1 / (4 * pi);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
            return true;
        }


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        double scattering_pdf(const ray& r_in, const hit_record& rec, const ray& scattered)
        const override {
            return 1 / (4 * pi);
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

      private:
        shared_ptr<texture> albedo;
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [class-isotropic-impsample]: <kbd>[material.h]</kbd>
    Isotropic material, modified for importance sampling]

</div>



Sampling Lights Directly
====================================================================================================
The problem with sampling uniformly over all directions is that lights are no more likely to be
sampled than any arbirary or unimportant direction. We could use shadow rays to solve for the direct
lighting at any given point. Instead, I’ll just use a PDF that sends more rays to the light. We can
then turn around and change that PDF to send more rays in whatever direction we want.

It’s really easy to pick a random direction toward the light; just pick a random point on the light
and send a ray in that direction. But we'll need to know the PDF, $p(\omega)$, so that we're not
biasing our render. But what is that?


Getting the PDF of a Light
---------------------------
For a light with a surface area of $A$, if we sample uniformly on that light, the PDF on the surface
is just $\frac{1}{A}$. How much area does the entire surface of the light take up if its projected
back onto the unit sphere? Fortunately, there is a simple correspondence, as outlined in this
diagram:

  ![Figure [shape-onto-pdf]: Projection of light shape onto PDF
  ](../images/fig-3.11-shape-onto-pdf.jpg)

If we look at a small area $dA$ on the light, the probability of sampling it is
  $\operatorname{p_q}(q) \cdot dA$.
On the sphere, the probability of sampling the small area $d\omega$ on the sphere is
  $\operatorname{p}(\omega) \cdot d\omega$.
There is a geometric relationship between $d\omega$ and $dA$:

    $$ d\omega = \frac{dA \cdot \cos(\theta)}{\operatorname{distance}^2(p,q)} $$

Since the probability of sampling $d\omega$ and $dA$ must be the same, then

    $$ \operatorname{p}(\omega) \cdot d\omega = \operatorname{p_q}(q) \cdot dA $$
    $$ \operatorname{p}(\omega)
       \cdot \frac{dA \cdot \cos(\theta)}{\operatorname{distance}^2(p,q)}
       = \operatorname{p_q}(q) \cdot dA $$

We know that if we sample uniformly on the light the PDF on the surface is $\frac{1}{A}$:

    $$ \operatorname{p_q}(q) = \frac{1}{A} $$
    $$ \operatorname{p}(\omega) \cdot \frac{dA \cdot \cos(\theta)}{\operatorname{distance}^2(p,q)}
       =  \frac{dA}{A} $$

So

  $$ \operatorname{p}(\omega) = \frac{\operatorname{distance}^2(p,q)}{\cos(\theta) \cdot A} $$


Light Sampling
---------------
We can hack our `ray_color()` function to sample the light in a very hard-coded fashion just to
check that we got the math and concept right:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class camera {
      ...
      private:
        color ray_color(const ray& r, int depth, const hittable& world) const {
            hit_record rec;

            // If we've exceeded the ray bounce limit, no more light is gathered.
            if (depth <= 0)
                return color(0,0,0);

            // If the ray hits nothing, return the background color.
            if (!world.hit(r, interval(0.001, infinity), rec))
                return background;

            ray scattered;
            color attenuation;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            double pdf;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
            color color_from_emission = rec.mat->emitted(rec.u, rec.v, rec.p);


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            if (!rec.mat->scatter(r, rec, attenuation, scattered, pdf))
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
                return color_from_emission;


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            auto on_light = point3(random_double(213,343), 554, random_double(227,332));
            auto to_light = on_light - rec.p;
            auto distance_squared = to_light.length_squared();
            to_light = unit_vector(to_light);

            if (dot(to_light, rec.normal) < 0)
                return color_from_emission;

            double light_area = (343-213)*(332-227);
            auto light_cosine = fabs(to_light.y());
            if (light_cosine < 0.000001)
                return color_from_emission;

            pdf = distance_squared / (light_cosine * light_area);
            scattered = ray(rec.p, to_light, r.time());

            double scattering_pdf = rec.mat->scattering_pdf(r, rec, scattered);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            color color_from_scatter =
                (attenuation * scattering_pdf * ray_color(scattered, depth-1, world)) / pdf;

            return color_from_emission + color_from_scatter;
        }
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [ray-color-lights]: <kbd>[camera.h]</kbd> Ray color with light sampling]

<div class='together'>
With 10 samples per pixel this yields:

  ![<span class='num'>Image 7:</span> Cornell box, sampling only the light, 10 samples per pixel
  ](../images/img-3.07-cornell-sample-light.jpg class='pixel')

This is about what we would expect from something that samples only the light sources, so this
appears to work.

</div>


Switching to Unidirectional Light
----------------------------------
The noisy pops around the light on the ceiling are because the light is two-sided
and there is a small space between light and ceiling. We probably want to have the light just emit
down. We can do that by letting the emitted member function of hittable take extra information:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class material {
      public:
        ...


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        virtual color emitted(
            const ray& r_in, const hit_record& rec, double u, double v, const point3& p
        ) const {
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
            return color(0,0,0);
        }
        ...
    };

    class diffuse_light : public material {
      public:
        ...


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        color emitted(const ray& r_in, const hit_record& rec, double u, double v, const point3& p)
        const override {
            if (!rec.front_face)
                return color(0,0,0);
            return emit->value(u, v, p);
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

        ...
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [emitted-directional]: <kbd>[material.h]</kbd> Material emission, directional]

<div class='together'>
This gives us:

  ![<span class='num'>Image 8:</span> Cornell box, light emitted only in the downward direction
  ](../images/img-3.08-cornell-lightdown.jpg class='pixel')

</div>



Mixture Densities
====================================================================================================
We have used a PDF related to $\cos(\theta)$, and a PDF related to sampling the light. We would like
a PDF that combines these.


The PDF Class
-------------
We've worked with PDFs in quite a lot of code already. I think that now is a good time to figure out
how we want to standardize our usage of PDFs. We already know that we are going to have a PDF for
the surface and a PDF for the light, so let's create a `pdf` base class. So far, we've had a `pdf()`
function that took a direction and returned the PDF's distribution value for that direction. This
value has so far been one of $1/4\pi$, $1/2\pi$, and $\cos(\theta)/\pi$. In a couple of our examples
we generated the random direction using a different distribution than the
distribution of the PDF. We covered this quite a lot in the chapter Playing with Importance
Sampling. In general, if we know the distribution of our random directions, we should use a PDF with
the same distribution. This will lead to the fastest convergence. With that in mind, we'll create a
`pdf` class that is responsible for generating random directions and determining the value of the
PDF.

From all of this, any `pdf` class should be responsible for

  1. returning a random direction weighted by the internal PDF distribution, and
  2. returning the corresponding PDF distribution value in that direction.

<div class='together'>
The details of how this is done under the hood varies for $\operatorname{pSurface}$ and
  $\operatorname{pLight}$, but that is exactly what class hierarchies were invented for!
It’s never obvious what goes in an abstract class, so my approach is to be greedy and hope a minimal
  interface works, and for `pdf` this implies:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    #ifndef PDF_H
    #define PDF_H

    #include "rtweekend.h"

    #include "onb.h"


    class pdf {
      public:
        virtual ~pdf() {}

        virtual double value(const vec3& direction) const = 0;
        virtual vec3 generate() const = 0;
    };

    #endif
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [class-pdf]: <kbd>[pdf.h]</kbd> The abstract pdf class]

</div>

<div class='together'>
We’ll see if we need to add anything else to `pdf` by fleshing out the subclasses. First, we'll
create a uniform density over the unit sphere:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class sphere_pdf : public pdf {
      public:
        sphere_pdf() { }

        double value(const vec3& direction) const override {
            return 1/ (4 * pi);
        }

        vec3 generate() const override {
            return random_unit_vector();
        }
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [class-uni-pdf]: <kbd>[pdf.h]</kbd> The uniform_pdf class]

</div>

<div class='together'>
Next, let’s try a cosine density:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class cosine_pdf : public pdf {
      public:
        cosine_pdf(const vec3& w) { uvw.build_from_w(w); }

        double value(const vec3& direction) const override {
            auto cosine_theta = dot(unit_vector(direction), uvw.w());
            return fmax(0, cosine_theta/pi);
        }

        vec3 generate() const override {
            return uvw.local(random_cosine_direction());
        }

      private:
        onb uvw;
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [class-cos-pdf]: <kbd>[pdf.h]</kbd> The cosine_pdf class]

</div>

<div class='together'>
We can try this cosine PDF in the `ray_color()` function:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class camera {
      ...
      private:
        ...
        color ray_color(const ray& r, int depth, const hittable& world) const {
            hit_record rec;

            // If we've exceeded the ray bounce limit, no more light is gathered.
            if (depth <= 0)
                return color(0,0,0);

            // If the ray hits nothing, return the background color.
            if (!world.hit(r, interval(0.001, infinity), rec))
                return background;

            ray scattered;
            color attenuation;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            double pdf_val;
            color color_from_emission = rec.mat->emitted(r, rec, rec.u, rec.v, rec.p);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            if (!rec.mat->scatter(r, rec, attenuation, scattered, pdf_val))
                return color_from_emission;


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            cosine_pdf surface_pdf(rec.normal);
            scattered = ray(rec.p, surface_pdf.generate(), r.time());
            pdf_val = surface_pdf.value(scattered.direction());
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            double scattering_pdf = rec.mat->scattering_pdf(r, rec, scattered);

            color color_from_scatter =
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
                (attenuation * scattering_pdf * ray_color(scattered, depth-1, world)) / pdf_val;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            return color_from_emission + color_from_scatter;
        }
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [ray-color-cos-pdf]: <kbd>[camera.h]</kbd> The ray_color function, using cosine pdf]

</div>

<div class='together'>
This yields an exactly matching result so all we’ve done so far is move some computation up into the
`cosine_pdf` class:

  ![<span class='num'>Image 9:</span> Cornell box with a cosine density PDF
  ](../images/img-3.09-cornell-cos-pdf.jpg class='pixel')

</div>


Sampling Directions towards a Hittable
---------------------------------------
Now we can try sampling directions toward a `hittable`, like the light.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    ...
    #include "hittable_list.h"
    ...
    class hittable_pdf : public pdf {
      public:
        hittable_pdf(const hittable& _objects, const point3& _origin)
          : objects(_objects), origin(_origin)
        {}

        double value(const vec3& direction) const override {
            return objects.pdf_value(origin, direction);
        }

        vec3 generate() const override {
            return objects.random(origin);
        }

      private:
        const hittable& objects;
        point3 origin;
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [class-hittable-pdf]: <kbd>[pdf.h]</kbd> The hittable_pdf class]

If we want to sample the light, we will need `hittable` to answer some queries that it doesn’t yet
have an interface for. The above code assumes the existence of two as-of-yet unimplemented functions
in the `hittable` class: `pdf_value()` and `random()`. We need to add these functions for the
program to compile. We could go through all of the `hittable` subclasses and add these functions,
but that would be a hassle, so we’ll just add two trivial functions to the `hittable` base class.
This breaks our previously pure abstract implementation, but it saves work. Feel free to write these
functions through to subclasses if you want a purely abstract `hittable` interface class.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class hittable {
      public:
        ...


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        virtual double pdf_value(const point3& o, const vec3& v) const {
            return 0.0;
        }

        virtual vec3 random(const vec3& o) const {
            return vec3(1, 0, 0);
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [hittable-plus2]: <kbd>[hittable.h]</kbd> The hittable class, with two new methods]

<div class='together'>
And then we change `quad` to implement those functions:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class quad : public hittable {
      public:
        quad(const point3& _Q, const vec3& _u, const vec3& _v, shared_ptr<material> m)
          : Q(_Q), u(_u), v(_v), mat(m)
        {
            auto n = cross(u, v);
            normal = unit_vector(n);
            D = dot(normal, Q);
            w = n / dot(n,n);


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            area = n.length();
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            set_bounding_box();
        }
        ...


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        double pdf_value(const point3& origin, const vec3& v) const override {
            hit_record rec;
            if (!this->hit(ray(origin, v), interval(0.001, infinity), rec))
                return 0;

            auto distance_squared = rec.t * rec.t * v.length_squared();
            auto cosine = fabs(dot(v, rec.normal) / v.length());

            return distance_squared / (cosine * area);
        }

        vec3 random(const point3& origin) const override {
            auto p = plane_origin + (random_double() * axis_A) + (random_double() * axis_B);
            return p - origin;
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

      private:
        point3 Q;
        vec3 u, v;
        shared_ptr<material> mat;
        aabb bbox;
        vec3 normal;
        double D;
        vec3 w;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        double area;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [quad-pdf]: <kbd>[quad.h]</kbd> quad with pdf]

</div>

We only need to add `pdf_value()` and `random()` to `quad` because we're using this to importance
sample the light, and the only light we have in our scene is a `quad`. if you want other light
geometries, or want to use a PDF with other objects, you'll need to implement the above functions
for the corresponding classes.

<div class='together'>
Add a `lights` parameter to the camera `render()` function:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class camera {
      public:
        ...
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        void render(const hittable& world, const hittable& lights) {
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
            initialize();

            std::cout << "P3\n" << image_width << ' ' << image_height << "\n255\n";

            int sqrt_spp = int(sqrt(samples_per_pixel));
            for (int j = 0; j < image_height; ++j) {
                std::clog << "\rScanlines remaining: " << (image_height - j) << ' ' << std::flush;
                for (int i = 0; i < image_width; ++i) {
                    color pixel_color(0,0,0);
                    for (int s_j = 0; s_j < sqrt_spp; ++s_j) {
                        for (int s_i = 0; s_i < sqrt_spp; ++s_i) {
                            ray r = get_ray(i, j, s_i, s_j);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
                            pixel_color += ray_color(r, max_depth, world, lights);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
                        }
                    }
                    write_color(std::cout, pixel_color, samples_per_pixel);
                }
            }

            std::clog << "\rDone.                 \n";
        }

        ...
      private:
        ...
        color ray_color(const ray& r, int depth, const hittable& world, const hittable& lights)
        const {
            ...

            ray scattered;
            color attenuation;
            double pdf_val;
            color color_from_emission = rec.mat->emitted(r, rec, rec.u, rec.v, rec.p);

            if (!rec.mat->scatter(r, rec, attenuation, scattered, pdf_val))
                return color_from_emission;


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            hittable_pdf light_pdf(light_ptr, rec.p);
            scattered = ray(rec.p, light_pdf.generate(), r.time());
            pdf_val = light_pdf.value(scattered.direction());
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            double scattering_pdf = rec.mat->scattering_pdf(r, rec, scattered);


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            color sample_color = ray_color(scattered, depth-1, world, lights);
            color color_from_scatter = (attenuation * scattering_pdf * sample_color) / pdf_val;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            return color_from_emission + color_from_scatter;
        }
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [ray-color-lights]: <kbd>[camera.h]</kbd> ray_color function with light PDF]

</div>

<div class='together'>
Create a light in the middle of the ceiling:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    int main() {
        ...

        // Box 2
        shared_ptr<hittable> box2 = box(point3(0,0,0), point3(165,165,165), white);
        box2 = make_shared<rotate_y>(box2, -18);
        box2 = make_shared<translate>(box2, vec3(130,0,65));
        world.add(box2);


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        // Light Sources
        hittable_list lights;
        auto m = shared_ptr<material>();
        lights.add(make_shared<quad>(point3(343,554,332), vec3(-130,0,0), vec3(0,0,-105), m));
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

        camera cam;
        ...


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        cam.render(world, lights);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [ray-color-hittable-pdf]: <kbd>[main.cc]</kbd> Adding a light to the Cornell box]

</div>

<div class='together'>
At 10 samples per pixel we get:

  ![<span class='num'>Image 10:</span> Cornell box, sampling a hittable light, 10 samples per pixel
  ](../images/img-3.10-hittable-light.jpg class='pixel')

</div>


The Mixture PDF Class
----------------------
As was briefly mentioned in the chapter Playing with Importance Sampling, we can create linear
mixtures of any PDFs to form mixture densities that are also PDFs. Any weighted average of PDFs is
also a PDF. As long as the weights are positive and add up to any one, we have a new PDF.

  $$ \operatorname{pMixture}() = w_0 p_0() + w_1 p_1() + w_2 p_2() + \ldots + w_{n-1} p_{n-1}() $$

  $$ 1 = w_0 + w_1 + w_2 + \ldots + w_{n-1} $$

For example, we could just average the two densities:

  $$ \operatorname{pMixture}(\omega_o)
     = \frac{1}{2} \operatorname{pSurface}(\omega_o) + \frac{1}{2} \operatorname{pLight}(\omega_o)
  $$

<div class='together'>
How would we instrument our code to do that? There is a very important detail that makes this not
quite as easy as one might expect. Generating the random direction for a mixture PDF is simple:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    if (random_double() < 0.5)
        pick direction according to pSurface
    else
        pick direction according to pLight
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

</div>

But solving for the PDF value of $\operatorname{pMixture}$ is slightly more subtle. We can't just

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    if (direction is from pSurface)
        get PDF value of pSurface
    else
        get PDF value of pLight
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

</div>

<div class='together'>
For one, figuring out which PDF the random direction came from is probably not trivial. We don't
have any plumbing for `generate()` to tell `value()` what the original `random_double()` was, so we
can't trivially say which PDF the random direction comes from. If we thought that the above was
correct, we would have to solve backwards to figure which PDF the direction could come from. Which
honestly sounds like a nightmare, but fortunately we don't need to do that. There are some
directions that both PDFs could have generated.
For example, a direction toward the light could have been generated by either
  $\operatorname{pLight}$ _or_ $\operatorname{pSurface}$.
It is sufficient for us to solve for the pdf value of $\operatorname{pSurface}$ and of
  $\operatorname{pLight}$ for a random direction and then take the PDF mixture weights to solve for
  the total PDF value for that direction.
The mixture density class is actually pretty straightforward:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class mixture_pdf : public pdf {
      public:
        mixture_pdf(shared_ptr<pdf> p0, shared_ptr<pdf> p1) {
            p[0] = p0;
            p[1] = p1;
        }

        double value(const vec3& direction) const override {
            return 0.5 * p[0]->value(direction) + 0.5 *p[1]->value(direction);
        }

        vec3 generate() const override {
            if (random_double() < 0.5)
                return p[0]->generate();
            else
                return p[1]->generate();
        }

      private:
        shared_ptr<pdf> p[2];
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [class-mixturep-df]: <kbd>[pdf.h]</kbd> The mixture_pdf class]

</div>

<div class='together'>
Now we would like to do a mixture density of the cosine sampling and of the light sampling. We can
plug it into `ray_color()`:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class camera {
      ...
      private:
        ...
        color ray_color(const ray& r, int depth, const hittable& world, const hittable& lights)
        const {
            ...

            if (!rec.mat->scatter(r, rec, attenuation, scattered, pdf_val))
                return color_from_emission;


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            auto p0 = make_shared<hittable_pdf>(light_ptr, rec.p);
            auto p1 = make_shared<cosine_pdf>(rec.normal);
            mixture_pdf mixed_pdf(p0, p1);

            scattered = ray(rec.p, mixed_pdf.generate(), r.time());
            pdf_val = mixed_pdf.value(scattered.direction());
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            double scattering_pdf = rec.mat->scattering_pdf(r, rec, scattered);

            color sample_color = ray_color(scattered, depth-1, world, lights);
            color color_from_scatter = (attenuation * scattering_pdf * sample_color) / pdf_val;

            return color_from_emission + color_from_scatter;
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [ray-color-mixture]: <kbd>[camera.h]</kbd> The ray_color function, using mixture PDF]

</div>

<div class='together'>
1000 samples per pixel yields:

  ![<span class='num'>Image 11:</span> Cornell box, mixture density of cosine and light sampling
  ](../images/img-3.11-cosine-and-light.jpg class='pixel')

</div>



Some Architectural Decisions
====================================================================================================
We won't write any code in this chapter. We’re at a crossroads and we need to make some
architectural decisions.

The mixture-density approach is an alternative to having more traditional shadow rays. These are
rays that check for an unobstructed path from an intersection point to a given light source. Rays
that intersect an object between a point and a given light source indicate that the intersection
point is in the shadow of that particular light source. The mixture-density approach is something
that I personally prefer, because in addition to lights, you can sample windows or bright cracks
under doors or whatever else you think might be bright -- or important. But you'll still see shadow
rays in most professional path tracers. Typically they'll have a predefined number of shadow rays
(_e.g_ 1, 4, 8, 16) where over the course of rendering, at each place where the path tracing ray
intersects, they'll send these terminal shadow rays to random lights in the scene to determine if
the intersection is lit by that random light. The intersection will either be lit by that light, or
completely in shadow, where more shadow rays lead to a more accurate illumination. After all of the
shadow rays terminate (either at a light or at an occluding surface), the inital path tracing ray
continues on and more shadow rays are sent at the next intersection. You can't tell the shadow rays
what is important, you can only tell them what is emissive, so shadow rays work best on simpler
scenes that don't have overly complicated photon distribution. That said, shadow rays terminate at
the first thing they run into and don't bounce around, so one shadow ray is cheaper than one path
tracing ray, which is the reason that you'll typically see a lot more shadow rays than path tracing
rays (_e.g_ 1, 4, 8, 16). You could choose shadow rays over mixture-density in a more restricted
scene; that’s a personal design preference. Shadow rays tend to be cheaper for a crude result than
mixture-density and is becoming increasingly common in realtime.

There are some other issues with the code.

The PDF construction is hard coded in the `ray_color()` function. We should clean that up.

We've accidentally broken the specular rays (glass and metal), and they are no longer supported.
The math would continue to work out if we just made their scattering function a delta function, but
that would lead to all kinds of floating point disasters. We could either make specular reflection
a special case that skips $f()/p()$, or we could set surface roughness to a very small -- but
nonzero -- value and have almost-mirrors that look perfectly smooth but that don’t generate NaNs. I
don’t have an opinion on which way to do it (I have tried both and they both have their advantages),
but we have smooth metal and glass code anyway, so we'll add perfect specular surfaces that just
skip over explicit $f()/p()$ calculations.

We also lack a real background function infrastructure in case we want to add an environment map or
a more interesting functional background. Some environment maps are HDR (the RGB components are
normalized floats rather than 0–255 bytes). Our output has been HDR all along; we’ve just been
truncating it.

Finally, our renderer is RGB. A more physically based one -- like an automobile manufacturer might
use -- would probably need to use spectral colors and maybe even polarization. For a movie
renderer, most studios still get away with RGB. You can make a hybrid renderer that has both modes,
but that is of course harder. I’m going to stick to RGB for now, but I will touch on this at the end
of the book.



Cleaning Up PDF Management
====================================================================================================
So far I have the `ray_color()` function create two hard-coded PDFs:

  1. `p0()` related to the shape of the light
  2. `p1()` related to the normal vector and type of surface

We can pass information about the light (or whatever `hittable` we want to sample) into the
`ray_color()` function, and we can ask the `material` function for a PDF (we would have to add
instrumentation to do that). We also need to know if the scattered ray is specular, and we can do
this either by asking the `hit()` function or the `material` class.


Diffuse Versus Specular
------------------------
One thing we would like to allow for is a material -- like varnished wood -- that is partially ideal
specular (the polish) and partially diffuse (the wood). Some renderers have the material generate
two rays: one specular and one diffuse. I am not fond of branching, so I would rather have the
material randomly decide whether it is diffuse or specular. The catch with that approach is that we
need to be careful when we ask for the PDF value, and `ray_color()` needs to be aware of whether
this ray is diffuse or specular. Fortunately, we have decided that we should only call the
`pdf_value()` if it is diffuse, so we can handle that implicitly.

<div class='together'>
We can redesign `material` and stuff all the new arguments into a class like we did for `hittable`:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
    class scatter_record {
      public:
        color attenuation;
        shared_ptr<pdf> pdf_ptr;
        bool skip_pdf;
        ray skip_pdf_ray;
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

    class material {
      public:
        ...

        virtual bool scatter(
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            const ray& r_in, const hit_record& rec, scatter_record& srec
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
        ) const {
            return false;
        }
        ...
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [material-refactor]: <kbd>[material.h]</kbd> Refactoring the material class]

</div>

<div class='together'>
The `lambertian` material becomes simpler:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class lambertian : public material {
      public:
        lambertian(const color& a) : albedo(make_shared<solid_color>(a)) {}
        lambertian(shared_ptr<texture> a) : albedo(a) {}


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        bool scatter(const ray& r_in, const hit_record& rec, scatter_record& srec) const override {
            srec.attenuation = albedo->value(rec.u, rec.v, rec.p);
            srec.pdf_ptr = make_shared<cosine_pdf>(rec.normal);
            srec.skip_pdf = false;
            return true;
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

        double scattering_pdf(const ray& r_in, const hit_record& rec, const ray& scattered) const {
            auto cosine = dot(rec.normal, unit_vector(scattered.direction()));
            return cosine < 0 ? 0 : cosine/pi;
        }

      private:
        shared_ptr<texture> albedo;
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [lambertian-scatter]: <kbd>[material.h]</kbd> New lambertian scatter() method]

</div>

<div class='together'>
As does the `isotropic` material:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class isotropic : public material {
      public:
        isotropic(color c) : albedo(make_shared<solid_color>(c)) {}
        isotropic(shared_ptr<texture> a) : albedo(a) {}


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        bool scatter(const ray& r_in, const hit_record& rec, scatter_record& srec) const override {
            srec.attenuation = albedo->value(rec.u, rec.v, rec.p);
            srec.pdf_ptr = make_shared<sphere_pdf>();
            srec.skip_pdf = false;
            return true;
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++


        double scattering_pdf(const ray& r_in, const hit_record& rec, const ray& scattered)
        const override {
            return 1 / (4 * pi);
        }

      private:
        shared_ptr<texture> albedo;
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [isotropic-scatter]: <kbd>[material.h]</kbd> New isotropic scatter() method]

</div>

<div class='together'>
And `ray_color()` changes are small:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class camera {
      ...
      private:
        ...
        color ray_color(const ray& r, int depth, const hittable& world, const hittable& lights)
        const {
            hit_record rec;

            // If we've exceeded the ray bounce limit, no more light is gathered.
            if (depth <= 0)
                return color(0,0,0);

            // If the ray hits nothing, return the background color.
            if (!world.hit(r, interval(0.001, infinity), rec))
                return background;


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            scatter_record srec;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
            color color_from_emission = rec.mat->emitted(r, rec, rec.u, rec.v, rec.p);


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            if (!rec.mat->scatter(r, rec, srec))
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
                return color_from_emission;


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            auto light_ptr = make_shared<hittable_pdf>(lights, rec.p);
            mixture_pdf p(light_ptr, srec.pdf_ptr);

            ray scattered = ray(rec.p, p.generate(), r.time());
            auto pdf_val = p.value(scattered.direction());
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            double scattering_pdf = rec.mat->scattering_pdf(r, rec, scattered);

            color sample_color = ray_color(scattered, depth-1, world, lights);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            color color_from_scatter = (srec.attenuation * scattering_pdf * sample_color) / pdf_val;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            return color_from_emission + color_from_scatter;
        }
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [ray-color-mixture]: <kbd>[camera.h]</kbd> The ray_color function, using mixture PDF]

</div>


Handling Specular
------------------
We have not yet dealt with specular surfaces, nor instances that mess with the surface normal. But
this design is clean overall, and those are all fixable. For now, I will just fix `specular`. Metal
and dielectric materials are easy to fix.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class metal : public material {
      public:
        metal(const color& a, double f) : albedo(a), fuzz(f < 1 ? f : 1) {}


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        bool scatter(const ray& r_in, const hit_record& rec, scatter_record& srec) const override {
            srec.attenuation = albedo;
            srec.pdf_ptr = nullptr;
            srec.skip_pdf = true;
            vec3 reflected = reflect(unit_vector(r_in.direction()), rec.normal);
            srec.skip_pdf_ray =
                ray(rec.p, reflected + fuzz*random_in_unit_sphere(), r_in.time());
            return true;
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

      private:
        color albedo;
        double fuzz;
    };

    ...

    class dielectric : public material {
      public:
        dielectric(double index_of_refraction) : ir(index_of_refraction) {}


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        bool scatter(const ray& r_in, const hit_record& rec, scatter_record& srec) const override {
            srec.attenuation = color(1.0, 1.0, 1.0);
            srec.pdf_ptr = nullptr;
            srec.skip_pdf = true;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
            double refraction_ratio = rec.front_face ? (1.0/ir) : ir;

            vec3 unit_direction = unit_vector(r_in.direction());
            double cos_theta = fmin(dot(-unit_direction, rec.normal), 1.0);
            double sin_theta = sqrt(1.0 - cos_theta*cos_theta);

            bool cannot_refract = refraction_ratio * sin_theta > 1.0;
            vec3 direction;

            if (cannot_refract || reflectance(cos_theta, refraction_ratio) > random_double())
                direction = reflect(unit_direction, rec.normal);
            else
                direction = refract(unit_direction, rec.normal, refraction_ratio);


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            srec.skip_pdf_ray = ray(rec.p, direction, r_in.time());
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
            return true;
        }

      ...
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [material-scatter]: <kbd>[material.h]</kbd> The metal and dielectric scatter methods]

Note that if the fuzziness is nonzero, this surface isn’t really ideally specular, but the implicit
sampling works just like it did before. We're effectively skipping all of our PDF work for the
materials that we're treating specularly.

<div class='together'>
`ray_color()` just needs a new case to generate an implicitly sampled ray:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class camera {
      ...
      private:
        ...
        color ray_color(const ray& r, int depth, const hittable& world, const hittable& lights)
        const {
            ...

            if (!rec.mat->scatter(r, rec, srec))
                return color_from_emission;


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
            if (srec.skip_pdf) {
                return srec.attenuation * ray_color(srec.skip_pdf_ray, depth-1, world, lights);
            }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

            auto light_ptr = make_shared<hittable_pdf>(lights, rec.p);
            mixture_pdf p(light_ptr, srec.pdf_ptr);

            ...
        }
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [ray-color-implicit]: <kbd>[camera.h]</kbd>
    Ray color function with implicitly-sampled rays]

</div>

We'll check our work by changing a block to metal. We'd also like to swap out one of the blocks for
a glass object, but we'll push that off for the next section. Glass objects are difficult to render
well, so we'd like to make a PDF for them, but we have some more work to do before we're able to do
that.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    int main() {
        ...
        // Light
        world.add(make_shared<quad>(point3(213,554,227), vec3(130,0,0), vec3(0,0,105), light));

        // Box 1
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        shared_ptr<material> aluminum = make_shared<metal>(color(0.8, 0.85, 0.88), 0.0);
        shared_ptr<hittable> box1 = box(point3(0,0,0), point3(165,330,165), aluminum);
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
        box1 = make_shared<rotate_y>(box1, 15);
        box1 = make_shared<translate>(box1, vec3(265,0,295));
        world.add(box1);

        // Box 2
        shared_ptr<hittable> box2 = box(point3(0,0,0), point3(165,165,165), white);
        box2 = make_shared<rotate_y>(box2, -18);
        box2 = make_shared<translate>(box2, vec3(130,0,65));
        world.add(box2);

        // Light Sources
        hittable_list lights;
        auto m = shared_ptr<material>();
        lights.add(make_shared<quad>(point3(343,554,332), vec3(-130,0,0), vec3(0,0,-105), m));

        ...
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [scene-cornell-al]: <kbd>[main.cc]</kbd> Cornell box scene with aluminum material]

<div class='together'>
The resulting image has a noisy reflection on the ceiling because the directions toward the box are
not sampled with more density.

  ![<span class='num'>Image 12:</span> Cornell box with arbitrary PDF functions
  ](../images/img-3.12-arbitrary-pdf.jpg class='pixel')

</div>


Sampling a Sphere Object
-------------------------
The noisiness on the ceiling could be reduced by making a PDF of the metal block. We would also want
a PDF for the block if we made it glass. But making a PDF for a block is quite a bit of work and
isn't terribly interesting, so let’s create a PDF for a glass sphere instead. It's quicker and makes
for a more interesting render. We need to figure out how to sample a sphere to determine an
appropriate PDF distribution. If we want to sample a sphere from a point outside of the sphere, we
can't just pick a random point on its surface and be done. If we did that, we would frequently pick
a point on the far side of the sphere, which would be occluded by the front side of the sphere. We
need a way to uniformly sample the side of the sphere that is visible from an arbitrary point. When
we sample a sphere’s solid angle uniformly from a point outside the sphere, we are really just
sampling a cone uniformly. The cone axis goes from the ray origin through the sphere center, with
the sides of the cone tangent to the sphere -- see illustration below. Let’s say the code has
`theta_max`. Recall from the Generating Random Directions chapter that to sample $\theta$ we have:

  $$ r_2 = \int_{0}^{\theta} 2 \pi f(\theta') \sin(\theta') d\theta' $$

Here $f(\theta')$ is an as-of-yet uncalculated constant $C$, so:

  $$ r_2 = \int_{0}^{\theta} 2 \pi C \sin(\theta') d\theta' $$

If we solve through the calculus:

  $$ r_2 = 2\pi \cdot C \cdot (1-\cos(\theta)) $$

So

  $$ cos(\theta) = 1 - \frac{r_2}{2 \pi \cdot C} $$

We are constraining our distribution so that the random direction must be less than $\theta_{max}$.
This means that the integral from 0 to $\theta_{max}$ must be one, and therefore $r_2 = 1$. We can
use this to solve for $C$:

  $$ r_2 = 2\pi \cdot C \cdot (1-\cos(\theta)) $$
  $$ 1 = 2\pi \cdot C \cdot (1-\cos(\theta_{max})) $$
  $$ C = \frac{1}{2\pi \cdot (1-\cos(\theta_{max})} $$

Which gives us an equality between $\theta$, $\theta_{max}$, and $r_2$:

  $$ \cos(\theta) = 1 + r_2 \cdot (\cos(\theta_{max})-1) $$

We sample $\phi$ like before, so:

  $$ z = \cos(\theta) = 1 + r_2 \cdot (\cos(\theta_{max}) - 1) $$
  $$ x = \cos(\phi) \cdot \sin(\theta) = \cos(2\pi \cdot r_1) \cdot \sqrt{1-z^2} $$
  $$ y = \sin(\phi) \cdot \sin(\theta) = \sin(2\pi \cdot r_1) \cdot \sqrt{1-z^2} $$

Now what is $\theta_{max}$?

  ![Figure [sphere-enclosing-cone]: A sphere-enclosing cone
  ](../images/fig-3.12-sphere-enclosing-cone.jpg)

We can see from the figure that $\sin(\theta_{max}) = R / length(\mathbf{c} - \mathbf{p})$. So:

  $$ \cos(\theta_{max}) = \sqrt{1 - \frac{R^2}{length^2(\mathbf{c} - \mathbf{p})}} $$

We also need to evaluate the PDF of directions. For a uniform distribution toward the sphere the PDF
is $1/\mathit{solid_angle}$. What is the solid angle of the sphere? It has something to do with the
$C$ above. It is -- by definition -- the area on the unit sphere, so the integral is

  $$ \mathit{solid angle} = \int_{0}^{2\pi} \int_{0}^{\theta_{max}} \sin(\theta)
       = 2 \pi \cdot (1-\cos(\theta_{max})) $$

It’s good to check the math on all such calculations. I usually plug in the extreme cases (thank you
for that concept, Mr. Horton -- my high school physics teacher). For a zero radius sphere
$\cos(\theta_{max}) = 0$, and that works. For a sphere tangent at $\mathbf{p}$,
$\cos(\theta_{max}) = 0$, and $2\pi$ is the area of a hemisphere, so that works too.


Updating the Sphere Code
-------------------------
The sphere class needs the two PDF-related functions:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class sphere : public hittable {
      public:
        ...


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        double pdf_value(const point3& o, const vec3& v) const override {
            // This method only works for stationary spheres.

            hit_record rec;
            if (!this->hit(ray(o, v), interval(0.001, infinity), rec))
                return 0;

            auto cos_theta_max = sqrt(1 - radius*radius/(center1 - o).length_squared());
            auto solid_angle = 2*pi*(1-cos_theta_max);

            return  1 / solid_angle;
        }

        vec3 random(const point3& o) const override {
            vec3 direction = center1 - o;
            auto distance_squared = direction.length_squared();
            onb uvw;
            uvw.build_from_w(direction);
            return uvw.local(random_to_sphere(radius, distance_squared));
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

      private:
        ...

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        static vec3 random_to_sphere(double radius, double distance_squared) {
            auto r1 = random_double();
            auto r2 = random_double();
            auto z = 1 + r2*(sqrt(1-radius*radius/distance_squared) - 1);

            auto phi = 2*pi*r1;
            auto x = cos(phi)*sqrt(1-z*z);
            auto y = sin(phi)*sqrt(1-z*z);

            return vec3(x, y, z);
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [sphere-pdf]: <kbd>[sphere.h]</kbd> Sphere with PDF]

<div class='together'>
We can first try just sampling the sphere rather than the light:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    int main() {
        ...

        // Light
        world.add(make_shared<quad>(point3(213,554,227), vec3(130,0,0), vec3(0,0,105), light));


        ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        // Box
        shared_ptr<hittable> box1 = box(point3(0,0,0), point3(165,330,165), white);
        box1 = make_shared<rotate_y>(box1, 15);
        box1 = make_shared<translate>(box1, vec3(265,0,295));
        world.add(box1);

        // Glass Sphere
        auto glass = make_shared<dielectric>(1.5);
        world.add(make_shared<sphere>(point3(190,90,190), 90, glass));
        ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

        // Light Sources
        hittable_list lights;
        auto m = shared_ptr<material>();
        lights.add(make_shared<quad>(point3(343,554,332), vec3(-130,0,0), vec3(0,0,-105), m));

        ...
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [sampling-sphere]: <kbd>[main.cc]</kbd> Sampling just the sphere]

</div>

<div class='together'>
This yields a noisy room, but the caustic under the sphere is good. It took five times as long as
sampling the light did for my code. This is probably because those rays that hit the glass are
expensive!

  ![<span class='num'>Image 13:</span> Cornell box with glass sphere, using new PDF functions
  ](../images/img-3.13-cornell-glass-sphere.jpg class='pixel')

</div>


Adding PDF Functions to Hittable Lists
---------------------------------------
We should probably just sample both the sphere and the light. We can do that by creating a mixture
density of their two distributions. We could do that in the `ray_color()` function by passing a list
of hittables in and building a mixture PDF, or we could add PDF functions to `hittable_list`. I
think both tactics would work fine, but I will go with instrumenting `hittable_list`.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    class hittable_list : public hittable {
      public:
        ...


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        double pdf_value(const point3& o, const vec3& v) const override {
            auto weight = 1.0/objects.size();
            auto sum = 0.0;

            for (const auto& object : objects)
                sum += weight * object->pdf_value(o, v);

            return sum;
        }

        vec3 random(const vec3& o) const override {
            auto int_size = static_cast<int>(objects.size());
            return objects[random_int(0, int_size-1)]->random(o);
        }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

        ...
    };
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [density-mixture]: <kbd>[hittable_list.h]</kbd> Creating a mixture of densities]

<div class='together'>
We assemble a list to pass to `render()` from `main()`:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    int main() {
        ...

        // Light Sources
        hittable_list lights;
        auto m = shared_ptr<material>();
        lights.add(make_shared<quad>(point3(343,554,332), vec3(-130,0,0), vec3(0,0,-105), m));
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        lights.add(make_shared<sphere>(point3(190, 90, 190), 90, m));
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

        ...
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [scene-density-mixture]: <kbd>[main.cc]</kbd> Updating the scene]

</div>

<div class='together'>
And we get a decent image with 1000 samples as before:

  ![<span class='num'>Image 14:</span> Cornell box using a mixture of glass & light PDFs
  ](../images/img-3.14-glass-and-light.jpg class='pixel')

</div>


Handling Surface Acne
----------------------
An astute reader pointed out there are some black specks in the image above. All Monte Carlo Ray
Tracers have this as a main loop:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    pixel_color = average(many many samples)
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

If you find yourself getting some form of acne in your renders, and this acne is white or black --
where one "bad" sample seems to kill the whole pixel -- then that sample is probably a huge number
or a `NaN` (Not A Number). This particular acne is probably a `NaN`. Mine seems to come up once in
every 10–100 million rays or so.

<div class='together'>
So big decision: sweep this bug under the rug and check for `NaN`s, or just kill `NaN`s and hope
this doesn't come back to bite us later. I will always opt for the lazy strategy, especially when I
know that working with floating point is hard. First, how do we check for a `NaN`? The one thing I
always remember for `NaN`s is that a `NaN` does not equal itself. Using this trick, we update the
`write_color()` function to replace any `NaN` components with zero:

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
    void write_color(std::ostream &out, color pixel_color, int samples_per_pixel) {
        auto r = pixel_color.x();
        auto g = pixel_color.y();
        auto b = pixel_color.z();


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
        // Replace NaN components with zero.
        if (r != r) r = 0.0;
        if (g != g) g = 0.0;
        if (b != b) b = 0.0;
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++

        // Divide the color by the number of samples and gamma-correct for gamma=2.0.
        auto scale = 1.0 / samples_per_pixel;
        r = sqrt(scale * r);
        g = sqrt(scale * g);
        b = sqrt(scale * b);

        // Write the translated [0,255] value of each color component.
        static const interval intensity(0.000, 0.999);
        out << static_cast<int>(256 * intensity.clamp(r)) << ' '
            << static_cast<int>(256 * intensity.clamp(g)) << ' '
            << static_cast<int>(256 * intensity.clamp(b)) << '\n';
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    [Listing [write-color-nan]: <kbd>[color.h]</kbd> NaN-tolerant write_color function]

</div>

<div class='together'>
Happily, the black specks are gone:

  ![<span class='num'>Image 15:</span> Cornell box with anti-acne color function
  ](../images/img-3.15-book3-final.jpg class='pixel')

</div>



The Rest of Your Life
====================================================================================================
The purpose of this book was to walk through all of the little details (dotting all the i's and
crossing all of the t's) necessary when organizing a physically based renderer’s sampling approach.
You should now be able to take all of this detail and explore a lot of different potential paths.

If you want to explore Monte Carlo methods, look into bidirectional and path spaced approaches such
as Metropolis. Your probability space won't be over solid angle, but will instead be over path
space, where a path is a multidimensional point in a high-dimensional space. Don’t let that scare
you -- if you can describe an object with an array of numbers, mathematicians call it a point in the
space of all possible arrays of such points. That’s not just for show. Once you get a clean
abstraction like that, your code can get clean too. Clean abstractions are what programming is all
about!

If you want to do movie renderers, look at the papers out of studios and Solid Angle. They are
surprisingly open about their craft.

If you want to do high-performance ray tracing, look first at papers from Intel and NVIDIA. They are
also surprisingly open.

If you want to do hard-core physically based renderers, convert your renderer from RGB to spectral.
I am a big fan of each ray having a random wavelength and almost all the RGBs in your program
turning into floats. It sounds inefficient, but it isn’t!

Regardless of what direction you take, add a glossy BRDF model. There are many to choose from, and
each has its advantages.

Have fun!

[Peter Shirley][]<br>
Salt Lake City, March, 2016



                               (insert acknowledgments.md.html here)



Citing This Book
====================================================================================================
Consistent citations make it easier to identify the source, location and versions of this work. If
you are citing this book, we ask that you try to use one of the following forms if possible.

Basic Data
-----------
  - **Title (series)**: “Ray Tracing in One Weekend Series”
  - **Title (book)**: “Ray Tracing: The Rest of Your Life”
  - **Author**: Peter Shirley, Trevor David Black, Steve Hollasch
  - **Version/Edition**: v4.0.0-alpha.1
  - **Date**: 2023-08-06
  - **URL (series)**: https://raytracing.github.io/
  - **URL (book)**: https://raytracing.github.io/books/RayTracingTheRestOfYourLife.html

Snippets
---------

  ### Markdown
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

  ### HTML
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    <a href='https://raytracing.github.io/books/RayTracingTheRestOfYourLife.html'>
        <cite>Ray Tracing: The Rest of Your Life</cite>
    </a>
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

  ### LaTeX and BibTex
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    ~\cite{Shirley2023RTW3}

    @misc{Shirley2023RTW3,
       title = {Ray Tracing: The Rest of Your Life},
       author = {Peter Shirley, Trevor David Black, Steve Hollasch},
       year = {2023},
       month = {August},
       note = {\small \texttt{https://raytracing.github.io/books/RayTracingTheRestOfYourLife.html}},
       url = {https://raytracing.github.io/books/RayTracingTheRestOfYourLife.html}
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

  ### BibLaTeX
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    \usepackage{biblatex}

    ~\cite{Shirley2023RTW3}

    @online{Shirley2023RTW3,
       title = {Ray Tracing: The Rest of Your Life},
       author = {Peter Shirley, Trevor David Black, Steve Hollasch},
       year = {2023},
       month = {August},
       url = {https://raytracing.github.io/books/RayTracingTheRestOfYourLife.html}
    }
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

  ### IEEE
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    “Ray Tracing: The Rest of Your Life.”
    raytracing.github.io/books/RayTracingTheRestOfYourLife.html
    (accessed MMM. DD, YYYY)
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

  ### MLA:
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Ray Tracing: The Rest of Your Life. raytracing.github.io/books/RayTracingTheRestOfYourLife.html
    Accessed DD MMM. YYYY.
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


[Peter Shirley]:      https://github.com/petershirley
[Steve Hollasch]:     https://github.com/hollasch
[Trevor David Black]: https://github.com/trevordblack



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