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<!DOCTYPE html>
<meta charset="utf-8">
<link rel="icon" type="image/png" href="../favicon.png">
<!-- Markdeep: https://casual-effects.com/markdeep/ -->
**Ray Tracing: The Next Week**
[Peter Shirley][], [Trevor David Black][], [Steve Hollasch][]
<br>
Version 4.0.3, (XXYear)-(XXMonNum)-(XXDay)
Version 4.1.0, (XXYear)-(XXMonNum)-(XXDay)
Version 5.0.0, (XXYear)-(XXMonNum)-(XXDay)
**DRAFT IN DEVELOPMENT**
<br>
Copyright 2018-(XXYear) Peter Shirley. All rights reserved.
Overview
====================================================================================================
In Ray Tracing in One Weekend, you built a simple brute force path tracer. In this installment we’ll
add textures, volumes (like fog), rectangles, instances, lights, and support for lots of objects
using a BVH. When done, you’ll have a “real” ray tracer.
A heuristic in ray tracing that many people--including me--believe, is that most optimizations
complicate the code without delivering much speedup. What I will do in this mini-book is go with the
simplest approach in each design decision I make. See [our Further Reading wiki page][wiki-further]
for additional project related resources. However, I strongly encourage you to do no
premature optimization; if it doesn’t show up high in the execution time profile, it doesn’t need
optimization until all the features are supported!
The two hardest parts of this book are the BVH and the Perlin textures. This is why the title
suggests you take a week rather than a weekend for this endeavor. But you can save those for last if
you want a weekend project. Order is not very important for the concepts presented in this book, and
without BVH and Perlin texture you will still get a Cornell Box!
See the [project README][readme] file for information about this project, the repository on GitHub,
directory structure, building & running, and how to make or reference corrections and contributions.
These books have been formatted to print well directly from your browser. We also include PDFs of
each book [with each release][releases], in the "Assets" section.
Thanks to everyone who lent a hand on this project. You can find them in the acknowledgments section
at the end of this book.
Motion Blur
====================================================================================================
When you decided to ray trace, you decided that visual quality was worth more than run-time. When
rendering fuzzy reflection and defocus blur, we used multiple samples per pixel. Once you have taken
a step down that road, the good news is that almost _all_ effects can be similarly brute-forced.
Motion blur is certainly one of those.
In a real camera, the shutter remains open for a short time interval, during which the camera and
objects in the world may move. To accurately reproduce such a camera shot, we seek an average of
what the camera senses while its shutter is open to the world.
Introduction of SpaceTime Ray Tracing
--------------------------------------
We can get a random estimate of a single (simplified) photon by sending a single ray at some random
instant in time while the shutter is open. As long as we can determine where the objects are
supposed to be at that instant, we can get an accurate measure of the light for that ray at that
same instant. This is yet another example of how random (Monte Carlo) ray tracing ends up being
quite simple. Brute force wins again!
<div class='together'>
Since the “engine” of the ray tracer can just make sure the objects are where they need to be for
each ray, the intersection guts don’t change much. To accomplish this, we need to store the exact
time for each ray:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
class ray {
public:
ray() {}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
ray(const point3& origin, const vec3& direction, double time)
: orig(origin), dir(direction), tm(time) {}
ray(const point3& origin, const vec3& direction)
: ray(origin, direction, 0) {}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
const point3& origin() const { return orig; }
const vec3& direction() const { return dir; }
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
double time() const { return tm; }
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
point3 at(double t) const {
return orig + t*dir;
}
private:
point3 orig;
vec3 dir;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
double tm;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
};
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [time-ray]: <kbd>[ray.h]</kbd> Ray with time information]
</div>
Managing Time
--------------
Before continuing, let's think about time, and how we might manage it across one or more successive
renders. There are two aspects of shutter timing to think about: the time from one shutter opening
to the next shutter opening, and how long the shutter stays open for each frame. Standard movie film
used to be shot at 24 frames per second. Modern digital movies can be 24, 30, 48, 60, 120 or however
many frames per second director wants.
Each frame can have its own shutter speed. This shutter speed need not be -- and typically isn't --
the maximum duration of the entire frame. You could have the shutter open for 1/1000th of a second
every frame, or 1/60th of a second.
If you wanted to render a sequence of images, you would need to set up the camera with the
appropriate shutter timings: frame-to-frame period, shutter/render duration, and the total number of
frames (total shot time). If the camera is moving and the world is static, you're good to go.
However, if anything in the world is moving, you would need to add a method to `hittable` so that
every object could be made aware of the current frame's time period. This method would provide a way
for all animated objects to set up their motion during that frame.
This is fairly straight-forward, and definitely a fun avenue for you to experiment with if you wish.
However, for our purposes right now, we're going to proceed with a much simpler model. We will
render only a single frame, implicitly assuming a start at time = 0 and ending at time = 1. Our
first task is to modify the camera to launch rays with random times in $[0,1]$, and our second task
will be the creation of an animated sphere class.
Updating the Camera to Simulate Motion Blur
--------------------------------------------
We need to modify the camera to generate rays at a random instant between the start time and the end
time. Should the camera keep track of the time interval, or should that be up to the user of the
camera when a ray is created? When in doubt, I like to make constructors complicated if it makes
calls simple, so I will make the camera keep track, but that’s a personal preference. Not many
changes are needed to camera because for now it is not allowed to move; it just sends out rays over
a time period.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
class camera {
...
private:
...
ray get_ray(int i, int j) const {
// Construct a camera ray originating from the defocus disk and directed at a randomly
// sampled point around the pixel location i, j.
auto offset = sample_square();
auto pixel_sample = pixel00_loc
+ ((i + offset.x()) * pixel_delta_u)
+ ((j + offset.y()) * pixel_delta_v);
auto ray_origin = (defocus_angle <= 0) ? center : defocus_disk_sample();
auto ray_direction = pixel_sample - ray_origin;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
auto ray_time = random_double();
return ray(ray_origin, ray_direction, ray_time);
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
}
...
};
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [time-camera]: <kbd>[camera.h]</kbd> Camera with time information]
Adding Moving Spheres
----------------------
Now to create a moving object. I’ll update the sphere class so that its center moves linearly from
`center1` at time=0 to `center2` at time=1. (It continues on indefinitely outside that time
interval, so it really can be sampled at any time.) We'll do this by replacing the 3D center point
with a 3D ray that describes the original position at time=0 and the displacement to the end
position at time=1.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
class sphere : public hittable {
public:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
// Stationary Sphere
sphere(const point3& static_center, double radius, shared_ptr<material> mat)
: center(static_center, vec3(0,0,0)), radius(std::fmax(0,radius)), mat(mat) {}
// Moving Sphere
sphere(const point3& center1, const point3& center2, double radius,
shared_ptr<material> mat)
: center(center1, center2 - center1), radius(std::fmax(0,radius)), mat(mat) {}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
...
private:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
ray center;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
double radius;
shared_ptr<material> mat;
};
#endif
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [moving-sphere]: <kbd>[sphere.h]</kbd> A moving sphere]
<div class='together'>
The updated `sphere::hit()` function is almost identical to the old `sphere::hit()` function:
we just need to now determine the current position of the animated center:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
class sphere : public hittable {
public:
...
bool hit(const ray& r, interval ray_t, hit_record& rec) const override {
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
point3 current_center = center.at(r.time());
vec3 oc = current_center - r.origin();
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
auto a = r.direction().length_squared();
auto h = dot(r.direction(), oc);
auto c = oc.length_squared() - radius*radius;
...
rec.t = root;
rec.p = r.at(rec.t);
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
vec3 outward_normal = (rec.p - current_center) / radius;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
rec.set_face_normal(r, outward_normal);
get_sphere_uv(outward_normal, rec.u, rec.v);
rec.mat = mat;
return true;
}
...
};
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [moving-sphere-hit]: <kbd>[sphere.h]</kbd> Moving sphere hit function]
</div>
Tracking the Time of Ray Intersection
--------------------------------------
Now that rays have a time property, we need to update the `material::scatter()` methods to account
for the time of intersection:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
class lambertian : public material {
...
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;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
scattered = ray(rec.p, scatter_direction, r_in.time());
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
attenuation = albedo;
return true;
}
...
};
class metal : public material {
...
bool scatter(const ray& r_in, const hit_record& rec, color& attenuation, ray& scattered)
const override {
vec3 reflected = reflect(r_in.direction(), rec.normal);
reflected = unit_vector(reflected) + (fuzz * random_unit_vector());
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
scattered = ray(rec.p, reflected, r_in.time());
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
attenuation = albedo;
return (dot(scattered.direction(), rec.normal) > 0);
}
...
};
class dielectric : public material {
...
bool scatter(const ray& r_in, const hit_record& rec, color& attenuation, ray& scattered)
const override {
...
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
scattered = ray(rec.p, direction, r_in.time());
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
return true;
}
...
};
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [material-time]: <kbd>[material.h]</kbd> Handle ray time in the material::scatter() methods]
Putting Everything Together
----------------------------
The code below takes the example diffuse spheres from the scene at the end of the last book, and
makes them move during the image render. Each sphere moves from its center $\mathbf{C}$ at time
$t=0$ to $\mathbf{C} + (0, r/2, 0)$ at time $t=1$:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
int main() {
hittable_list world;
auto ground_material = make_shared<lambertian>(color(0.5, 0.5, 0.5));
world.add(make_shared<sphere>(point3(0,-1000,0), 1000, ground_material));
for (int a = -11; a < 11; a++) {
for (int b = -11; b < 11; b++) {
auto choose_mat = random_double();
point3 center(a + 0.9*random_double(), 0.2, b + 0.9*random_double());
if ((center - point3(4, 0.2, 0)).length() > 0.9) {
shared_ptr<material> sphere_material;
if (choose_mat < 0.8) {
// diffuse
auto albedo = color::random() * color::random();
sphere_material = make_shared<lambertian>(albedo);
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
auto center2 = center + vec3(0, random_double(0,.5), 0);
world.add(make_shared<sphere>(center, center2, 0.2, sphere_material));
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
} else if (choose_mat < 0.95) {
...
}
...
camera cam;
cam.aspect_ratio = 16.0 / 9.0;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
cam.image_width = 400;
cam.samples_per_pixel = 100;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
cam.max_depth = 50;
cam.vfov = 20;
cam.lookfrom = point3(13,2,3);
cam.lookat = point3(0,0,0);
cam.vup = vec3(0,1,0);
cam.defocus_angle = 0.6;
cam.focus_dist = 10.0;
cam.render(world);
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [scene-spheres-moving]: <kbd>[main.cc]</kbd> Last book's final scene, but with moving spheres]
<div class='together'>
This gives the following result:
<div id="image-bouncing-spheres">
</div>
</div>
Bounding Volume Hierarchies
====================================================================================================
This part is by far the most difficult and involved part of the ray tracer we are working on. I am
sticking it in this chapter so the code can run faster, and because it refactors `hittable` a
little, and when I add rectangles and boxes we won't have to go back and refactor them.
Ray-object intersection is the main time-bottleneck in a ray tracer, and the run time is linear with
the number of objects. But it’s a repeated search on the same scene, so we ought to be able to make
it a logarithmic search in the spirit of binary search. Because we are sending millions to billions
of rays into the same scene, we can sort the objects in the scene, and then each ray intersection
can be a sublinear search. The two most common methods of sorting are to 1) subdivide the space, and
2) subdivide the objects. The latter is usually much easier to code up, and just as fast to run for
most models.
The Key Idea
-------------
The key idea of creating bounding volumes for a set of primitives is to find a volume that fully
encloses (bounds) all the objects. For example, suppose you computed a sphere that bounds 10
objects. Any ray that misses the bounding sphere definitely misses all ten objects inside. If the
ray hits the bounding sphere, then it might hit one of the ten objects. So the bounding code is
always of the form:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
if (ray hits bounding object)
return whether ray hits bounded objects
else
return false
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Note that we will use these bounding volumes to group the objects in the scene into subgroups. We
are *not* dividing the screen or the scene space. We want any given object to be in just one
bounding volume, though bounding volumes can overlap.
Hierarchies of Bounding Volumes
--------------------------------
To make things sub-linear we need to make the bounding volumes hierarchical. For example, if we
divided a set of objects into two groups, red and blue, and used rectangular bounding volumes, we’d
have:
![Figure [bvol-hierarchy]: Bounding volume hierarchy](../images/fig-2.01-bvol-hierarchy.jpg)
<div class='together'>
Note that the blue and red bounding volumes are contained in the purple one, but they might overlap,
and they are not ordered -- they are just both inside. So the tree shown on the right has no concept
of ordering in the left and right children; they are simply inside. The code would be:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
if (hits purple)
hit0 = hits blue enclosed objects
hit1 = hits red enclosed objects
if (hit0 or hit1)
return true and info of closer hit
return false
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
</div>
Axis-Aligned Bounding Boxes (AABBs)
------------------------------------
To get that all to work we need a way to make good divisions, rather than bad ones, and a way to
intersect a ray with a bounding volume. A ray bounding volume intersection needs to be fast, and
bounding volumes need to be pretty compact. In practice for most models, axis-aligned boxes work
better than the alternatives (such as the spherical bounds mentioned above), but this design choice
is always something to keep in mind if you encounter other types of bounding models.
From now on we will call axis-aligned bounding rectangular parallelepipeds (really, that is what
they need to be called if we're being precise) _axis-aligned bounding boxes_, or AABBs. (In the
code, you will also come across the naming abbreviation "bbox" for "bounding box".) Any method you
want to use to intersect a ray with an AABB is fine. And all we need to know is whether or not we
hit it; we don’t need hit points or normals or any of the stuff we need to display the object.
<div class='together'>
Most people use the “slab” method. This is based on the observation that an n-dimensional AABB is
just the intersection of $n$ axis-aligned intervals, often called “slabs”. Recall that an interval
is just the points within two endpoints, for example, $x$ such that $3 \leq x \leq 5$, or more
succinctly $x$ in $[3,5]$. In 2D, an AABB (a rectangle) is defined by the overlap two intervals:
![Figure [2d-aabb]: 2D axis-aligned bounding box](../images/fig-2.02-2d-aabb.jpg)
</div>
To determine if a ray hits one interval, we first need to figure out whether the ray hits the
boundaries. For example, in 1D, ray intersection with two planes will yield the ray parameters $t_0$
and $t_1$. (If the ray is parallel to the planes, its intersection with any plane will be
undefined.)
![Figure [ray-slab]: Ray-slab intersection](../images/fig-2.03-ray-slab.jpg)
How do we find the intersection between a ray and a plane? Recall that the ray is just defined by a
function that--given a parameter $t$--returns a location $\mathbf{P}(t)$:
$$ \mathbf{P}(t) = \mathbf{Q} + t \mathbf{d} $$
This equation applies to all three of the x/y/z coordinates. For example, $x(t) = Q_x + t d_x$. This
ray hits the plane $x = x_0$ at the parameter $t$ that satisfies this equation:
$$ x_0 = Q_x + t_0 d_x $$
So $t$ at the intersection is given by
$$ t_0 = \frac{x_0 - Q_x}{d_x} $$
We get the similar expression for $x_1$:
$$ t_1 = \frac{x_1 - Q_x}{d_x} $$
<div class='together'>
The key observation to turn that 1D math into a 2D or 3D hit test is this: if a ray intersects the
box bounded by all pairs of planes, then all $t$-intervals will overlap. For example, in 2D the
green and blue overlapping only happens if the ray intersects the bounded box:
![Figure [ray-slab-interval]: Ray-slab $t$-interval overlap
](../images/fig-2.04-ray-slab-interval.jpg)
In this figure, the upper ray intervals do not overlap, so we know the ray does _not_ hit the 2D box
bounded by the green and blue planes. The lower ray intervals _do_ overlap, so we know the lower ray
_does_ hit the bounded box.
</div>
Ray Intersection with an AABB
------------------------------
The following pseudocode determines whether the $t$ intervals in the slab overlap:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
interval_x ← compute_intersection_x (ray, x0, x1)
interval_y ← compute_intersection_y (ray, y0, y1)
return overlaps(interval_x, interval_y)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
<div class='together'>
That is awesomely simple, and the fact that the 3D version trivially extends the above is why people
love the slab method:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
interval_x ← compute_intersection_x (ray, x0, x1)
interval_y ← compute_intersection_y (ray, y0, y1)
interval_z ← compute_intersection_z (ray, z0, z1)
return overlaps(interval_x, interval_y, interval_z)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
</div>
There are some caveats that make this less pretty than it first appears. Consider again the 1D
equations for $t_0$ and $t_1$:
$$ t_0 = \frac{x_0 - Q_x}{d_x} $$
$$ t_1 = \frac{x_1 - Q_x}{d_x} $$
First, suppose the ray is traveling in the negative $\mathbf{x}$ direction. The interval $(t_{x0},
t_{x1})$ as computed above might be reversed, like $(7, 3)$ for example. Second, the denominator
$d_x$ could be zero, yielding infinite values. And if the ray origin lies on one of the slab
boundaries, we can get a `NaN`, since both the numerator and the denominator can be zero. Also, the
zero will have a ± sign when using IEEE floating point.
The good news for $d_x = 0$ is that $t_{x0}$ and $t_{x1}$ will be equal: both +∞ or -∞, if not
between $x_0$ and $x_1$. So, using min and max should get us the right answers:
$$ t_{x0} = \min(
\frac{x_0 - Q_x}{d_x},
\frac{x_1 - Q_x}{d_x})
$$
$$ t_{x1} = \max(
\frac{x_0 - Q_x}{d_x},
\frac{x_1 - Q_x}{d_x})
$$
The remaining troublesome case if we do that is if $d_x = 0$ and either $x_0 - Q_x = 0$ or
$x_1 - Q_x = 0$ so we get a `NaN`. In that case we can arbitrarily interpret that as either hit or
no hit, but we’ll revisit that later.
Now, let’s look at the pseudo-function `overlaps`. Suppose we can assume the intervals are not
reversed, and we want to return true when the intervals overlap. The boolean `overlaps()` function
computes the overlap of the $t$ intervals `t_interval1` and `t_interval2`, and uses that to
determine if that overlap is non-empty:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
bool overlaps(t_interval1, t_interval2)
t_min ← max(t_interval1.min, t_interval2.min)
t_max ← min(t_interval1.max, t_interval2.max)
return t_min < t_max
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
If there are any `NaN`s running around there, the compare will return false, so we need to be sure
our bounding boxes have a little padding if we care about grazing cases (and we probably _should_
because in a ray tracer all cases come up eventually).
<div class='together'>
To accomplish this, we'll first add a new `interval` function `expand`, which pads an interval by a
given amount:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
class interval {
public:
...
double clamp(double x) const {
if (x < min) return min;
if (x > max) return max;
return x;
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
interval expand(double delta) const {
auto padding = delta/2;
return interval(min - padding, max + padding);
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
static const interval empty, universe;
};
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [interval-expand]: <kbd>[interval.h]</kbd> interval::expand() method]
</div>
<div class='together'>
Now we have everything we need to implement the new AABB class.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
#ifndef AABB_H
#define AABB_H
class aabb {
public:
interval x, y, z;
aabb() {} // The default AABB is empty, since intervals are empty by default.
aabb(const interval& x, const interval& y, const interval& z)
: x(x), y(y), z(z) {}
aabb(const point3& a, const point3& b) {
// Treat the two points a and b as extrema for the bounding box, so we don't require a
// particular minimum/maximum coordinate order.
x = (a[0] <= b[0]) ? interval(a[0], b[0]) : interval(b[0], a[0]);
y = (a[1] <= b[1]) ? interval(a[1], b[1]) : interval(b[1], a[1]);
z = (a[2] <= b[2]) ? interval(a[2], b[2]) : interval(b[2], a[2]);
}
const interval& axis_interval(int n) const {
if (n == 1) return y;
if (n == 2) return z;
return x;
}
bool hit(const ray& r, interval ray_t) const {
const point3& ray_orig = r.origin();
const vec3& ray_dir = r.direction();
for (int axis = 0; axis < 3; axis++) {
const interval& ax = axis_interval(axis);
const double adinv = 1.0 / ray_dir[axis];
auto t0 = (ax.min - ray_orig[axis]) * adinv;
auto t1 = (ax.max - ray_orig[axis]) * adinv;
if (t0 < t1) {
if (t0 > ray_t.min) ray_t.min = t0;
if (t1 < ray_t.max) ray_t.max = t1;
} else {
if (t1 > ray_t.min) ray_t.min = t1;
if (t0 < ray_t.max) ray_t.max = t0;
}
if (ray_t.max <= ray_t.min)
return false;
}
return true;
}
};
#endif
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [aabb]: <kbd>[aabb.h]</kbd> Axis-aligned bounding box class]
</div>
Constructing Bounding Boxes for Hittables
------------------------------------------
We now need to add a function to compute the bounding boxes of all the hittables. Then we will make
a hierarchy of boxes over all the primitives, and the individual primitives--like spheres--will live
at the leaves.
Recall that `interval` values constructed without arguments will be empty by default. Since an
`aabb` object has an interval for each of its three dimensions, each of these will then be empty by
default, and therefore `aabb` objects will be empty by default. Thus, some objects may have empty
bounding volumes. For example, consider a `hittable_list` object with no children. Happily, the way
we've designed our interval class, the math all works out.
Finally, recall that some objects may be animated. Such objects should return their bounds over the
entire range of motion, from time=0 to time=1.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
#include "aabb.h"
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
class material;
...
class hittable {
public:
virtual ~hittable() = default;
virtual bool hit(const ray& r, interval ray_t, hit_record& rec) const = 0;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
virtual aabb bounding_box() const = 0;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
};
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [hittable-bbox]: <kbd>[hittable.h]</kbd> Hittable class with bounding box]
For a stationary sphere, the `bounding_box` function is easy:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
class sphere : public hittable {
public:
// Stationary Sphere
sphere(const point3& static_center, double radius, shared_ptr<material> mat)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
: center(static_center, vec3(0,0,0)), radius(std::fmax(0,radius)), mat(mat)
{
auto rvec = vec3(radius, radius, radius);
bbox = aabb(static_center - rvec, static_center + rvec);
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
...
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
aabb bounding_box() const override { return bbox; }
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
private:
ray center;
double radius;
shared_ptr<material> mat;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
aabb bbox;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
...
};
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [sphere-bbox]: <kbd>[sphere.h]</kbd> Sphere with bounding box]
For a moving sphere, we want the bounds of its entire range of motion. To do this, we can take the
box of the sphere at time=0, and the box of the sphere at time=1, and compute the box around those
two boxes.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
class sphere : public hittable {
public:
...
// Moving Sphere
sphere(const point3& center1, const point3& center2, double radius,
shared_ptr<material> mat)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
: center(center1, center2 - center1), radius(std::fmax(0,radius)), mat(mat)
{
auto rvec = vec3(radius, radius, radius);
aabb box1(center.at(0) - rvec, center.at(0) + rvec);
aabb box2(center.at(1) - rvec, center.at(1) + rvec);
bbox = aabb(box1, box2);
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
...
};
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [moving-sphere-bbox]: <kbd>[sphere.h]</kbd> Moving sphere with bounding box]
<div class='together'>
Now we need a new `aabb` constructor that takes two boxes as input. First, we'll add a new interval
constructor to do this:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
class interval {
public:
double min, max;
interval() : min(+infinity), max(-infinity) {} // Default interval is empty
interval(double _min, double _max) : min(_min), max(_max) {}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
interval(const interval& a, const interval& b) {
// Create the interval tightly enclosing the two input intervals.
min = a.min <= b.min ? a.min : b.min;
max = a.max >= b.max ? a.max : b.max;
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
double size() const {
...
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [interval-from-intervals]: <kbd>[interval.h]</kbd> Interval constructor from two intervals]
</div>
<div class='together'>
Now we can use this to construct an axis-aligned bounding box from two input boxes.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
class aabb {
public:
...
aabb(const point3& a, const point3& b) {
...
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
aabb(const aabb& box0, const aabb& box1) {
x = interval(box0.x, box1.x);
y = interval(box0.y, box1.y);
z = interval(box0.z, box1.z);
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
...
};
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [aabb-from-two-aabb]: <kbd>[aabb.h]</kbd> AABB constructor from two AABB inputs]
</div>
Creating Bounding Boxes of Lists of Objects
--------------------------------------------
Now we'll update the `hittable_list` object, computing the bounds of its children. We'll update the
bounding box incrementally as each new child is added.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
#include "aabb.h"
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
#include "hittable.h"
#include <vector>
class hittable_list : public hittable {
public:
std::vector<shared_ptr<hittable>> objects;
...
void add(shared_ptr<hittable> object) {
objects.push_back(object);
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
bbox = aabb(bbox, object->bounding_box());
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
}
bool hit(const ray& r, interval ray_t, hit_record& rec) const override {
...
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
aabb bounding_box() const override { return bbox; }
private:
aabb bbox;
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
};
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [hit-list-bbox]: <kbd>[hittable_list.h]</kbd> Hittable list with bounding box]
The BVH Node Class
-------------------
A BVH is also going to be a `hittable` -- just like lists of `hittable`s. It’s really a container,
but it can respond to the query “does this ray hit you?”. One design question is whether we have two
classes, one for the tree, and one for the nodes in the tree; or do we have just one class and have
the root just be a node we point to. The `hit` function is pretty straightforward: check whether the
box for the node is hit, and if so, check the children and sort out any details.
<div class='together'>
I am a fan of the one class design when feasible. Here is such a class:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
#ifndef BVH_H
#define BVH_H
#include "aabb.h"
#include "hittable.h"
#include "hittable_list.h"
class bvh_node : public hittable {
public:
bvh_node(hittable_list list) : bvh_node(list.objects, 0, list.objects.size()) {
// There's a C++ subtlety here. This constructor (without span indices) creates an
// implicit copy of the hittable list, which we will modify. The lifetime of the copied
// list only extends until this constructor exits. That's OK, because we only need to
// persist the resulting bounding volume hierarchy.
}
bvh_node(std::vector<shared_ptr<hittable>>& objects, size_t start, size_t end) {
// To be implemented later.
}
bool hit(const ray& r, interval ray_t, hit_record& rec) const override {
if (!bbox.hit(r, ray_t))
return false;
bool hit_left = left->hit(r, ray_t, rec);
bool hit_right = right->hit(r, interval(ray_t.min, hit_left ? rec.t : ray_t.max), rec);
return hit_left || hit_right;
}
aabb bounding_box() const override { return bbox; }
private:
shared_ptr<hittable> left;
shared_ptr<hittable> right;
aabb bbox;
};
#endif
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [bvh]: <kbd>[bvh.h]</kbd> Bounding volume hierarchy]
</div>
Splitting BVH Volumes
----------------------
The most complicated part of any efficiency structure, including the BVH, is building it. We do this
in the constructor. A cool thing about BVHs is that as long as the list of objects in a `bvh_node`
gets divided into two sub-lists, the hit function will work. It will work best if the division is
done well, so that the two children have smaller bounding boxes than their parent’s bounding box,
but that is for speed not correctness. I’ll choose the middle ground, and at each node split the
list along one axis. I’ll go for simplicity:
1. randomly choose an axis
2. sort the primitives (`using std::sort`)
3. put half in each subtree
When the list coming in is two elements, I put one in each subtree and end the recursion. The
traversal algorithm should be smooth and not have to check for null pointers, so if I just have one
element I duplicate it in each subtree. Checking explicitly for three elements and just following
one recursion would probably help a little, but I figure the whole method will get optimized later.
The following code uses three methods--`box_x_compare`, `box_y_compare`, and `box_z_compare`--that
we haven't yet defined.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
#include "aabb.h"
#include "hittable.h"
#include "hittable_list.h"
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
#include <algorithm>
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
class bvh_node : public hittable {
public:
...
bvh_node(std::vector<shared_ptr<hittable>>& objects, size_t start, size_t end) {
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
int axis = random_int(0,2);
auto comparator = (axis == 0) ? box_x_compare
: (axis == 1) ? box_y_compare
: box_z_compare;
size_t object_span = end - start;
if (object_span == 1) {
left = right = objects[start];
} else if (object_span == 2) {
left = objects[start];
right = objects[start+1];
} else {
std::sort(std::begin(objects) + start, std::begin(objects) + end, comparator);
auto mid = start + object_span/2;
left = make_shared<bvh_node>(objects, start, mid);
right = make_shared<bvh_node>(objects, mid, end);
}
bbox = aabb(left->bounding_box(), right->bounding_box());
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
}
...
};
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[Listing [bvh-node]: <kbd>[bvh.h]</kbd> Bounding volume hierarchy node]
<div class='together'>
This uses a new function: `random_int()`:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
...
inline double random_double(double min, double max) {
// Returns a random real in [min,max).
return min + (max-min)*random_double();
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
inline int random_int(int min, int max) {
// Returns a random integer in [min,max].
return int(random_double(min, max+1));
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
