1.1 Primary and Secondary Rays

A ray is a semi-infinite line; it is typically stored as a point called the ray origin, $\mathbf{r_o}$; and a direction vector $\vec r_d$. Then the ray itself is the set of points $\{\mathbf{r_o} + t \vec r_d \;|\; t \in \mathbb{R}^{+}\}$, and the goal of raytracing is to find the point in the ray on the surface of another object which is closest to the ray origin (that is, with minimal $t$).

Ray tracing creates one or more rays per pixel. Those rays are intersected with objects in the scene and then, generally, several secondary rays are generated from those intersection points, and intersected with the scene again, and then more are generated, etc., until you get tired of shooting rays, at which point you do some sort of direct illumination and call it good.

There are a number of ways to generate rays from pixels. For all of them we need the camera position and orientation, given by the eye position $\mathbf{e}$ and the forward, up, and right directions $\vec f$, $\vec u$, and $\vec r$. The ray origin is generally just $\mathbf{e}$. The most common choice for ray direction is designed to replicate the standard rasterization perspective look: given normalized device coordinates for a pixel $(x,y)$, its ray direction is $\vec f \cos(\theta) + (x \vec r + y \vec u)\sin(\theta)$, where $\theta$ is the field of view. Fish-eye $\left(x \vec r + y \vec u + \sqrt{1-x^2-y^2}\vec f\right)$ and cylindrical $\left(y \vec u + \cos(x) \vec f + \sin(x) \vec r\right)$ projections are also used in some settings.

2.1 Ray-Plane Intersection

Planes can be described in a number of ways; principle among them are implicit equations $A x + B y + C z + D = 0$, point-and-normal form ($\vec n$, $\mathbf{p}$), and three-point form ($\mathbf{p_0}$, $\mathbf{p_1}$, $\mathbf{p_2}$). We will use the point-normal version, noting that $$\vec n \equiv (A,B,C) \equiv (\overrightarrow{\mathbf{p_1}-\mathbf{p_0}}) \times (\overrightarrow{\mathbf{p_2}-\mathbf{p_0}})$$ and that the point $\left(-\frac{D}{A},0,0\right)$ is on the plane.

The distance between the ray origin $\mathbf{r_o}$ and a plane is $(\overrightarrow{\mathbf{r_o}-\mathbf{p}}) \cdot \vec n\frac{1}{\|\vec n\|}$. The distance the ray travels toward the plane per unit $t$ is $\vec r_d \cdot \vec n\frac{-1}{\|\vec n\|}$. Setting these equal to one another we get $t = \frac{(\overrightarrow{\mathbf{p}-\mathbf{r_o}}) \cdot \vec n}{\vec r_d \cdot \vec n}$. If $t$ is positive, we can use that $t$ in the ray equation to find the intersection point $\mathbf{p'} = t \vec r_d + \mathbf{r_o}$. If $t$ is not positive there is no intersection.

2.2 Ray-Sphere Intersection

Given a sphere with center $\mathbf{c}$ and radius $r$, we first evaluate if the ray originates inside the sphere ($\|\overrightarrow{\mathbf{c}-\mathbf{r_o}}\|^2 < r^2$) or not. We then find the $t$ value of the point where the ray comes closest to the center of the sphere, $t_c = \frac{(\overrightarrow{\mathbf{c}-\mathbf{r_o}}) \cdot \vec r_d}{\|\vec r_d\|}$. If the ray origin is outside and $t_c$ is negative, there is no intersection. Otherwise we proceed to find the squared distance of closest approach $d^2 = \|\mathbf{r_o} + t_c \vec r_d - \mathbf{c}\|^2$. If $d^2 > r^2$, which can only happen if the ray originates outside the sphere, then there is no intersection; otherwise we find how far from the point of closest approach the point of intersection is as $t_{\text{offset}} = \frac{\sqrt{r^2 - d^2}}{\|\vec r_d\|}$. If the origin is inside, the point of intersection is $\mathbf{r_o} + (t_c + t_{\text{offset}}) \vec r_d$; otherwise, it is $\mathbf{r_o} + (t_c - t_{\text{offset}}) \vec r_d$.

2.3 Ray-AABB Intersection

It is rare to want to create images of axis-aligned bounding boxes (AABBs), but it is easy to find ray-AABB intersections and easy to find the AABB that contains a set of objects, so most raytracers try AABB intersections for sets of nearby objects before checking for intersections with the objects within the AABB.

AABBs consist of six axis-aligned planes. For the axis-aligned case, the ray-plane intersection becomes quite simple because the normal has only one non-zero element; thus, for the plane, e.g., $x = a$, the $t$ value of intersection is $t_{x=a} = \frac{a - \mathbf{r_o}_x}{\overrightarrow{r_d}_x}$. The ray then intersects the AABB if and only if there is some positive $t$ between all six planes; for minimum point $(a,b,c)$ and maximum point $(A,B,C)$, we have $$[0,\infty) \cap [t_{x=a}, t_{x=A}] \cap [t_{y=b}, t_{y=B}] \cap [t_{z=c}, t_{z=C}] \ne \emptyset.$$ It is usually not important to know the $t$-value for an AABB intersection, but if needed it is simply the smallest $t$ in the interval defined above.

3.1 Inverse Sphere Mapping

For a sphere, if the point of intersection is $\mathbf{p}$ then the normal simply points from the center to the point of intersection, $\vec n = \frac{1}{r}\left(\overrightarrow{\mathbf{p} - \mathbf{c}}\right)$. From that it is easy to derive that the longitude is $\mathtt{atan2}(n_x, n_z)$ and the latitude is $\mathtt{atan2}\left(n_y, \sqrt{n_x^2 + n_z^2}\right)$.

3.2 Inverse Triangle Mapping

For a triangle, the typical inverse mapping gives you the Barycentric coordinates of the point of intersection. Barycentric coordinates are three numbers, one per vertex of the triangle, which state how close to each of the three vertices the point in question is. In particular, given an intersection point of $\mathbf{p}$ and vertices $\mathbf{p_0}$, $\mathbf{p_1}$, and $\mathbf{p_2}$, the barycentric coordinates $(b_0, b_1, b_2)$ satisfy the two properties that

1. they sum to 1, and
2. $\mathbf{p} = b_0\mathbf{p_0} + b_1\mathbf{p_1} + b_2\mathbf{p_2}$.

Every point in the same plane as the triangle has a unique set of Barycentric coordinates, and all three coordinates are positive if and only if the point is within the triangle’s bounds.

There are many techniques for inverse mapping a triangle. The one I present here is not the most common, but is easy and efficient as long as you compute and store the information only once. First, observe that since $b_i$ is the nearness to point $\mathbf{p_i}$, it is also the distance from the edge joining the other two points. This distance can be found directly by using a dot product with a vector perpendicular to this edge.

in that image, $b_1 = \vec e_1 \cdot (\mathbf{p}-\mathbf{p_0})$ because $\vec e_1$ points directly away from the edge between $\mathbf{p_{i\ne1}}$. Similarly, $b_2 = \vec e_2 \cdot (\mathbf{p}-\mathbf{p_0})$ and $b_0 = 1-b_1 - b_2$. It thus suffices to find $\vec e_1$ and $\vec e_2$ in order to find the barycentric coordinates. This may be done as $$\begin{split} \vec a_1 &= \overrightarrow{\mathbf{p_2}-\mathbf{p_0}} \times \vec n\\ \vec a_2 &= \overrightarrow{\mathbf{p_1}-\mathbf{p_0}} \times \vec n\\ \vec e_1 &= \frac{1}{\vec a_1 \cdot \overrightarrow{\mathbf{p_1}-\mathbf{p_0}}} \vec a_1 \\ \vec e_2 &= \frac{1}{\vec a_2 \cdot \overrightarrow{\mathbf{p_2}-\mathbf{p_0}}} \vec a_2 \end{split}$$ These two $\vec e_i$ vectors can be precomputed and stored along with the normal $\vec n$ in each triangle data structure to allow rapid ray-triangle intersections. Note that this also suffices for the inside-outside test needed to turn a ray-plane intersection into a ray-triangle intersection; a point is inside a triangle if and only if all three barycentric coordinates are between zero and one.

Because $\mathbf{p} = b_0\mathbf{p_0} + b_1\mathbf{p_1} + b_2\mathbf{p_2}$, we can use the barycentric coordinates to find all the information stored in each vertex interpolated to any point on the interior of the triangle: $\vec{p} = b_0\vec{p_0} + b_1\vec{p_1} + b_2\vec{p_2}$. You can then use that information just as you would interpolated fragment values during rasterization.

4.1 Shadows

A point is in shadow relative to a particular light source if the ray $(\mathbf p, \vec \ell)$ intersects anything closer than the light source itself.

4.2 Reflection

A mirrored object’s color is given by the ray tracing result of the ray with origin $\mathbf p$ and direction $2(\vec n \cdot \vec e)\vec n - \vec e$. A partially mirrored object mixes that color result with a standard lighting computation at $\mathbf p$. One reflection ray might generate another; a cutoff number of recursions is necessary to prevent infinite loops.

4.3 Transparency

Transparency is somewhat more complicated, relying on Snell’s Law. For it to make sense, every surface needs to be a boundary between two materials, which is not trivially true in the case of triangles nor intersecting objects. However, assuming that we know that the ray is traveling from a material with index of refraction $n_1$ for index of refraction $n_2$, we can derive a rule for finding the transmitted ray.

The cosine of the entering ray is $(\vec e \cdot \vec n)$, meaning that it’s sine is $\sqrt{1-(\vec e \cdot \vec n)^2}$, or $\|-\vec e -(\vec e \cdot \vec n)\vec n\|_2$. The sine of the outgoing vector thus needs to be $\frac{n_1}{n_2}\sqrt{1-(\vec e \cdot \vec n)}$; if that is greater than 1, we have total internal refraction and use the reflection equation instead; otherwise, the cosine of the outgoing ray is $\sqrt{1-(\frac{n_1}{n_2})^2(1-(\vec e \cdot \vec n))}$. Putting this all together, we have

4.4 Global Illumination

The standard lighting models pretend like the world is divided into a small number of light emitters and a vast supply of things that emit no light. This is obviously not true; if nothing but light sources gave off photons, we could only see the light sources themselves. With global illumination, you try to discover the impact of light bouncing off of the wall, floor, and other objects by creating a number of secondary rays sampling the diffuse reflection of each object.
Ideally, the distribution of rays should parallel the chosen model of diffuse lighting (see Section~) and should be so dense as to be intractable for any reasonable scene. Much of the research in photo-realistic rendering is devoted to finding shortcuts and techniques that make this process require fewer rays for the same visual image quality.