Notes for February 11 class -- Introduction to Ray Tracing

 WebGL Cheat Sheet Here is a compact guide to WebGL The part that is relevant to GLSL for fragment shaders is on the last page. Note that GLSL fragment shaders do not allow recursion.

 Gamma correction Displays are adjusted for human perception using a "gamma curve", since people can perceive a vary large range of brightness. For example, the image to the right shows the values 0...255 on the horizontal axis, the resulting actual displayed brightness on the virtual axis. Various displays differ, but this adjustment is generally approximately x2 We need to do all our math in linear brightness, and then do a gamma correction. Roughly: c → sqrt(c) Output brightness Input values 0 ... 255

 Ray tracing: Forming a ray At each pixel, a ray from the origin V = (0,0,0) shoots into the scene, hitting a virtual screen which is on the z = -f plane. We refer to f as the "focal length" of this virtual camera. The ray toward any pixel aims at: (x, y, -f), where -1 ≤ x ≤ +1 and -1 ≤ y ≤ +1. So the ray direction W = normalize( vec3(x, y, -f) ). We need to see what gets hit along the ray: V + Wt, where t > 0

 Ray tracing: Defining a sphere                                                                                                                                                                                                 We can describe a sphere in a GLSL vector of length 4: `vec4 sphere = vec4(x,y,z,r);` where (x,y,z) is the center of the sphere, and r is the sphere radius. As we discussed in class, the components of a vec4 in GLSL can be accessed in one of two ways: ```v.x, v.y, v.z, v.w // when thought of as a 4D point v.r, v.g, v.b, v.a // when thought of as a color + alpha ``` So to access the value of your sphere's radius, you will want to refer to it as sphere.w (not sphere.r).

 Ray tracing: Finding where a ray hits a sphere ((V-s.xyz) + Wt)2 = s.r2 (Wt + (V-s.xyz))2 - s.r2 = 0 Generally, if a and b are vectors, then a • b = ( ax * bx + ay * by + az * bz ) This "inner product" also equals: |a| * |b| * cos(θ) Multiplying the terms out, we need to solve the following quadratic equation: (W • W) t2 + 2 (W • (V-s.xyz)) t + ((V-s.xyz) • (V-s.xyz)) - r2 = 0 Since W is unit length, the first term in this equation, W • W, is just 1.

 Ray tracing: Finding the nearest intersection Compute the intersection to all spheres in the scene. The visible sphere is the one with smallest positive t, if any.

 Ray tracing: Computing the surface point Once we know the value of t, we can just plug it into the ray equation to get the location of the surface point on the sphere that is visible at this ray, as shown in the equation below and in the figure to the right: S = V + W t

 Ray tracing: Computing the surface normal We now need to compute the surface normal in order to compute how the sphere interacts with light to produce a final color at this pixel. The "surface normal" is the unit length vector that is perpendicular to the surface of the sphere -- the "up" direction if you are standing on the surface. For a sphere, we can get this by subtracting the center of the sphere from the surface point S, and then dividing by the radius of the sphere: N = (S - s.xyz) / s.r

 Ray tracing: A simple shader A simple light source at infinity can be defined as an rgb color, together with an xyz light direction: ``` vec3 lightColor; // rgb vec3 lightDirection; // xyz ``` A diffuse simple surface material can be defined by an rgb "ambient" component, which is independent of any particular light source, and an rgb "diffuse" component, which is added for each light source: ``` vec3 ambient; // rgb vec3 diffuse; // rgb ``` The figure to the right shows a diffuse sphere, where the point at each pixel is computed as: ambient + lightColor * diffuse * max(0, normal • lightDirection)

 Ray tracing: Multiple light sources If the scene contains more than one light source, then pixel color for points on the sphere can be computed by: ambient + ∑n ( lightColorn * diffuse * max(0, normal • lightDirectionn) ) where n iterates over the lights in the scene.

 Homework Implement simple ray tracer for spheres Implement simple diffuse shading Feel free to start with this code. Extra credit: multiple spheres Hint: You can create and use an array of spheres as: ``` vec4 spheres[3]; // Note that I am explicitly setting the size of the array. ... for (int i = 0 ; i < 3 ; i++) { // and also the size of the loop. ... spheres[i] ... } ``` If you try this extra credit, make sure that at each pixel you choose the sphere with the smallest value of t. Extra credit: multiple light sources Extra credit: procedural textures Make something cool and fun, try to create something interactive (using uCursor) and/or animated (using uTime).