CS488 - Introduction to Computer Graphics - Lecture 21

Comments and Questions

Review

Shading

Ray Tracing

The idea is simple:

for each pixel
  calculate the eye/pixel ray
  project the ray into the scene
  find the first intersection
  apply illumination and shading calculations
  colour the pixel with the result

Eye-pixel ray

E + t*(P - E)

Object in scene

Set of points {Q} satisfying

f(Q) = 0
This is the surface of an object. (How do you know?)
Why don't we care about inside points?

Solve the equation

f(E + t*(P-E)) = 0 for t.
For example, all points, Q, on the surface of a sphere, centred at C, of radius r, satisfies |Q-C| = r
Thus, we solve | E - C + t*(P-E) | = r
- which amounts to (E - C + t*(P-E)).(E - C + t*(P-E)) = r^2
- This is the quadratic equation
  - t^2 | P-E |^2 + 2t (E-C).(P-E) + | E-C |^2 - r^2 = 0
  which has zero, one or two solutions.
Most of the time there is no solution

The surface might be defined by an algorithm, then use a root-finding method

For example, float f(E + t*(P - E)) returns distance of point from surface
Submit the function to your favourite root-finder

Object is planar, e.g. a triangle

Equation of the plane is \sum_i ai*xi = b. Where do the parameters come from?

Normal vector is n = (a0, a1, a2, 0)
Take the cross product of any two edges n = (Vn - Vn+1) x (Vm - Vm+1). Now we know the ai.
Substitute and vertex Vn = (v0, v1, v2)
b = \sum ai*vi

Substitute eye ray into the plane equation and solve for t

This point is on the plane: is it inside the polygon?
Triangle: calculate barycentric coordinates; if < 0 outside.
Convex polygon: do n half-plane tests
Either: determine coordinates of the intersection point(s) in the plane of the polygon
- Clip the point against the polygon edges

Another idea: intersect in object coordinates

In object coordinates each point is given by the object definition.
In world coordinates each point in object coordinates is Q = MQ', where Q' is the point in object coordinates
- M is an affine transformation, therefore invertible
  f'(Q') is the object definition in object coordinates
Use invM to transform E,P to E',P', then solve the equation for t
Find intersection point in world coordinates as E + t*(P-E)

Shading for Ray Tracing

When we have found the intersection point we need to calculate a colour.

We know

The eye ray, V
Surface properties
The normal vector
- from a planar primitive
- by interpolation from vertex normals
- by calculation from equation of surface