CS488 - Introduction to Computer Graphics - Lecture 20

Comments and Questions

Review

Body reflection
Surface reflection

Shading

Terminology

Artist: shading
- Add lighting effects that convey shape
- Masaccio
Psychologist: shading
- `Shape from shading'
Utah-graphics: shading
- interpolate within polygons to eliminate faceting
Renderman: shader
- function applied at a pixel to determine colour once geometry and illumination are known
GPU: shaders
- vertex shader
- geometry shader
- pixel shader

Meta-terminology

What an umbrella does to a person from the rain or the sun

We are doing Utah-graphics

Flat Shading

Shade entire polygon one colour

Which colour?
The one calculated from
- illumination
- normal
- view point
- surface properties
For which pixel
- average colour of all pixels in polygon -- (What?!!?)
- some particular pixel

Gouraud Shading

Calculate at vertices and interpolate linearly

derivative discontinuities at polygon edges
Mach bands

Interpolating within triangles

Barycentric coordinates
- Three vertices: V1, V2, V3
- Inside point: P = a1*V1 + a2*V2 + a3*V3, subject to a1 + a2 + a3 = 1
  - ai = area of triangle opposite Vi / area of entire triangle
  - a1 = (V3 - V2) x (P - V2) / area of entire triangle
For practical interpolation we care about P -> P + v
- a1 -> a1 + (V3 - V2) x v / area of entire triangle
C(P) = a1 * C(V1) + ...
- C(P) is the determinant of a matrix, if you care
The calculations are affine-invariant, so you could
- transform a triangle into a standard triangle with an affine transformation
- shade it
- transform it back
BUT, the calculations are NOT projection-invariant
- Why?
- Graphics hardware can adjust so that you can shade in display space

Interpolating within arbitrary polygons is a bad idea

Divide into triangles
But if you must, the algorithm is similar to scan-converting an arbitrary polygon.

Phong Shading

Interpolate the vertex normals, light vectors, and view vectors

Do the lighting calculation at each pixel

Highlights are more precise

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
Most of the time there is no solution

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

Object is planar, e.g. a triangle

Equation of the plane is \sum_i ai*xi = b