babternatives to babametric burbaces

| 14-24: advanced shading stuff | 30-32: mesh and widgets (?) (arcball, mover) | 33-36: bezier curves | 42-45: bezier patches | 55-61: alternative to parametric surfaces

alternatives to parametric surfaces (3)

subdivision surfaces, implicit surfaces, particle systems

subdivision surfaces def +

• who came up with them? (2)

• how do they work? (3 steps)

• what do the surfaces become?

blend via recursive smoothing of a polygonal mesh

1. ed catmull, jim clark (silicon graphics guy)

2. e.g. for a rectangle defined by 4 points:

   1. add midpoints on each edge

   2. move each point halfway toward its cw neighbor

   3. repeat

3. surfaces become more smooth with each iteration

implicit surfaces def +

• compare to what?

• how are they different?

• what is the implicit equation for a circle/ what does it mean?

blend via summation of primitives

1. parametric, where you have a single alpha and p = cos alpha, sin alpha

2. parametric: give me parameters, i give you the point
   implicit: give me a test so i can check if a point is on it

3. {p } : px^2 + py^2 − 1 = 0 → set of all p that fit this equation, exact same circle as parametric

   1. if = 0, on circle. positive = outside circle, negative = inside circle

use of implicit surfaces

1. allows you to blend!! see blinn’s blobbies, merging two sphere together. way easier than trying to do it parametrically

2. blue slime blob bob

in implicit surfaces, how do you know where there is an intersection?

if one point is positive and another is negative, you know there MUST be a boundary there somewhere, need to find where it equals 0, repeatedly subdivide and keep testing until you get 0

particle systems def

individual points acting independently, individually displayed