bebier babtches

| 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

how to make bezier patch?

• how many control points

• what are the two curve called and what are they made of

• how do you find an arbitrary point (u, v) in the patch?

refer to image for #3

1. 16

2. t curve made of s-points, s-curve made of t-points

3. to find e.g. a point (u, v) go up each of the s curves (red lines) until you get the point where s = u. Using these four points (red points), create the t-curve. Find where t = v and yay you have the point you want

given a point u, v with parameters s, t on a bezier patch, how can you find the surface normal? (N(s,t) = ?)

how exact is this answer?

find the tangents of the s curve and the t curve and cross product them together

it is super precise!! it is accurate!! NOT an approximation!!

what key concept does the utah teapot demonstrate??

bezier curves/patches!! can see the control mesh is all straight lines, but turns into a curved shape!

how to split a bezier patch? what happens to the control points?

subdivided into two sub-patches along s-direction (top to bottom down the middle), then split those two patches in the t-direction (left to right horizontally). the result is 1 patch has been split into 4 subpatches

old control points thrown out, new control points have to be calculated

continuity in bezier patches?

same as for curves:

bezier patch boundaries lined up → c0 continuity, position only

tangents/slope line up, control points before and after boundary form a stright line with the boundary points → c1 continuity