   Next: Barycentric Coordinates Up: Splines Previous: Bézier Splines

# Tensor Product Patches

Tensor Product Patches:
• The control polygon is the polygonal mesh with vertices • The patch basis functions are products of curve basis functions where Properties:
• Patch basis functions sum to one • Patch basis functions are nonnegative on   Surface patch is in the convex hull of the control points Surface patch is affinely invariant
(Transform the patch by transforming the control points)

Subdivision, Recursion, Evaluation:
• As for curves in each variable separately and independently
• Tangent plane is not produced!
• Normals must be computed from partial derivatives.

Partial Derivatives:
Ordinary derivative in each variable separately:  Each of the above is a tangent vector in a parametric direction
Surface is regular at each (s,t) where these two vectors are linearly independent
The (unnormalized) surface normal is given at any regular point by (the sign dictates what is the outward pointing normal)

In particular, the cross-boundary tangent is given by
(e.g. for the s=0 boundary): (and similarly for the other boundaries)

Smoothly Joined Patches:
Can be achieved by ensuring that (and correspondingly for the other boundaries)

Rendering:
• Divide up into polygons:
A.
By stepping and joining up sides and diagonals to produce a triangular mesh
B.
By subdividing and rendering the control polygon

CS488/688: Introduction to Interactive Computer Graphics
University of Waterloo
Computer Graphics Lab

cs488@cgl.uwaterloo.ca