1. 09/18/02 Dinesh Manocha, COMP258 Parametric Patches Tensor product or rectangular patches are of the form: P(u,w) = u,w [0,1]. The number of control points is (m+1)(n+1) Triangular patches have triangular domain. They are of the form: P(r,s,t) = r,s,t 0 It has (n+1)(n+2)/2 control points

2. 09/18/02 Dinesh Manocha, COMP258 Trimmed Patches Arise in applications involving surface intersections, visibility (silhouettes), illumination etc. The domain is irregular Boundary or trimming curves are used to delimit a subset of points on the patch In most applications, trimming curves correspond to high degree algebraic curves –Evaluate points on these curves using numerical methods –Fit spline curve(s) to these points Trimmed domain is represented using piecewise spline curves Point Classification:Check whether a point is in the trimmed domain, compute number of intersections with a line

3. 09/18/02 Dinesh Manocha, COMP258 Hermite Patches A bicubic Hermite patch is given as: P( u,w ) =, where u,w [0,1] In matrix form it is given as P( u,w ) = U A W T, where U = [u 3 u 2 u 1], W = [w 3 w 2 w 1] & A = [ ], A is a 4 X 4 X 3 matrix, 0 i 3, 0 j 3, It has 48 algebraic coefficients

4. 09/18/02 Dinesh Manocha, COMP258 Bicubic Hermite Patches A bicubic Hermite patch is specified using: –4 corner points: P 00, P 01, P 10, P 11 –4 boundary curves: P u0, P u1, P 0w, P 1w (each is a cubic curve) Use Hermite interpolation to specify the boundary curves: P u0 = F[ P 00 P 10 P u 00 P u 10 ] T P u1 = F[ P 01 P 11 P u 01 P u 11 ] T P 0w = F[ P 00 P 01 P w 00 P w 01 ] T P 1w = F[ P 00 P 11 P w 10 P w 11 ] T

5. 09/18/02 Dinesh Manocha, COMP258 Bicubic Hermite Patches Boundary curve constraints: 12 of the 16 vectors needed to specify the geometric coefficients Other 4 vectors are specified using twist vectors at each corner point as: at u = 0, w = 0 at u = 1, w = 0 and similarly These twist vectors determine how the tangent vectors change along the boundary curves

6. 09/18/02 Dinesh Manocha, COMP258 Bicubic Hermite Patches Given the boundary conditions and control points, the patch is given as:, where, are the Hermite basis functions, and P 00 P 01 P 00 w P 01 w B = P 10 P 11 P 10 w P 11 w P 00 u P 01 u P 00 uw P 01 uw P 10 u P 11 u P 10 uw P 11 uw

7. 09/18/02 Dinesh Manocha, COMP258 Hermite Patches Given the boundary conditions and control points, the patch is given as:, or it can be given in tensor product representation as:

8. 09/18/02 Dinesh Manocha, COMP258 Composite Hermite Surfaces Given as a collection of individual patches Continuity: Given two patches: P(u,w) & Q(u,w) –C 0 or G 0 continuity: Means same boundary curves: P(1,w) = Q(0,w) – G 1 continuity: The coefficients of auxiliary curves used to define tangent vectors must be scalar multiples, i.e.: If these conditions are satisfied, we find that