Parametric Patches Tensor product or rectangular patches are

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Parametric Patches • Tensor product or rectangular patches are of the form: P(u, w)

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 It has (n+1)(n+2)/2 control points 09/18/02 Dinesh Manocha, COMP 258 0

Trimmed Patches • Arise in applications involving surface intersections, visibility (silhouettes), illumination etc. •

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 09/18/02 Dinesh Manocha, COMP 258

Hermite Patches • A bicubic Hermite patch is given as: P(u, w) = ,

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 WT, where U = [u 3 A= [ u 2 u 1], W = [w 3 ], A is a 4 X 3 matrix, 0 It has 48 algebraic coefficients 09/18/02 Dinesh Manocha, COMP 258 i w 2 w 1] & 3, 0 j 3,

Bicubic Hermite Patches • A bicubic Hermite patch is specified using: – 4 corner

Bicubic Hermite Patches • A bicubic Hermite patch is specified using: – 4 corner points: P 00 , P 01 , P 10 , P 11 – 4 boundary curves: Pu 0 , Pu 1 , P 0 w , P 1 w (each is a cubic curve) • Use Hermite interpolation to specify the boundary curves: Pu 0 = F[P 00 Pu 1 = P 10 Pu 00 Pu 10 ]T F[P 01 P 11 P 0 w = F[P 00 P 01 P 1 w = F[P 00 P 11 09/18/02 Pu 01 Pu 11 ]T Pw 00 Pw 01 ]T Pw 10 Pw 11 ]T Dinesh Manocha, COMP 258

Bicubic Hermite Patches • Boundary curve constraints: 12 of the 16 vectors needed to

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 09/18/02 Dinesh Manocha, COMP 258

Bicubic Hermite Patches • Given the boundary conditions and control points, the patch is

Bicubic Hermite Patches • Given the boundary conditions and control points, the patch is given as: , where , are the Hermite basis functions, and B 09/18/02 = P 00 P 01 P 00 w P 01 w P 10 P 11 P 10 w P 11 w P 00 u P 10 u P 01 u P 11 u P 00 uw P 10 uw P 01 uw P 11 uw Dinesh Manocha, COMP 258

Hermite Patches • Given the boundary conditions and control points, the patch is given

Hermite Patches • Given the boundary conditions and control points, the patch is given as: , or it can be given in tensor product representation as: 09/18/02 Dinesh Manocha, COMP 258

Composite Hermite Surfaces Given as a collection of individual patches • Continuity: Given two

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 09/18/02 Dinesh Manocha, COMP 258