Parametric Patches

1
Parametric Patches

• Tensor product or rectangular patches are of the
  form
• P(u,w) u,w 0,1.
• The number of control points is (m1)(n1)
• Triangular patches have triangular domain. They
  are of the form
• P(r,s,t) r,s,t 0
• It has (n1)(n2)/2 control points

2
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 ClassificationCheck whether a point is in
  the trimmed domain, compute number of
  intersections with a line

3
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 u3 u2 u 1, W w3 w2
  w 1
• A , A is a 4 X 4 X 3 matrix, 0
  i 3, 0 j 3,
• It has 48 algebraic coefficients

4
Bicubic Hermite Patches

• A bicubic Hermite patch is specified using
• 4 corner points P00 , P01 , P10 , P11
• 4 boundary curves Pu0 , Pu1 , P0w , P1w (each
  is a cubic curve)
• Use Hermite interpolation to specify the boundary
  curves
• Pu0 FP00 P10 Pu00 Pu10 T
• Pu1 FP01 P11 Pu01 Pu11 T
• P0w FP00 P01 Pw00 Pw01 T
• P1w FP00 P11 Pw10 Pw11 T

5
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
Bicubic Hermite Patches

• Given the boundary conditions and control
  points, the patch is given as ,
• where
• ,
• are the Hermite basis functions,
• and
• P00 P01 P00w P01w
• B P10 P11 P10w P11w
• P00u P01u P00uw P01uw
• P10u P11u P10uw P11uw

7
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
Composite Hermite Surfaces

• Given as a collection of individual patches
• Continuity Given two patches P(u,w) Q(u,w)
• C0 or G0 continuity Means same boundary curves
• P(1,w) Q(0,w)
• G1 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