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Splines
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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