On-Line Geometric Modeling Notes
Overview
A general method can be specified to subdivide a Bézier patch. This method is specified unlike the matrix methods, as it is based upon the definition of the patch as a set of curves..
To get a postscript version of these notes look here.
The Method for Subdivision
We recall that, if we take the analytic equation
of a Bézier patch,
fix u and group factors appropriately, we
obtain
We notice that portion of the equation inside the brackets is the
representation of a Bézier curve. If we let 
be the value
inside the brackets, i.e.
Then
That is, the quantities 
form the control points of another
Bézier curve, and together for all u and v, they form the
surface.
If, then, we subdivide each of the m rows of the 
matrix,
it implies that the 
s in the above equation represent only
points from the first half of the patch (with respect to u).
The following illustration shows the result of subdividing the rows in
the 
case.
The second half of the patch can be obtained in a similar fashion. The first and second half of the patch, with respect to v, can be obtained by subdividing the columns.
Summary
So, using only curve methods, and by subdividing the rows or columns of the control point array, we can effectively subdivide a Bézier patch. This is the most frequently used algorithm in software implementations of subdivision and can be utilized for Bézier patches of arbitrary degree.
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This document maintained by
Ken Joy
All contents copyright (c) 1996, 1997 |
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