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On-Line Geometric Modeling Notes
BÉZIER PATCH SUBDIVISION


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..

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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

$\displaystyle {\bf P} (u,v) \: = \: \sum_{j=0}^{m}
\left[ \sum_{i=0}^n {\bf P} _{i,j}
B_{i,n}(u) \right] B_{j,m}(v)
$

We notice that portion of the equation inside the brackets is the representation of a Bézier curve. If we let $ {\bf Q} _j(u)$ be the value inside the brackets, i.e.

$\displaystyle {\bf Q} _j(u) \: = \: \sum_{i=0}^n {\bf P} _{i,j} B_{i,n}(u)
$

Then

$\displaystyle {\bf P} (u,v) \: = \: \sum_{j=0}^{m} {\bf Q} _j(u) B_{j,m}(v)
$

That is, the quantities $ {\bf Q} _j(u)$ 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 $ {\bf P} _{i,j}$ matrix, it implies that the $ {\bf Q} _j$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 $ 4 \times 4$ case.

\includegraphics {figures/bezier-patch-6}

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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Ken Joy
2000-11-28