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1.3.6 Bézier surfaces

A tensor product surface patch is
formed by moving a curve through space while allowing deformations in
that curve. This can be thought of as allowing each control point
to sweep a curve in space. If this surface is represented
using Bernstein polynomials, a Bézier surface patch
is formed, with the following formula:

(1.56)

Here, the set of straight lines drawn between consecutive control
points
is referred to as the control net.
It is easy to see that boundary iso-parametric curves (
,
,
and
) have the same control points as the corresponding
boundary points on the net. An example of a bi-quadratic Bézier
surface with its control net can be seen in
Fig. 1.8. Since a Bézier surface is a direct
extension of univariate Bézier curve to its bivariate form, it
inherits many of the properties of the Bézier curve described in
Sect. 1.3.4 such as:

Geometry invariance property.

End points geometric property.

Convex hull property.

However, no variation diminishing property is known for Bézier
surface patches.

The surface patches treated in this book are mostly topologically
quadrilateral. However we sometimes need to use topologically
triangular patches. In such cases, we may collapse one boundary curve
of a quadrilateral patch into a single point to form a three-sided
patch as shown in Fig. 1.9. Such a triangular patch
is said to be degenerate
[116,92]. Alternatively one
could arrange for two partial derivatives
and
at one of the corners of a
quadrilateral patch (1.56) to be collinear to create
degenerate patches
[92].
The differential geometry of degenerated patches is studied in
[116,453,457].

Figure 1.8:
A bi-quadratic Bézier surface with control net

Figure 1.9:
Octant of ellipsoid, represented by a degenerate patch

Next: 1.4 B-spline curves and
Up: 1.3 Bézier curves and
Previous: 1.3.5 Algorithms for Bézier
Contents Index
December 2009