next up previous contents index
Next: 1.4 B-spline curves and Up: 1.3 Bézier curves and Previous: 1.3.5 Algorithms for Bézier   Contents   Index


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 $ {\bf b}_i$ to sweep a curve in space. If this surface is represented using Bernstein polynomials, a Bézier surface patch is formed, with the following formula:
$\displaystyle {\bf r}(u,v) = \sum_{i=0}^m\sum_{j=0}^n{\bf b}_{ij}B_{i,m}(u)B_{j,n}(v), \;\;\;\;\;\; 0 \leq u,v \leq 1\;.$     (1.56)

Here, the set of straight lines drawn between consecutive control points $ {\bf b}_{ij}$ is referred to as the control net. It is easy to see that boundary iso-parametric curves ($ u=0$ , $ u=1$ , $ v=0$ and $ v=1$ ) 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:

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 $ {\bf r}_u$ and $ {\bf r}_v$ 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
\begin{figure}\centerline{\psfig{figure=fig/bez_surf.ps,height=2.5in}}\end{figure}

Figure 1.9: Octant of ellipsoid, represented by a degenerate patch
\begin{figure}\vspace*{-15mm} \centerline{ \hspace*{-20mm} \psfig{figure=fig/ell_bez_oct.ps,height=2.5in} \vspace*{7mm}}\end{figure}

next up previous contents index
Next: 1.4 B-spline curves and Up: 1.3 Bézier curves and Previous: 1.3.5 Algorithms for Bézier   Contents   Index
December 2009