# Parametric Bernstein/Bezier Curves and Tensor Product Surfaces

### http://www2.eecs.berkeley.edu/Pubs/TechRpts/1990/CSD-90-571.pdf

This tutorial describes parametric Bernstein/Bezier curves and parametric tensor-product Bernstein/Bezier surfaces. The parametric representation is described, the Bezier curve representation is explained, and the mathematics are presented. The key properties of Bezier curves are discussed.

The single Bezier curve is extended to a composite Bezier curve using parametric continuity. Then the more general geometric continuity is defined, first for order two (G^2), and then for arbitrary order n (G^n). Composite Bezier curves are stitched together with G^1 and G^2 continuity using constraints on the control vertices and using geometric constructions.

The subdivision of Bezier curves is then derived along with a discussion of the associated geometric construction, the deCasseljau Algorithm, and flatness testing.

Then, the Bezier curve is generalized to a tensor-product surface. Finally, the rational Bezier curve and rational tensor-product surface are discussed.

BibTeX citation:

```@techreport{Barsky:CSD-90-571,
Author = {Barsky, Brian A.},
Title = {Parametric Bernstein/Bezier Curves and Tensor Product Surfaces},
Institution = {EECS Department, University of California, Berkeley},
Year = {1990},
Month = {May},
URL = {http://www2.eecs.berkeley.edu/Pubs/TechRpts/1990/5232.html},
Number = {UCB/CSD-90-571},
Abstract = {This tutorial describes parametric Bernstein/Bezier curves and parametric tensor-product Bernstein/Bezier surfaces. The parametric representation is described, the Bezier curve representation is explained, and the mathematics are presented. The key properties of Bezier curves are discussed. <p>The single Bezier curve is extended to a composite Bezier curve using parametric continuity. Then the more general geometric continuity is defined, first for order two (<i>G</i>^2), and then for arbitrary order <i>n</i> (G^<i>n</i>). Composite Bezier curves are stitched together with <i>G</i>^1 and <i>G</i>^2 continuity using constraints on the control vertices and using geometric constructions. <p>The subdivision of Bezier curves is then derived along with a discussion of the associated geometric construction, the deCasseljau Algorithm, and flatness testing. <p>Then, the Bezier curve is generalized to a tensor-product surface. Finally, the rational Bezier curve and rational tensor-product surface are discussed.}
}
```

EndNote citation:

```%0 Report
%A Barsky, Brian A.
%T Parametric Bernstein/Bezier Curves and Tensor Product Surfaces
%I EECS Department, University of California, Berkeley
%D 1990
%@ UCB/CSD-90-571
%U http://www2.eecs.berkeley.edu/Pubs/TechRpts/1990/5232.html
%F Barsky:CSD-90-571
```