Bézier surface

Bézier patch

A parametric polynomial surface in $ \mathbf R ^ {3} $ that can be expressed in terms of certain Bernstein polynomials defined over a rectangle or triangle. Bézier surfaces (also called Bézier patches) are used in the field of computer-aided geometric design (CAGD); for historical remarks see Bézier spline. There are two types of Bézier surfaces: tensor product Bézier surfaces and triangular Bézier surfaces.

1) Given degrees $ m,n > 0 $ and points $ \mathbf b _ {ij } \in \mathbf R ^ {3} $, $ i = 0 \dots m $ and $ j = 0 \dots n $, the parametric surface $ \mathbf q : {[ 0,1 ] ^ {2} } \rightarrow {\mathbf R ^ {3} } $ defined by

$$ \mathbf q ( u,v ) = \sum _ {i = 0 } ^ { m } \sum _ {j = 0 } ^ { n } {B _ {i} ^ {m} ( u ) B _ {j} ^ {n} ( v ) \cdot \mathbf b _ {ij } } , u,v \in [ 0,1 ] , $$

is called a tensor product Bézier surface, where $ B _ {i} ^ {m} ( u ) $ and $ B _ {j} ^ {n} ( v ) $ are Bernstein polynomials of degree $ m $, respectively $ n $( see also Bézier spline). The points $ \mathbf b _ {ij } $ are called control points or Bézier points. The net of control points is said to be the control net or Bézier net of the surface $ \mathbf q $. Analogously as for Bézier curves (see Bézier curve), tensor product Bézier surfaces having an arbitrary rectangle $ [ o,r ] \times [ s,t ] $ as domain may be introduced.

Many properties of Bézier curves carry over to tensor product Bézier surfaces. Any isoparametric curve $ v = v _ {0} = \textrm{ const } $ of $ \mathbf q $ is a Bézier curve $ \mathbf p ( u ) = \mathbf q ( u,v _ {0} ) = \sum _ {i = 0 } ^ {m} ( \sum _ {j = 0 } ^ {n} B _ {j} ^ {n} ( v _ {0} ) \mathbf b _ {ij } ) \cdot B _ {i} ^ {m} ( u ) $ of degree $ m $. The Bézier points of $ \mathbf p $ can be obtained by applying $ m + 1 $ de Casteljau algorithms using Bézier points $ \mathbf b _ {ij } $, $ j = 0 \dots n $( see Bézier spline). A point $ \mathbf q ( u _ {0} ,v _ {0} ) $ on the surface is then obtained by performing one more de Casteljau algorithm. Furthermore, the surface lies in the convex hull of its Bézier points. The boundary curves of the surface are Bézier curves whose Bézier points are the corresponding boundary points of the Bézier net.

2) Triangular Bézier surfaces have a triangle as domain, therefore they are also called Bézier triangles. Thus, let there be given an arbitrary, non-degenerate triangle with vertices $ \mathbf r, \mathbf s, \mathbf t \in \mathbf R ^ {2} $. Then any point $ \mathbf p \in \mathbf R ^ {2} $ has unique barycentric coordinates $ u,v,w \in \mathbf R $ with respect to $ \mathbf r $, $ \mathbf s $, $ \mathbf t $, i.e., $ \mathbf p = u \mathbf r + v \mathbf s + w \mathbf t $ with $ u + v + w = 1 $. For any $ n > 0 $, the bivariate Bernstein polynomials $ B _ {ijk } ^ {n} $ are defined by

$$ B _ {ijk } ^ {n} ( u,v,w ) = { \frac{n! }{i! j! k! } } u ^ {i} v ^ {j} w ^ {k} $$

for all $ i,j,k \in \mathbf N _ {0} $, $ i + j + k = n $, $ u + v + w = 1 $. Now, given points $ \mathbf b _ {ijk } \in \mathbf R ^ {3} $, a triangular Bézier surface of degree $ n $ is defined by

$$ \mathbf q ( u,v,w ) = \sum _ {i + j + k = n } B _ {ijk } ^ {n} ( u,v,w ) \cdot \mathbf b _ {ijk } , $$

where $ u + v + w = 1 $ and $ u,v,w \geq 0 $; these conditions characterize the barycentric coordinates of any point $ \mathbf p $ of the closed triangle $ \mathbf r, \mathbf s, \mathbf t $.

The notions of control (or Bézier) point and net are defined as above. Triangular Bézier surfaces have similar properties as tensor product Bézier surfaces; in particular, there exists a modified de Casteljau algorithm, the surface is contained in the convex hull of its Bézier points, and the boundary curves of the surface are Bézier curves of degree $ n $ whose Bézier points are the corresponding boundary points of the Bézier net.

In analogy to Bézier curves, it is possible to build up smooth, complex surfaces from a number of rectangular or triangular Bézier patches. This leads to spline surfaces. For rational Bézier surfaces see the references below; higher-dimensional Bézier surfaces are treated in [a4].

References

[a1]	W. Boehm, G. Farin, J. Kahmann, "A survey of curve and surface methods in CAGD" Computer Aided Geometric Design , 1 (1984) pp. 1–60
[a2]	J. Encarnaçao, W. Straßer, R. Klein, "Datenverarbeitung 1. Gerätetechnik, Programmierung und Anwendung graphischer Systeme" , R. Oldenbourg (1996)
[a3]	G. Farin, "Curves and surfaces for computer aided geometric design. A practical guide" , Acad. Press (1993) (Edition: Third)
[a4]	J. Hoschek, D. Lasser, "Grundlagen der geometrischen Datenverarbeitung" , Teubner (1992) (Edition: Second)
[a5]	M.E Mortensen, "Geometric modeling" , Wiley (1985)