Bezier surface

Tensor product Bezier surface and its control network (blue)

In geometry, Bezier surfaces are surfaces that are defined as spatial generalizations of Bezier curves . There are essentially two ways of generalization. This lists: ${\displaystyle \mathbb {R} ^{3}}$

• Tensor product bezier surfaces . Products of Bernstein polynomials are used.
• Triangular bezier surfaces . Bernstein polynomials for barycentric coordinates are introduced.

Bezier surfaces play an essential role in the modeling of freeform surfaces in the areas of computer graphics and computer-aided design .

Tensor product bezier surface

definition

It is a Bezier curve in whose control points of another parameter dependent, and that they should even be on Bezier curves: . So describes ${\displaystyle {\bf {b}}^{m}(u)=\sum _{i=0}^{m}{\bf {b}}_{i}B_{i}^{m}(u)}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle v}$ ${\displaystyle \ {\bf {b}}_{i}(v)=\sum _{j=0}^{n}{\bf {b}}_{ij}B_{j}^{n}(v)}$

• ${\displaystyle {\bf {b}}^{m,n}(u,v)=\sum _{i=0}^{m}\sum _{j=0}^{n}{\bf {b}}_{ij}B_{i}^{m}(u)B_{j}^{n}(v),\quad u,v\in [0,1]}$

an area to the control points or the control network associated (m, n) -Tensorprodukt-Bezierfläche . The surface contains the points and the parameter curves ( or are constant), in particular the boundary curves, are Bezier curves. ${\displaystyle {\bf {b}}_{ij}}$ ${\displaystyle \ {\bf {b}}_{00},\ {\bf {b}}_{m0},\ {\bf {b}}_{0n},\ {\bf {b}}_{mn}\ }$ ${\displaystyle u}$ ${\displaystyle v}$

Note that a -tensor product Bezier surface contains straight lines, but is generally not flat. For example, for ${\displaystyle (1,1)}$

${\displaystyle {\bf {b}}_{00}=(0,0,0)^{T},\ {\bf {b}}_{10}=(1,0,0)^{T},\ {\bf {b}}_{01}=(0,1,0)^{T},\ {\bf {b}}_{11}=(1,1,1)^{T}}$ the area with the parametric representation

${\displaystyle {\bf {x}}={\bf {b}}^{1,1}(u,v)={\bf {b}}_{00}(1-u)(1-v)+{\bf {b}}_{10}u(1-v)+{\bf {b}}_{01}(1-u)v+{\bf {b}}_{11}uv}$

${\displaystyle =\cdots =\left(u,v,uv\right)^{T}}$

This is part of the hyperbolic paraboloid with the equation . ${\displaystyle z=xy}$

The Casteljau algorithm

The basic idea of ​​the Casteljau algorithm for curves is the linear interpolation of pairs of points. If one transfers this idea to tensor product Bezier surfaces, one has to define a {\ bf bi} linear interpolation for four points. As with curves, it can be read in the simplest case: A (1,1) -tensor product Bezier surface on the four points has the following representation: ${\displaystyle {\bf {b}}_{00},{\bf {b}}_{10},{\bf {b}}_{01},{\bf {b}}_{11}}$

${\displaystyle {\bf {b}}^{1,1}(u,v)=(1-u)(1-v){\bf {b}}_{00}+u(1-v){\bf {b}}_{10}+(1-u)v{\bf {b}}_{01}+uv{\bf {b}}_{11}}$

Or in matrix form:

${\displaystyle {\bf {b}}^{1,1}(u,v)=(1-u,u)\left({\begin{array}{ll}{\bf {b}}_{00}&{\bf {b}}_{01}\\{\bf {b}}_{10}&{\bf {b}}_{11}\end{array}}\right){1-v \choose v}}$

One starts with a control network and determines (as in the case of curves) for and a pair of parameters intermediate vectors that arise through bilinear interpolation: ${\displaystyle (n\times n)}$ ${\displaystyle r=1,2,..,n}$ ${\displaystyle (u,v)}$

${\displaystyle {\bf {b}}_{i,j}^{r}=(1-u,u)\left({\begin{array}{ll}{\bf {b}}_{i,j}^{r-1}&{\bf {b}}_{i,j+1}^{r-1}\\{\bf {b}}_{i+1,j}^{r-1}&{\bf {b}}_{i+1,j+1}^{r-1}\end{array}}\right){1-v \choose v},}$

where is. Then be the point that is assigned to the parameter pair. ${\displaystyle {\bf {b}}_{i,j}^{0}:={\bf {b}}_{i,j}}$ ${\displaystyle {\bf {b}}_{0,0}^{n}}$ ${\displaystyle (u,v)}$

If is, the second index is constant from on and only linear interpolation is carried out (as with Bezier curves). ${\displaystyle m>n}$ ${\displaystyle r=n}$ ${\displaystyle j=0}$

• The point is then the area point. ${\displaystyle {\bf {b}}_{0,0}^{m}}$

The procedure is analogous if is. ${\displaystyle m<n}$

Grade increase

It is often advantageous if there is a -tensor product Bezier surface . If this is not the case, this can be achieved with the help of suitable grade elevations. ${\displaystyle (m,n)}$ ${\displaystyle m=n}$

The degree increase of on the tensor product bezier surface ${\displaystyle (m,n)}$ ${\displaystyle (m+1,n)}$

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

leads to the increases in degrees for the Bezier curves in square brackets: ${\displaystyle n+1}$

${\displaystyle \sum _{i=0}^{m}{\bf {b}}_{ij}B_{i}^{m}(u)=\sum _{i=0}^{m+1}{\bf {b}}_{ij}^{(1,0)}B_{i}^{m+1}(u),\quad j=0,...n}$

With

${\displaystyle {\bf {b}}_{ij}^{(1,0)}:=(1-{\frac {i}{m+1}}){\bf {b}}_{i,j}+{\frac {i}{m+1}}{\bf {b}}_{i-1,j},\quad i=0,...,m+1.}$

Derivations of a Bezier surface

The partial derivative of the tensor product Bezier surface

${\displaystyle {\bf {b}}^{m,n}(u,v)=\sum _{j=0}^{n}\sum _{i=0}^{m}{\bf {b}}_{ij}B_{i}^{m}(u)B_{i}^{n}(v)}$

after is ${\displaystyle u}$

${\displaystyle {\frac {\partial }{\partial u}}{\bf {b}}^{m,n}(u,v)=\sum _{j=0}^{n}\left[{\frac {\partial }{\partial u}}\sum _{i=0}^{m}{\bf {b}}_{ij}B_{i}^{m}(u)\right]B_{i}^{n}(v).}$

With the result for the derivation of a Bezier curve we get:

• ${\displaystyle {\frac {\partial }{\partial u}}{\bf {b}}^{m,n}(u,v)=m\sum _{j=0}^{n}\left[\sum _{i=0}^{m-1}\Delta ^{1,0}{\bf {b}}_{ij}B_{i}^{m-1}(u)\right]B_{i}^{n}(v),}$

whereby . The partial derivative with respect to and all higher derivatives are obtained analogously . ${\displaystyle \Delta ^{1,0}{\bf {b}}_{i,j}:={\bf {b}}_{i+1,j}-{\bf {b}}_{i,j}}$ ${\displaystyle v}$

Since the vectors are tangent vectors of the edge curves beginning at the point , is ${\displaystyle \Delta ^{1,0}{\bf {b}}_{0,0},\Delta ^{0,1}{\bf {b}}_{0,0}}$ ${\displaystyle {\bf {b}}_{0,0}}$

• ${\displaystyle \Delta ^{1,0}{\bf {b}}_{0,0}\times \Delta ^{0,1}{\bf {b}}_{0,0}}$

a normal vector of the surface at this point if both are linearly independent. Ie the tangential plane in the corner points of a tensor product Bezier surface is generally spanned by the corner point and its neighboring points in the control network.

Triangular bezier surfaces

Motivation and Definition

A formal generalization of the Bernstein polynomials to functions with two variables would start from the relationship . So that the terms that occur are all positive, they must lie in the triangle . Two of the three sides of the triangle play a special role as intervals on the coordinate axes. To avoid this preference one introduces homogeneous coordinates with the condition . are called barycentric coordinates . The generalized Bernstein polynomials result from the expansion of to: ${\displaystyle 1=(u+v+(1-u-v))^{n}=\cdots }$ ${\displaystyle (u,v)}$ ${\displaystyle (0,0),(1,0),(0,1)}$ ${\displaystyle u,v,w}$ ${\displaystyle u+v+w=1,u,v,w\geq 0}$ ${\displaystyle u,v,w}$ ${\displaystyle (u+v+w)^{n}}$

• ${\displaystyle B_{ijk}^{n}(u,v,w):={\frac {n!}{i!j!k!}}u^{i}v^{j}w^{k}}$

with and . ${\displaystyle i+j+k=n,\ i,j,k\geq 0}$ ${\displaystyle u+v+w=1,\ u,v,w\geq 0}$

Control points of a triangle bezier surface

With the abbreviations and is ${\displaystyle {\bf {I}}:=(i,j,k),\ {\bf {u}}:=(u,v,w)}$ ${\displaystyle |{\bf {I}}|:=i+j+k,\ |{\bf {u}}|:=u+v+w}$

${\displaystyle B_{\bf {I}}^{n}({\bf {u}}):={\frac {n!}{i!j!k!}}u^{i}v^{j}w^{k},\quad |{\bf {u}}|=1\quad {\text{und}}\quad \sum _{|{\bf {I}}|=n}B_{\bf {I}}^{n}=1.}$

Is now

${\displaystyle {\bf {b}}_{00n},{\bf {b}}_{10,n-1},...{\bf {b}}_{n00},{\bf {b}}_{01,n-1},{\bf {b}}_{11,n-2},...,{\bf {b}}_{n-1,10},...,{\bf {b}}_{0n0}}$

a triangular network of points of the , the control points , so is ${\displaystyle \mathbb {R} ^{3}}$

• ${\displaystyle {\bf {b}}^{n}({\bf {u}}):=\sum _{|{\bf {I}}|=n}{\bf {b}}_{\bf {I}}B_{\bf {I}}^{n}({\bf {u}})}$

the corresponding triangular Bezier surface .

The figure shows the arrangement of the points for the case . ${\displaystyle n=4}$

De Casteljau algorithm

In order to be able to clearly formulate the Casteljau algorithm for triangular Bezier surfaces, the following abbreviations are introduced:

${\displaystyle {\bf {e}}_{1}:=(1,0,0),\;{\bf {e}}_{2}:=(0,1,0),\;{\bf {e}}_{3}:=(0,0,1)\ }$and . ${\displaystyle \;{\bf {o}}:=(0,0,0)}$

Let us now be a triangular network of points in and a parameter vector in barycentric coordinates. Then be for and ${\displaystyle \{{\bf {b}}_{\bf {I}}||{\bf {I}}|=n\}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle {\bf {u}}}$ ${\displaystyle r=1,...,n}$ ${\displaystyle {\bf {I}}=n-r}$

${\displaystyle {\bf {b}}_{\bf {I}}^{r}:=u{\bf {b}}_{{\bf {I}}+{\bf {e}}_{1}}^{r-1}({\bf {u}})+v{\bf {b}}_{{\bf {I}}+{\bf {e}}_{2}}^{r-1}({\bf {u}})+w{\bf {b}}_{{\bf {I}}+{\bf {e}}_{3}}^{r-1}({\bf {u}})}$

with then is ${\displaystyle {\bf {b}}_{\bf {I}}^{0}({\bf {u}}):={\bf {b}}_{\bf {I}}.}$

• ${\displaystyle {\bf {b}}_{\bf {o}}^{n}({\bf {u}})}$ a point of the triangular bezier surface.

The proof that the Casteljau algorithm really delivers a point of the triangular Bezier surface uses the recursion formulas for Bernstein polynomials (analogous to the curve case):

• ${\displaystyle B_{\bf {I}}^{n}({\bf {u}})=uB_{{\bf {I}}-{\bf {e}}_{1}}^{n-1}({\bf {u}})+vB_{{\bf {I}}-{\bf {e}}_{2}}^{n-1}({\bf {u}})+wB_{{\bf {I}}-{\bf {e}}_{3}}^{n-1}({\bf {u}}),\quad |{\bf {I}}|=n.}$

Please refer to the literature for further details.

Individual evidence

1. ^ Farin: Curves and Surfaces for CAGD
2. ↑ Hoschek & Lasser: Fundamentals of geometric data processing
3. ↑ Farin p. 254
4. ↑ Farin p. 310
5. ↑ Farin p. 307
6. ↑ Farin p. 306

literature

• Gerald Farin: Curves and Surfaces for CAGD. A practical guide. 5th Ed. Academic Press, San Diego 2002, ISBN 1-55860-737-4
• J. Hoschek, D. Lasser: Fundamentals of geometric data processing , Vieweg + Teubner Verlag, 1989, ISBN 978-3-519-02962-5
• David Salomon: Curves and Surfaces for Computer Graphics . Springer Science + Business Media, Inc., 2006, ISBN 0-387-24196-5
• Boaswan Dzung Wong: Bézier curves: drawn and calculated . Orell Füssli Verlag, Zurich 2003, ISBN 3-280-04021-3
• Wolfgang Boehm, Gerald Farin, Jürgen Kahmann: A survey of curve and surface methods in CAGD , Comput. Aided Geom. Des. 1, pp. 1-60, 1984