Coons area

Coons surface into a curved square

A Coons surface (engl .: Coons patch ) is in the geometry of a surface element , which interpolates the four edge corners of a curved spatial quadrangle. It is used in geometric data processing ( CAD ) to assemble a larger area from many patches. The method was introduced in 1967 by Steven Anson Coons .

Description of the method

problem

Given: The edge curves of a square patch:

${\displaystyle {\mathbf {x} }={\mathbf {x} }(u,0)\ ,\ {\mathbf {x} }={\mathbf {x} }(u,1)\ ,\ {\mathbf {x} }={\mathbf {x} }(0,v)\ ,\ {\mathbf {x} }={\mathbf {x} }(1,v)\ ,\ u,v\in [0,1]\;.}$

Wanted: A surface that interpolates the four curves. ${\displaystyle {\mathbf {x} }={\mathbf {x} }(u,v)}$

Method of Coons

The ruled surface

${\displaystyle {\mathbf {r} }_{1}=(1-v)\;{\bf {x}}(u,0)\;+v\;{\bf {x}}(u,1)\ }$interpolates the two curves and ${\displaystyle \ {\mathbf {x} }={\mathbf {x} }(u,0)\ ,\ {\mathbf {x} }={\mathbf {x} }(u,1)\ }$

${\displaystyle {\mathbf {r} }_{2}=(1-u)\;{\bf {x}}(0,v)\;+\;u\;{\bf {x}}(1,v)\ }$interpolates the curves . ${\displaystyle \ {\mathbf {x} }={\mathbf {x} }(0,v)\ ,\ {\mathbf {x} }={\mathbf {x} }(1,v)}$

bilinear Coons area generation

Each of these ruled surfaces therefore only ever interpolates two of the four edge curves. The idea of ​​Coons is to add a simple correction term to the sum of the two parametric representations so that the area ${\displaystyle {\bf {r}}_{12}(u,v)}$

• ${\displaystyle \ {\mathbf {x} }={\bf {r}}_{1}(u,v)+{\bf {r}}_{2}(u,v)-{\bf {r}}_{12}(u,v)}$

all four curves interpolated. It's easy to calculate that

${\displaystyle {\mathbf {r} }_{12}=(1-u){\big (}(1-v){\mathbf {x} }(0,0)+v{\mathbf {x} }(0,1){\big )}\ +\ u{\big (}(1-v){\mathbf {x} }(1,0)+v{\mathbf {x} }(1,1){\big )}}$

meets these requirements. The correction surface is the hyperbolic paraboloid through the four corner points.

Due to the bilinearity of the correction surface, the Coons surface constructed in this way is also called the bilinear Coons surface .

In practice, individual patches are put together. Bilinear Coons areas then only guarantee simple continuity of the total area. In order to achieve tangential plane continuity ( continuity ), bicubic Coons surfaces are used. ${\displaystyle G^{1}}$

Individual evidence

1. ^ Surfaces for computer-aided design of space forms, Technical Report MAC-TR-41, Project MAC, MIT, June 1967.
2. ^ G. Farin: Curves and Surfaces for Computer Aided Geometric Design , Academic Press, 1990, ISBN 0-12-249051-7 , p. 346
3. ^ I. Grieger: Graphische Datenverarbeitung , Springer-Verlag, 2013, ISBN 3642844413 , 9783642844416, p. 84