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[[Image:Example of coons surface.svg|thumb|400px|right|Sample Coons patch]]

In [[mathematics]], a '''Coons patch''', is a type of [[manifold]] [[parametrization]] used in [[computer graphics]] to smoothly join other [[surface]]s together, and in [[computational mechanics]] applications, particularly in [[finite element method]] and [[boundary element method]], to mesh problem domains into elements.

Coons patches are named after [[Steven Anson Coons]].

==Bilinear blending==
Given four [[space]] curves ''c''<sub>0</sub>(''s''), ''c''<sub>1</sub>(''s''), ''d''<sub>0</sub>(''t''), ''d''<sub>1</sub>(''t'') which meet at four corners ''c''<sub>0</sub>(0) = ''d''<sub>0</sub>(0), ''c''<sub>0</sub>(1) = ''d''<sub>1</sub>(0), ''c''<sub>1</sub>(0) = ''d''<sub>0</sub>(1), ''c''<sub>1</sub>(1) = ''d''<sub>1</sub>(1); [[linear interpolation]] can be used to interpolate between ''c''<sub>0</sub> and ''c''<sub>1</sub>, that is

:<math>L_c(s,t)=(1-t) c_0(s)+ t c_1(s) \, </math>

and between ''d''<sub>0</sub>, ''d''<sub>1</sub>

:<math>L_d(s,t)=(1-s) d_0(t)+ s d_1(t) \, </math> 

producing two [[ruled surface]]s defined on the unit square. 

The [[bilinear interpolation]] on the four corner points is another surface 

:<math> B(s,t) = c_0(0) (1-s)(1-t) + c_0(1) s(1-t) +  c_1(0) (1-s)t + c_1(1) s t. \, </math>

A '''bilinearly blended Coons patch''' is the surface 

:<math>L_c(s,t)+L_d(s,t)-B(s,t). \, </math>

==Bicubic blending==
Although the bilinear Coons patch exactly meets its four boundary curves, it does not necessarily have the same [[tangent plane]] at those curves as the surfaces to be joined, leading to creases in the joined surface along those curves. To fix this problem, the linear interpolation can be replaced with [[cubic Hermite spline]]s with the weights chosen to match the partial derivatives at the corners. This forms a '''bicubically blended Coons patch'''.

==See also==

* [[Surface]]
* [[Atlas (topology)]]
* [[Interpolation]]

==References==

*{{cite web|title=Surface Construction Schemes | author=Weiqing Gu | url=http://www.math.hmc.edu/~gu/math142/mellon/Application_to_CAGD/Surface_Construction_Schem.html | accessdate= 8 April 2012}}

[[Category:Multivariate interpolation]]
[[Category:Splines]]

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