# Coons patch

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Sample Coons patch

In [mathematics](/source/Mathematics), a **Coons patch**, is a type of [surface patch](/source/Surface_patch) or [manifold](/source/Manifold) [parametrization](/source/Parametrization_(geometry)) used in [computer graphics](/source/Computer_graphics) to smoothly join other [surfaces](/source/Surface_(topology)) together, and in [computational mechanics](/source/Computational_mechanics) applications, particularly in [finite element method](/source/Finite_element_method) and [boundary element method](/source/Boundary_element_method), to mesh problem domains into elements.

Coons patches are named after [Steven Anson Coons](/source/Steven_Anson_Coons), and date to 1967.[1]

## Bilinear blending

Given four [space](/source/Space) curves *c*0(*s*), *c*1(*s*), *d*0(*t*), *d*1(*t*) which meet at four corners *c*0(0) = *d*0(0), *c*0(1) = *d*1(0), *c*1(0) = *d*0(1), *c*1(1) = *d*1(1); [linear interpolation](/source/Linear_interpolation) can be used to interpolate between *c*0 and *c*1, that is

- L c ( s , t ) = ( 1 − t ) c 0 ( s ) + t c 1 ( s ) {\displaystyle L_{c}(s,t)=(1-t)c_{0}(s)+tc_{1}(s)}

and between *d*0, *d*1

- L d ( s , t ) = ( 1 − s ) d 0 ( t ) + s d 1 ( t ) {\displaystyle L_{d}(s,t)=(1-s)d_{0}(t)+sd_{1}(t)}

producing two [ruled surfaces](/source/Ruled_surface) defined on the unit square.

The [bilinear interpolation](/source/Bilinear_interpolation) on the four corner points is another surface

- 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 . {\displaystyle 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)st.}

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

- C ( s , t ) = L c ( s , t ) + L d ( s , t ) − B ( s , t ) . {\displaystyle C(s,t)=L_{c}(s,t)+L_{d}(s,t)-B(s,t).}

## Bicubic blending

Although the bilinear Coons patch exactly meets its four boundary curves, it does not necessarily have the same [tangent plane](/source/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 splines](/source/Cubic_Hermite_spline) with the weights chosen to match the partial derivatives at the corners. This forms a **bicubically blended Coons patch**.

## See also

- [Surface](/source/Surface_(topology))

- [Atlas (topology)](/source/Atlas_(topology))

- [Interpolation](/source/Interpolation)

## References

1. **[^](#cite_ref-coons-tr-41_1-0)** Coons, Steven A. (June 1967). [*Surfaces for Computer-Aided Design of Space Forms, MAC-TR-41*](https://hdl.handle.net/1721.1/149362). Cambridge, MA: MIT/LCS. Retrieved 2 June 2024.

Weiqing Gu. ["Surface Construction Schemes"](https://web.archive.org/web/20100806222407/http://www.math.hmc.edu/~gu/math142/mellon/Application_to_CAGD/Surface_Construction_Schem.html). Archived from [the original](http://www.math.hmc.edu/~gu/math142/mellon/Application_to_CAGD/Surface_Construction_Schem.html) on 2010-08-06. Retrieved 6 August 2010.

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Adapted from the Wikipedia article [Coons patch](https://en.wikipedia.org/wiki/Coons_patch) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Coons_patch?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.