Transfinite interpolation

In numerical analysis, transfinite interpolation is a means to construct functions over a planar domain in such a way that they match a given function on the boundary. This method is applied in geometric modelling and in the field of finite element method.[1] The transfinite interpolation method, first introduced by William J. Gordon and Charles A. Hall,[2] receives its name due to how a function belonging to this class is able to match the primitive function at a nondenumerable number of points.[3] In the authors' words:

“  | We use the term ‘transfinite’ to describe the general class of interpolation schemes studied herein since, unlike the classical methods of higher dimensional interpolation which match the primitive function F at a finite number of distinct points, these methods match F at a non-denumerable (transfinite) number of points.  | ”

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Transfinite interpolation is similar to the Coons patch, invented in 1967. [4]

## Formula

With parametrized curves c→1(u), c→3(u) describing one pair of opposite sides of a domain, and c→2(v), c→4(v) describing the other pair. the position of point (u,v) in the domain is

S→(u,v)=(1−v)c→1(u)+vc→3(u)+(1−u)c→2(v)+uc→4(v)−[(1−u)(1−v)P→1,2+uvP→3,4+u(1−v)P→1,4+(1−u)vP→3,2]

where, e.g., P→1,2 is the point where curves c→1 and c→2 meet.

## References

1. ↑ Dyken, Christopher; Floater, Michael S. (2009). "Transfinite mean value interpolation". Computer Aided Geometric Design 1 (26): 117–134. doi:10.1016/j.cagd.2007.12.003.
2. ↑ Gordon, William; Hall, Charles (1973). "Construction of curvilinear coordinate systems and application to mesh generation". International Journal for Numerical Methods in Engineering 7 (4): 461–477. doi:10.1002/nme.1620070405.
3. ↑ Gordon, William; Thiel, Linda (1982). "Transfinite mapping and their application to grid generation". Applied Mathematics and Computation 10–11 (10): 171–233. doi:10.1016/0096-3003(82)90191-6.
4. ↑ Steven A. Coons, Surfaces for computer-aided design of space forms, Technical Report MAC-TR-41, Project MAC, MIT, June 1967.

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