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.

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

Formula

With parametrized curves ${\displaystyle {\vec {c}}_{1}(u)}$, ${\displaystyle {\vec {c}}_{3}(u)}$ describing one pair of opposite sides of a domain, and ${\displaystyle {\vec {c}}_{2}(v)}$, ${\displaystyle {\vec {c}}_{4}(v)}$ describing the other pair. the position of point (u,v) in the domain is

${\displaystyle {\begin{array}{rcl}{\vec {S}}(u,v)&=&(1-v){\vec {c}}_{1}(u)+v{\vec {c}}_{3}(u)+(1-u){\vec {c}}_{2}(v)+u{\vec {c}}_{4}(v)\\&&-\left[(1-u)(1-v){\vec {P}}_{1,2}+uv{\vec {P}}_{3,4}+u(1-v){\vec {P}}_{1,4}+(1-u)v{\vec {P}}_{3,2}\right]\end{array}}}$

where, e.g., ${\displaystyle {\vec {P}}_{1,2}}$ is the point where curves ${\displaystyle {\vec {c}}_{1}}$ and ${\displaystyle {\vec {c}}_{2}}$ meet.

References

1. Dyken, Christopher; Floater, Michael S. (2009). "Transfinite mean value interpolation". Computer Aided Geometric Design. 1 (26): 117–134. CiteSeerX 10.1.1.137.4822. 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.