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.