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]

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

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

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