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.

”

Formula

With parametrized curves $\vec{c}_1(u)$, $\vec{c}_3(u)$ describing one pair of opposite sides of a domain, and $\vec{c}_2(v)$, $\vec{c}_4(v)$ describing the other pair. the position of point (u,v) in the domain is

$\begin{array}{rcl} \vec{S}(u,v)&=&(1-v)\vec{c}_1(u)+v\vec{c}_3(u)+(1-u)\vec{c}_2(v)+u\vec{c}_4(v)\\ && - \left[ (1-u)(1-v)\vec{P}_{1,2}+uv\vec{P}_{3,4}+u(1-v)\vec{P}_{1,4}+(1-u)v\vec{P}_{3,2} \right] \end{array}$

where, e.g., $\vec{P}_{1,2}$ is the point where curves $\vec{c}_1$ and $\vec{c}_2$ 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: 461–177. doi:10.1002/nme.1620070405.
3. Gordon, William; Thiel, Linda (1982). "Transfinite mapping and their application to grid generation". Applied Mathematics and Computation (10): 171–233. doi:10.1016/0096-3003(82)90191-6.