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 [math]\displaystyle{ \vec{c}_1(u) }[/math], [math]\displaystyle{ \vec{c}_3(u) }[/math] describing one pair of opposite sides of a domain, and [math]\displaystyle{ \vec{c}_2(v) }[/math], [math]\displaystyle{ \vec{c}_4(v) }[/math] describing the other pair. the position of point (u,v) in the domain is

[math]\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} }[/math]

where, e.g., [math]\displaystyle{ \vec{P}_{1,2} }[/math] is the point where curves [math]\displaystyle{ \vec{c}_1 }[/math] and [math]\displaystyle{ \vec{c}_2 }[/math] meet.

References

↑ 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.
↑ 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.
↑ 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.
↑ 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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[Dyken2009-1] 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.

[Hall73-2] 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.

[Gordon82-3] 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.

[coons-4] Steven A. Coons, Surfaces for computer-aided design of space forms, Technical Report MAC-TR-41, Project MAC, MIT, June 1967.