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. This method 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.^[1]

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

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

where, e.g., ${\displaystyle {\vec {P}}_{1,4}}$  is the point where curves ${\displaystyle {\vec {c}}_{1}}$  and ${\displaystyle {\vec {c}}_{4}}$  meet.

References

1. Gordon, William; Thiel, Linda (1982), "Transfinite mapping and their application to grid generation", in Thomson, Joe (ed.), Numerical grid generation, pp. 171–233 {{citation}}: Missing or empty |title= (help)

• Dyken, C., Floater, M. "Transfinite mean value interpolation", Computer Aided Geometric Design, Volume 26, Issue 1, January 2009, Pages 117–134