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.

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

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

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

References

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