In
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
, 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 model
__NOTOC__
Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes.
The shapes studied in geometric modeling are mostly two- or three-dimensio ...
ling and in the field of
finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
.
The transfinite interpolation method, first introduced by William J. Gordon and Charles A. Hall,
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.
In the authors' words:
Transfinite interpolation is similar to the
Coons patch
In mathematics, a Coons patch, is a type of surface patch or manifold parametrization used in computer graphics to smoothly join other surfaces together, and in computational mechanics applications, particularly in finite element method and bo ...
, invented in 1967.
[Steven A. Coons, Surfaces for computer-aided design of space forms, Technical Report MAC-TR-41, Project MAC, MIT, June 1967. ]
Formula
With parametrized curves
,
describing one pair of opposite sides of a domain, and
,
describing the other pair. the position of point (u,v) in the domain is
where, e.g.,
is the point where curves
and
meet.
References
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Interpolation
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