Transfinite interpolation: Difference between revisions

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In [[numerical analysis]], '''transfinite interpolation''' is a means to construct [[Function (mathematics)|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]]ling and in the field of [[finite element method]].<ref name="Dyken2009"/>

The transfinite interpolation method, first introduced by William J. Gordon and Charles A. Hall,<ref name="Hall73"/> 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.<ref name="Gordon82"/>{{Citation

| first = William

| last = Gordon

| author-link =

| first2 = Linda

| last2 = Thiel

| author2-link =

| editor-last =Thomson

| editor-first = Joe

| editor2-last =

| editor2-first =

| contribution = Transfinite mapping and their application to grid generation

| contribution-url =

| series = Numerical grid generation

| year =1982

| pages =171–233

| place =

| publisher =

| url =

| doi =

| id = }}</ref>

In the authors' words:

{{centered pull quote| 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. <ref name="coons">Steven A. Coons, Surfaces for computer-aided design of space forms, Technical Report MAC-TR-41, Project MAC, MIT, June 1967. </ref>

== Formula ==

Line 31 ⟶ 15:

<math>

\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}_{1,42}+uv\vec{P}_{2,3,4}+u(1-v)\vec{P}_{1,24}+(1-u)v\vec{P}_{3,42}

\right]

\end{array}

</math>

where, e.g., <math>\vec{P}_{1,42}</math> is the point where curves <math>\vec{c}_1</math> and <math>\vec{c}_4_2</math> meet.

== References ==

<references>

<ref name="Hall73">{{cite journal

| first1 = William

| last1 = Gordon

| author-linkfirst2 = Charles

| first2 = Gordon

| last2 = Hall

| title = Construction of curvilinear coordinate systems and application to mesh generation

| journal = International Journal for Numerical Methods in Engineering

| volume = 7

| pagesissue = 461-1774

| pages = 461–477

| year = 1973

| doi=10.1002/nme.1620070405

| bibcode = 1973IJNME...7..461G

}}

}}

</ref>

<ref name="Gordon82">{{cite journal

| firstfirst1 = William

| lastlast1 = Gordon

| first2 = Linda

| last2 = Thiel

| contributiontitle = Transfinite mapping and their application to grid generation

| journal = Applied Mathematics and Computation

| year =1982

| pages =171–233

| number = 10

| url =

| doi = 10.1016/0096-3003(82)90191-6

| volume=10–11}}

</ref>

<ref name="Dyken2009">{{cite journal

| first1 = Christopher

| last1 = Dyken

| first2 = GordonMichael S.

| last2 = Floater

| title = Transfinite mean value interpolation

| journal = Computer Aided Geometric Design

| number = 26

| volume = 1

| year = 2009

| pages = 117–134

| doi = 10.1016/j.cagd.2007.12.003| citeseerx = 10.1.1.137.4822

}}

</ref>

</references>

* Dyken, C., Floater, M. "Transfinite mean value interpolation", ''Computer Aided Geometric Design'', Volume 26, Issue 1, January 2009, Pages 117&ndash;134

[[Category:Interpolation]]