Function Repository Resource:

# PolygonInterpolation

Interpolate over values given at the vertices of a polygon

Contributed by: Jan Mangaldan
 ResourceFunction["PolygonInterpolation"][{p1,…,pn},{f1,…,fn},p] finds an interpolation of the function values fi corresponding to the polygon vertices pi at the point p.

## Details

ResourceFunction["PolygonInterpolation"][Polygon[{p1,,pn}],{f1,,fn},p] is equivalent to ResourceFunction["PolygonInterpolation"][{p1,,pn},{f1,,fn},p].
ResourceFunction["PolygonInterpolation"] uses a generalization of barycentric coordinates to interpolate over a polygon.

## Examples

### Basic Examples (2)

Compute an interpolation of vertex values at the center of a equilateral triangle:

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Calculate the vertices of a pentagon:

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Compute the value of a function at each vertex:

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Find the value of an interpolation of the values at a point within the polygon:

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### Scope (2)

A triangle:

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Define values at the triangle's vertices for a linear function:

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Evaluate the interpolant at a single point:

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Evaluate the interpolant at multiple points:

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Plot the interpolant over the triangle:

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A star-shaped Polygon:

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Values at the polygon's vertices:

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Evaluate the interpolant at a single point:

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Evaluate the interpolant at multiple points:

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Plot the interpolant along with the original function over the star-shaped polygon:

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### Applications (3)

Vertices for a bean-shaped polygon, and colors specified as RGB components:

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Visualize the polygon colored using PolygonInterpolation:

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Compare with the result of using the VertexColors option of Polygon:

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### Neat Examples (4)

Define a polygon using FindShortestTour over grid points:

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See the polygon:

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Define an arbitrary function and map it over all the vertices:

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For 64 vertices, visualizing the results takes a few seconds:

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## Version History

• 1.0.0 – 14 January 2021