Biinvariant Generalized Barycentric Coordinates on Lie Groups

by Jan Hakenberg published as viXra:2002.0584 – February 29th, 2020

Inverse Distance Weighting
[1968 Shepard]
Inverse Distance Coordinates
viXra:2002.0929
Biinvariant Coordinates

Figure: Basis functions of inverse distance weighting, inverse distance coordinates, and biinvariant coordinates with exponent 2 for an example set of six points in the unit square.

Abstract: We construct biinvariant generalized barycentric coordinates for scattered sets of points in any Lie group. The coordinates are invariant under left-action, right-action, and inversion, and satisfy the Lagrange property. The construction does not utilize a metric on the Lie group, unlike inverse distance coordinates. Instead, proximity is determined in a vector space of higher dimensions than the group using the Euclidean norm. The coordinates that we propose are an inverse to the unique, biinvariant weighted average in the Lie group.

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The author was partially supported by personal savings accumulated during his employment at ETH in 2017–2019. He'd like to thank everyone who worked to make this opportunity available to him.
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