# Isometries and Equivalences Between Point Configurations, Extended To $\varepsilon$-diffeomorphisms

## TL;DR

This paper extends the Orthogonal Procrustes Problem to approximate mappings called $\varepsilon$-diffeomorphisms, providing examples and an algorithm for partitioning point configurations to facilitate their construction.

## Contribution

It introduces the concept of $\varepsilon$-diffeomorphisms to the Orthogonal Procrustes Problem and proposes an algorithm for partitioning point configurations for easier map construction.

## Key findings

- Examples where complete maps cannot be constructed despite matching distributions
- An algorithm for partitioning configurations into polygons
- Extension of the Orthogonal Procrustes Problem to $\varepsilon$-diffeomorphisms

## Abstract

In this announcement, we deal with the Orthogonal Procrustes Problem, in which two point configurations are compared in order to construct a map to optimally align the two sets. This extends this to $\varepsilon$-diffeomorphisms, introduced by [1] Damelin and Fefferman. Examples will be given for when complete maps can not be constructed, for if the distributions do match, and finally an algorithm for partitioning the configurations into polygons for convenient construction of the maps. A revision of this announcement is the memoir preprint: arXiv: 2103.09748, [0], submitted for consideration for publication.