Nonlinear Invariants of Planar Point Clouds Transformed by Matrices

TL;DR

This paper develops nonlinear invariants for planar point clouds under linear transformations using Lie theory, providing solutions to PDEs and practical invariants for various transformation sets, verified through case studies.

Contribution

It introduces a novel Lie theory-based method to derive nonlinear invariants of planar point clouds under linear transformations, including practical invariants for specific transformation sets.

Findings

Derived invariants for four-parameter transformations

Identified invariants for one-parameter transformation sets

Validated invariants through case studies and simulations

Abstract

The goal of this paper is to present invariants of planar point clouds, that is functions which take the same value before and after a linear transformation of a planar point cloud via a $2 \times 2$ invertible matrix. In the approach we adopt here, these invariants are functions of two variables derived from the least squares straight line of the planar point cloud under consideration. A linear transformation of a point cloud induces a nonlinear transformation of these variables. The said invariants are solutions to certain Partial Differential Equations, which are obtained by employing Lie theory. We find cloud invariants in the general case of a four$-$parameter transformation matrix, as well as, cloud invariants of various one$-$parameter sets of transformations which can be practically implemented. Case studies and simulations which verify our findings are also provided.