Nonlinear Invariants of Planar Point Clouds Transformed by Matrices

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.