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The goal of this paper is to present invariants of planar point clouds, thatis functions which take the same value before and after a linear transformationof a planar point cloud via a $2 \times 2$ invertible matrix. In the approachwe adopt here, these invariants are functions of two variables derived from theleast squares straight line of the planar point cloud under consideration. Alinear transformation of a point cloud induces a nonlinear transformation ofthese variables. The said invariants are solutions to certain PartialDifferential Equations, which are obtained by employing Lie theory. We findcloud invariants in the general case of a four$-$parameter transformationmatrix, as well as, cloud invariants of various one$-$parameter sets oftransformations which can be practically implemented. Case studies andsimulations which verify our findings are also provided.

Nonlinear Invariants of Planar Point Clouds Transformed by Matrices