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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.


Invariants, Nonlinear Transformations, Lie Theory, Point Cloud, OCR, Image Analysis, Computational Geometry


Invariants, Nonlinear Transformations, Lie Theory, Point Cloud, OCR, Image Analysis, Computational Geometry.

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