  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.
