An algebra of geometric shapes

A simple algebra of shapes with 2D planar regions is developed. The fact that a 2D region can be completely described by a one-dimensional, closed-boundary curve if it is homogeneous is used in the presented approach, which first converts the spatial description of the closed curve into an equivalent Fourier series description and then uses the Fourier-description to define binary composition operations that combine two planar shapes to form another planar shape. It is shown how the geometric system comprising the set of all planar shapes and the composition operations can be mapped onto the algebraic system of linear/vector space.<>