AN ALGEBRA OF POLYGONS THROUGH THE NOTION OF NEGATIVE SHAPES

A new notion of negative shape is introduced in order to develop an algebraic system of geometric shapes within which one can add and subtract shapes exactly as one adds and subtracts within the integer number system. Concentrating on polygonal shapes in 2 dimensions, we show that this simple extension of our commonsense concept of geometric shapes opens up many new areas with a great potential for understanding and developing 2-dimensional geometry and geometric algorithms. In the course of this pursuit the concept of a new equivalence relation on convex polygons evolves that also appears to be significant in understanding convex polygons, particularly various symmetries in them. In constructing the algebraic system of shapes we use the Minkowski addition operation (in mathematical morphology dilation) as the composition Operation. © 1991.