Comparison of geometric figures

Abstract

Although the geometric equality of figures has already been studied thoroughly, little work has been done about the comparison of unequal figures. We are used to compare only similar figures but would it be meaningful to compare non similar ones? In this paper we attempt to build a context where it is possible to compare even non similar figures. Adopting Klein's view for the Euclidean Geometry, we defined a relation "<=" as: S<=T whenever there is a rigid motion f so that f(S) is a subset of T. This relation is not an order because there are figures (subsets of the plane) so that S<=T, T<=S and S, T not geometrically equal. Our goal is to avoid this paradox and to track down non-trivial classes of figures where the relation "<=" becomes, at least, a partial order. Such a class will be called a good class of figures. A reasonable question is whether the figures forming a good class have certain properties and whether the algebra of these figures is also a good class. Therefore we classified the figures into those that cause the paradox mentioned above and those that never cause it. The last ones are called good figures. Although simple, the definition of the good figure was difficult to handle, therefore we introduced a more technical, but intrinsic and handy definition, that of the strongly good figure. With these tools we constructed a new context, where we expanded our perspective about the geometric comparison not only in the Euclidean but also in the Hyperbolic and in the Elliptic Geometry. Eventually, there are still some open and quite challenging issues, which we present them at the last part of the paper.