Containment of a pair of rotating objects within a container of minimal area or perimeter

Cutting and packing problems arise in many fields of applications and theory. When dealing with irregular objects, an important subproblem is the identification of the optimal clustering of two objects. Within this paper we consider the case, where two irregular one-connected objects whose frontier formed by circular and/or line segments and which can be free rotated, should be placed such that the enclosing rectangle or circle has minimal area, or alternatively, has minimal perimeter. We propose a solution strategy which is based on the concept of phi-functions and provide some examples. Moreover, for the sake of completeness, we give a comprehensive collection of basic phi-functions.