Representations of the support function of convex bodies and operations on sets with these representations

In this thesis, the representations of the support function of convex bodies and some basic properties are investigated. The relationships between some algebraic operations on support functions and geometric operations on convex sets are expressed. It is also shown that some other representation schemes belong to the same class as the support function representation. It is discussed how non-convex shapes can be represented with a support function-like representation. In particular, some geometric methods providing boundary-representation of convex polygons and facilitating the computation of Minkowski operations are studied. For this purpose, calculation methods with support function vectors, slope diagram representation and boundary sum operation are discussed. Then, Minkowski operations of convex polyhedrons are investigated. Illustrative examples on calculations with slope diagrams for convex polyhedrons are given. In the sequel of the study, detailed results are expressed for convex polygons. First, a decomposition of the set of convex polygons is given in order to arrive at the concept of "negative shape". The concept of negative shape is reached with a vector representation of the equivalence classes in this decomposition. An extended boundary addition technique, which allows easy computation of Minkowski operations of positive and negative convex polygons is investigated. The generalizations of negative shape and boundary addition to simple polygons is studied and examined on some examples.