Cayley-Klein Geometries at Work

Based on the measurement in a Cayley-Klein geometry, we can now define specific geometric objects and relations. For instance a circle may be defined as the set of all points that have a constant distance to a given point. Being orthogonal may be defined as a certain angle relation between two lines. In each type of a Cayley-Klein geometry the objects and relations will have very specific properties. In this chapter we will deal with aspects of elementary geometry in the context of Cayley-Klein geometries. Following the spirit of this book, we will again focus on (algebraic and geometric representations of) geometric primitive operations, on incidence theorems, and on invariance properties. Again we try to present the definitions and statements in a way that they apply as generally as possible to degenerate Cayley Klein geometries. Still some statements may break down if the geometric configurations or the underlying geometry becomes too degenerate. Since we do not want to spend most of the exposition mainly with pathological degenerate cases, we will base our definitions whenever possible on constructive approaches that allow us to explicitly calculate the objects involved.