�ber drei zwangl�ufig gegeneinander bewegte Cayley/Klein-Ebenen

Abstract

If three Euclidean planes move relatively to each other, the three poles of rotation are either identical or pairwise distinct and collinear. In the second case the distances of the poles are in the ratio of the motions' angular velocities. These known facts of Euclidean kinematics can be generalized in a largely uniform way to plane Cayley/Klein motions with finite poles. For the angular velocities we give a representation which is valid for all considered Cayley/Klein motions. Application of the duality principle of the projective plane yields a proposition about concurrent fixed lines. We also generalize the generation of a pair of envelope curves of a Euclidean motion as paths of a point of a third moved plane.