Semi-Darboux Motions of Rational Curves with Equivalent Trajectories

A geometry in the sense of F. Klein’s Erlanger Programm (1872) is a pair (S,G) consisting of a set S and a group G of bijections μ : G → G. The set S is the scene of the geometry, its elements are usually called points. The group G is the geometry’s transformation group. An geometric motion in the geometry (S,G) is a map μ : D → G, t 7→ τt of some domain D ∈ R into the transformation group G. The image set {τt(x) | t ∈ D} of a point x ∈ S is called the trajectory of x. It shall be denoted by Tx,μ or simply by Tx (if it is clear which motion μ we refer to). Research on geometric motions began in the 19th century, when French geometers developed the theory for Euclidean three-space. Since then, investigations of motions in spaces of arbitrary dimension, in projective spaces and in diverse non-Euclidean spaces followed. Important tools for their investigation come from other mathematical disciplines such as differential geometry and the theory of matrices. A technique designed especially for the application to geometric motions are kinematic maps. Generally speaking, a kinematic map associates the transformations of the group G to the elements of some point space P, thus allowing to apply results from point geometry in P to the geometry of motions. Some examples and references can be found in [9]. Kinematic maps turned out to be powerful tools for theoretical considerations as well as for practical applications. For the present paper, the investigation of projective Darboux motions according to A. Karger and W. Rath will be most important. Therefore, we will shortly outline their approach.