Das kinematische Bild einer Regelfläche des einfach isotropen Raumes

Abstract

TheClifford parallelism of the three-dimensional isotropic space J ⁽¹⁾₃ induces thekinematic mapping of an element of surface in J ⁽¹⁾₃ onto a pair of points in a fixed plane π₀ . By identifying a regular surface Φ with the manifold of its osculating elements of surface thekinematic image of Φ is defined and equivalent to an area preserving transformation of the plane π₀ . In this paper we will examine the connections between the isotropic invariants of a skew ruled surface Φ and its kinematic image. Since the striction line y of Φ exactly consists of the singular points of Φ the kinematic images of the osculating planes of y are considered in addition. In this way we obtain a correspondence between the theory of ruled surfaces and the elementary geometry of two pairs of plane curves with a common middle curve.