An introduction to line geometry with applications

Abstract

The article presents a brief tutorial on classical line geometry and investigates new aspects of line geometry which arise in connection with a computational treatment. These mainly concern approximation and interpolation problems in the set of lines or line segments in Euclidean three-space. In particular, we study the approximation of data lines by, in a certain sense, ‘linear’ families of lines. These sets are, for instance linear complexes and linear congruences. An application is the reconstruction of helical surfaces or surfaces of revolution from scattered data points. This is based on the fact that the normals of these surfaces lie in linear complexes; in particular, normals of surfaces of revolution intersect the axis of revolution. Approximation with linear complexes or congruences is also useful in detecting singular positions of serial or parallel robots. These are positions where the robot should be a rigid system but possesses an undesirable and unexpected instantaneous self motion.