Using Line Congruences for Parameterizing Special Algebraic Surfaces

Abstract

Surfaces in line space are called line congruences. We consider several special line congruences forming a fibration of the three–dimensional space. These line congruences correspond to certain special algebraic surfaces. Using rational mappings associated with the line congruences, it is possible to generate rational curves and surfaces on them. This approach is demonstrated for quadric surfaces, cubic ruled surfaces, and for Veronese surfaces and their images in three–dimensional space (quadratic triangular Bezier surfaces).