Drawing Ruled Surfaces Using the Dual De Boor Algorithm

Abstract

This paper outlines emerging methods, presently under investigation, for constructing ruled surfaces and solid-body trajectories in a Computer Aided Geometric Design (CAGD) context. The underlying principle is that of the standard De Boor Algorithm [4,2,15] for constructing B-spline curves on the plane, or in 3-dimensional space. But the coordinates of all 3-vector points including the control points to be interpolated are now taken to be dual, not real, numbers. When the six resulting number components are interpreted as Pl ű cker coordinates [12], each dual point finds interpretation as a line, and each curve as a ruled surface, in 3-dimensional space. The construction of a duplicated surface is discussed along with some new methods based on a rigid body character. The problem of drawing infinitely long lines, on a screen, as represented by Pl űcker coordinates, is touched upon.