Geometric design of rational Bezier line congruences and ruled surfaces using line geometry

This paper presents a method for constructing rational Bezier line congruences and ruled surfaces suitable for Computer Aided Geometric Design based on line geometry. Directed lines in the Euclidean three-space are represented by vectors with three homogeneous components over the ring of dual numbers. A projective deCasteljau algorithm is presented for construction of rational Bezier line congruences and ruled surfaces. In the case of ruled surface patches an intrinsic representation of a ruled parch based on segmentation of the ruling about the striction curve of the surface is presented. This leads to a coordinate independent representation of such surface patches. An algorithm is presented that would allow geometric construction of the striction points on rulings of a ruled surface at each intermediate step in the deCasteljau's algorithm. This provides for a geometric method for constructing a coordinate independent representation of a rational ruled surface patch which can be used for geometric design purposes.