Motion synthesis for computer-aided geometric design

This thesis deals with parametrically defined geometric shapes such as Bezier curves and surfaces and parametrically defined Cartesian motions of a rigid body. The main purpose of this thesis is to extend the application domain of methods in the field of Computer Aided Geometric Design from the geometry of points to the geometries of planes, lines, and rigid body displacements. The thesis also seeks to develop methods for shape design from kinematics of rigid body motions.; Essential to the development of this thesis are the representations of planes, lines, and rigid-body displacements as points in the coordinate spaces of the respective geometric entities. The notion of projective space planes an important role in developing and utilizing these representations. The projective duality between points and planes is used to study the enveloping surfaces of one- and two-parameter family of planes. This leads to the development of developable rational Bezier surfaces and dual tensor-product surfaces. The representation of a line-segment in terms of Plucker line coordinates and Study's dual vector allows one to study the problem of designing ruled surfaces as a curve design problem in the space of line coordinates. The representation of a spatial rigid-body displacement in terms of dual-quaternion coordinates leads to two types of algorithms for spatial motion synthesis. One is developed by combining CAGD methods such as the deCastejau algorithm and geometric continuity with the geometry of a unit dual hypersphere. The other type of algorithm is obtained by applying projective algorithms in CAGD to the space of dual quaternions. This results in one-degree-of-freedom rational Bezier and B-spline motions as well as two-degree-of-freedom rational tensor-product Bezier motions. The rational Bezier and B-spline motions are then used to study the trajectories and enveloping surfaces of a moving object. This leads to the development of a special class of tensor-product Bezier surfaces as the enveloping surfaces of rational motions of planes and developable surfaces such as cylinders. The results have not only theoretical interest in CAGD and kinematics but also applications in CAD/CAM, Computer Graphics, and Robotics.