Subdivision On Manifolds And PB-splines

Spline and subdivision are two important modeling methods in CAGD and they are closely related with each other. Subdivision is playing an important role in the field of Computer Graphics since Catmull-Clark scheme was firstly proposed in 1978. Most subdivision schemes that are widely used today, like 3 point binary univariate scheme, Loop bivariate scheme and so on, can be derived from univariate or bivariate spline functions. The thesis constructs point-based splines based onthe Clough-Tocher split of triangular domain. The spline bases have star-shape supports and are degree 4 polynomials over each C-T macro-element, and are C~1 continuous along the boundary edges. By investigating 1-disc and 2-disc PB splines, we derive the so called 'PB-spline subdivision' scheme on arbitrary triangular meshes. A parameter α is introduced to adjust the convergence of the scheme and the continuity of the limit surface. By different configuration of α, C~0, C~1 and C~2 continuity of the scheme is achieved.Curve design on manifold is another important topic that has many applications in rigid body motion design in robotics, manufacturing geometry, and computer animation. Many methods have been proposed in literature, such as affine splines, quaternion representation, and motion planning process. However these approaches are all global methods and are not suitable for multi-resolution analysis. We introduce the concept of subdivision in curve design on manifolds, and for any given initial control points on the manifold, we can generate a smooth subdivision curve on the manifold by conformal mapping, MLS projection and orthogonal projection respectively.