Browse Theses

In the field of computer aided geometric design, it is often necessary to construct curves and surfaces that meet each other smoothly. A great deal of work has been published on the mathematics of smooth curves and surfaces. Geometric continuity—that is, continuity of shape only, without regard to how the geometric entities are parameterized—is a more significant problem than the parametric continuity of the underlying functions. It is more significant both because it is a more appropriate measure of smoothness for computer aided geometric design, and because it is more difficult to quantify than true parametric continuity.

Polynomial spline functions have long been used to describe the curves and surfaces used in computer aided geometric design, both for academic research and for industrial applications. More recently, rational functions have been added to the repertoire. These are a superset of polynomials, and have more flexibility. For example, the conic sections can be parameterized precisely using rational quadratic curves, but not by polynomial curves of any degree.

Rationals offer much more flexibility than just the conic sections, however. Unfortunately, most of the added flexibility has not been exploited to date. The main reason for this is that rational functions can be quite difficult to work with: the quotient rule for rational derivatives is a simple example of the difficulty. Constructions using rational surfaces are rare, and they have usually used overly-restrictive sufficient conditions for continuity, instead of the fully general necessary and sufficient conditions. As such, they have not captured the full expressive power of the rational representation. Some attempts have been made to express the necessary and sufficient conditions for rational surface continuity, but the forms of the statements of the conditions that have been presented in the literature have not enabled much practical application.

In this dissertation we derive the necessary and sufficient conditions for the tangent plane continuity of rational surfaces. The statement of the conditions resulting from our derivation differs from any such statement previously presented, and it does indeed allow practical constructions: we give an example of a useful surface construction that has never been accomplished with previous continuity conditions.

This investigation has also produced other useful related results. A common polynomial surface construction problem is analyzed in some depth, and it is shown exactly what its limits are and what is necessary to create the desired surface. Also, in the investigation of the positional continuity of rational surfaces, a generalization of the well-known degree elevation algorithm for Bézier curves is derived. This generalization is shown to be an aid in determining the coincidence of two rational curves, which is equivalent to the positional continuity of rational surfaces. It also consolidates the operations of reparameterization and degree elevation of rational curves into a single mathematical formulation. into a single mathematical formulation.