Free-form Shape Modeling Using Cyclides Splines

Abstract

Dupin cyclides are classical surfaces discovered by the French mathematician Charles Dupin in the early 19th century. These surfaces have been extensively studied for surface representation for about three decades since Ralph Martin introduced them to surface modeling in early 1980s. Cyclides have several remarkable properties; for instance, they are low-degree algebraic surfaces (degree 4 or less) and have rational bi-quadratic parameterization. Furthermore, the offsets of a cyclide are again cyclides. However, despite all these advantages cyclide could potentially offer for shape modeling, all previous attempts at using cyclides to model free-form surfaces have been unsuccessful because of the relative inflexibility of cyclide patches. Therefore, it is widely believed that cyclides do not have enough freedom to represent free-form shapes. The applications of cyclides are currently limited to modeling blend surfaces or canal surfaces.

I shall propose an effective approach to modeling free-form shapes using fair, smooth cyclide splines. The key ideas behind the approach are vertex relaxation and global optimization. Specifically, the inflexibility of cyclides is overcome by relaxing the vertices of cyclide patches in a constrained optimization framework. I shall present results on using cyclide splines for free-form surface fitting and general free-form shape modeling, thus proposing cyclide splines as the first practical free-form surface representation with the exact offset property.