Cyclides in engineering design: Algorithmic and computational aspects.

Abstract

Dupin cyclides have shown significant promise for applications in geometric modeling. While the mathematical and geometric properties of the surface are well known, the techniques and computational tools required for design and analysis of engineering objects received little attention. This research is an effort to design and develop algorithms and mathematical techniques for efficient use of cyclide surfaces for modeling geometrically complex objects. Fundamental definitions and intrinsic geometric properties of cyclides have been exploited for developing compact representations and efficient manipulation schemes. New auxiliary representations for modeling with cyclide pieces are presented. A rational parametric representation and an equivalent NURBS representation are developed for the parabolic cyclide. A suite of computational procedures for interrogating cyclide surfaces has been derived from the geometric and algebraic properties of cyclides. Methods for designing with tubular pieces of the surface for object modeling and volume representation are introduced. A new approach for generating composite surfaces using cyclide patches is presented. This research strongly emphasizes the algorithms and techniques that expand the application domain of cyclides. The issues of portability and handling of degenerate cases have been addressed. The main techniques developed during this research have been implemented in a CAD environment.