Chapter 23 - Cyclides

Abstract

This chapter provides an overview of Duplin Cyclides. Dupin cyclides belong to both differential geometry as well as projective geometry. In differential geometry, they can be defined as surfaces having constant main curvatures along the corresponding curvature lines. They are algebraic surfaces of fourth or third order being simultaneously enveloped by two families of spheres. Dupin cyclides were recovered for computer-aided geometric design (CAGD) purposes; they extended the class of surfaces so far used in geometric modeling. The chapter describes three main types of Duplin cyclides: tori, general Duplin cyclides, and parabolic Dupin cyclides, along with their parameter representation. Dupin cyclides are special supercyclides with additional properties coming from the Euclidean structure of the space. The chapter also distinguishes among the three classes of supercyclides: general supercylides, semi-parabolic supercylides, and parabolic supercylides. It describes the way cyclides can be used as blendings, which are the most useful applications of cyclides to CAGD and geometric modeling.