Page 81 - Geometric Modeling and Algebraic Geometry

5

Canal Surfaces Defined by Quadratic Families
of Spheres

Rimvydas Krasauskas and Severinas Zube

Faculty of Mathematics and Informatics,
Vilnius University
rimvydas.krasauskas@maf.vu.lt
Summary. This paper is devoted to quadratic canal surfaces, i.e. surfaces that are envelopes
of quadratic families of spheres. They are generalizations of Dupin cyclides but are more
flexible as blending surfaces between natural quadrics . The classification of quadratic canal
surfaces is given from the point of view of Laguerre geometry. Their properties that are impor-
tant for geometric modeling are studied: rational parametrizations of minimal degree, B´ ezier
representations, and implicit equations.

5.1 Introduction

Natural quadrics, i.e. spheres, circular cylinders and circular cones, are perhaps the
most popular surfaces in geometric modeling. They can be characterized as en-
velopes of linear (or constant) families of spheres in space. An other exceptional
property of natural quadrics is that their offset surfaces are of the same type. Usually
Dupin cyclides are used as blending surfaces between natural quadrics. For example,
any two circular cones with a common inscribed sphere can be blended by a part of
Dupin cyclide bounded by two circles as it was shown by Pratt [9] (see Fig. 5.1). Cy-
clides are envelopes of special quadratic families of spheres, and they are offset stable
as well. Here we consider envelopes of most general quadratic families of spheres
and call them quadratic canal surfaces. The main motivation is the possibility to use
patches of these surfaces for blending of natural quadrics.
In Section 5.2 we briefly remaind elements of Laguerre geometry. Section 5.3
is devoted to the classification of conics in the Laguerre space. Cases when conics
define quadratic canal surfaces that can be tangent to natural cones along non-trivial
curves are determined. In Section 5.4 we find rational parametrizations of such canal
surfaces of minimal degree. Their B´ ezier representations and implicit equations are
considered in Sections 5.5 and 5.6. Conclusions and possible applications are dis-
cussed Section 5.7.