A Laguerre geometric approach to rational offsets

1. Pythagorean - Hodograph Curves in Practical Use.

This paper studies algebraic and geometric properties of curve?curve, curve?surface, and surface?surface bisectors. The computation is in general difficult since the bisector is determined by solving a system of nonlinear equations. Geometric considerations will help us to determine several distinguished curve and surface pairs which possess elementary computable bisectors. Emphasis is on low-degree rational curves and surfaces, since they are of particular interest in surface modeling.

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Geometric Properties of Bisector Surfaces

This chapter defines quadric Surfaces or quadrics, and discusses their classifications under different groups of transformations—that is, Euclidean, affine, and projective transformations. Quadric surfaces are surfaces defined by algebraic equations of degree two. It explains why quadrics are widely used in computer-aided geometric design (CAGD). The chapter discusses the rational quadratic parameterization of a whole quadric surface and of a surface patch on a quadric. A simple way to derive a rational quadratic parameterization of a quadric is to use a stereographic projection of the quadric to establish a bi-rational mapping among points on a plane and points on the quadric. The properties of the generalized stereographic projection for a sphere are discussed. A rational quadratic surface whose algebraic degree is higher than two is called a “Steiner surface.” The chapter also describes the properties of quadrics concerning the applications in geometric modeling: fitting, blending, and offsetting. It reviews two algebraic methods developed for general quadrics: intersection and interference.

Modeling and Processing with Quadric Surfaces

G1 Hermite interpolation by PH cubics revisited

In this paper, we study two problems related to quadrics. One is the representation of triangular and quadrangular patches on quadrics: a necessary and sufficient condition is given for triangular patches on quadrics to be representable as rational quadratic triangular Bezier surfaces. The other is the rationality of the offsets to quadrics: the offsets of regular quadrics are shown to be rational and the algorithms are given for computing their rational parameterizations.

Rational parameterization of quadrics and their offsets

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Rational parameterization of quadrics and their offsets