Modeling and Processing with Quadric 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.

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