Conics and Circles

Abstract

The conic sections and circles play a fundamental role in CADCAM applications. Undoubtedly one of the greatest advantages of NURBS is their capability of precisely representing conic sections and circles, as well as free-form curves and surfaces. We assume a knowledge of conics and circles; the purpose of this chapter is to study them in the framework of their representation as NURBS curves. In Section 7.2 we review various forms and properties of conics which are required in subsequent sections. Section 7.3 covers the quadratic rational Bézier representation of conic and circular arcs; Section 7.4 introduces infinite control points. In Sections 7.5 and 7.6 we present algorithms for constructing the NURBS representation of arbitrary circles and conics, respectively, including full circles and ellipses. Section 7.7 covers conversions between the various representation forms, and Section 7.8 gives examples of higher order circle representations.