Accurate parametrization of conics by NURBS

One argument often given to explain the popularity of NURBS (nonuniform rational B-spline) is that it permits the definition of free-form curves and surfaces (as do most spline models). It also provides an exact representation of conic sections and thus of a large set of curves and surfaces used intensively in CAD: circular arcs, circles, cylinders, cones, spheres, surfaces of revolution and so forth. Nevertheless, few published works discuss the mathematical properties behind the representation of conics by NURBS except for two monographs by Piegl and Tiller (1995) and Farin (1995). The article does not pretend to fill this theoretical lack but rather deals with the following problems: all known NURBS representations of curves and surfaces based on conics have only a C^l continuity. Moreover, no technique exists that would eventually allow us to find a parametrization with a higher level of continuity. The parametrization resulting from the NURBS representation of conics can deviate significantly from the ideal are length (that is, uniform) parametrization. The only known solution to reduce this deviation is to increase the number of control points of the spline by using refinement algorithms, for instance, but such a process converges only slowly to the uniform parametrization. The solution proposed uses an original reparametrization process called zigzag reparametrization, based on a specific family of rational polynomials