From conics to NURBS: A tutorial and survey

The main geometric features of the nonuniform rational B-splines (NURBS) curve and surface representations are described. It is shown that most of these features are already exhibited by conics, which are a special case of NURBS. The properties typical of NURBS are discussed without dwelling on properties already present in polynomial curves. Conic sections and their representations using rational Bezier curves are reviewed. Cubic NURB curves, geometrical rational splines, rational and B-spline surfaces, and rational Bezier triangles are discussed.<<ETX>>

[1]  L. Piegl.  Modifying the shape of rational B-splines. part2: surfaces , 1989 .

[2]  Brian A. Barsky,et al.  Rational continuity: parametric, geometric, and Frenet frame continuity of rational curves , 1989, TOGS.

[3]  Wolfgang Boehm,et al.  Visual continuity , 1988 .

[4]  Les A. Piegl,et al.  On NURBS: A Survey , 2004 .

[5]  Gerald E. Farin,et al.  The octant of a sphere as a non-degenerate triangular Bézier patch , 1987, Comput. Aided Geom. Des..

[6]  G. Farin.  Algorithms for rational Bézier curves , 1983 .

[7]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[8]  Kenneth James Versprille.  Computer-aided design applications of the rational b-spline approximation form. , 1975 .

[9]  Wolfgang Böhm.  Rational geometric splines , 1987, Comput. Aided Geom. Des..

[10]  G. Farin.  Visually C2 cubic splines , 1982 .

[11]  Rida T. Farouki,et al.  Real rational curves are not 'unit speed' , 1991, Comput. Aided Geom. Des..

[12]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[13]  L. Ramshaw,et al.  Projectively invariant classes of geometric continuity for CAGD , 1989 .

[14]  Les A. Piegl.  The sphere as a rational Bézier surface , 1986, Comput. Aided Geom. Des..

[15]  J. A. Gregory.  Geometric continuity , 1989 .

[16]  Wolfgang Böhm,et al.  Smooth rational curves , 1990, Comput. Aided Des..

[17]  Laszlo Piegl,et al.  On the use of infinite control points in CAGD , 1987, Comput. Aided Geom. Des..

[18]  Richard R. Patterson,et al.  Projective transformations of the parameter of a Bernstein-Bézier curve , 1985, TOGS.

[19]  E. T. Y. Lee,et al.  Möbius reparametrizations of rational B-splines , 1991, Comput. Aided Geom. Des..

[20]  L. Piegl,et al.  Curve and surface constructions using rational B-splines , 1987 .

[21]  Dieter Lasser,et al.  Grundlagen der geometrischen Datenverarbeitung , 1989 .

[22]  I. Faux,et al.  Computational Geometry for Design and Manufacture , 1979 .

[23]  Tim N. T. Goodman.  Constructing piecewise rational curves with Frenet frame continuity , 1990, Comput. Aided Geom. Des..

[24]  Wendelin L. F. Degen,et al.  Some remarks on Bézier curves , 1988, Comput. Aided Geom. Des..

[25]  Helmut Pottman.  Locally controllable conic splines with curvature continuity , 1991 .