Parametric circles and spheres

Formulations for parametric circles and spheres in terms of rational Gaussian (RaG) curves and surfaces are introduced. With the proposed formulations, a full circle is generated by interpolating a closed RaG curve to the vertices of an equilateral triangle, and a full sphere is generated by interpolating a closed RaG surface to the vertices of an octahedron with equilateral triangular faces. Generation of circles and spheres in this manner is very intuitive and easy to remember as the weights are all 1 and the nodes are all unique and uniformly spaced.

[1]  A. Ardeshir Goshtasby,et al.  Geometric modelling using rational Gaussian curves and surfaces , 1995, Comput. Aided Des..

[2]  Berthold K. P. Horn.  Extended Gaussian images , 1984, Proceedings of the IEEE.

[3]  Knut Mørken,et al.  Best Approximation of Circle Segments by Quadratic Bézier Curves , 1991, Curves and Surfaces.

[4]  Wolfgang Böhm,et al.  A survey of curve and surface methods in CAGD , 1984, Comput. Aided Geom. Des..

[5]  James F. Blinn.  How Many Ways Can You Draw a Circle? , 1987, IEEE Computer Graphics and Applications.

[6]  Les A. Piegl,et al.  A menagerie of rational B-spline circles , 1989, IEEE Computer Graphics and Applications.

[7]  Wayne Tiller,et al.  Rational B-Splines for Curve and Surface Representation , 1983, IEEE Computer Graphics and Applications.

[8]  Gary J. Herron,et al.  Smooth closed surfaces with discrete triangular interpolants , 1985, Comput. Aided Geom. Des..

[9]  Tom Lyche,et al.  Good approximation of circles by curvature-continuous Bézier curves , 1990, Comput. Aided Geom. Des..

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

[11]  L. Schumaker,et al.  Curves and Surfaces , 1991, Lecture Notes in Computer Science.

[12]  Les A. Piegl,et al.  Algorithms for Computing Conic Splines , 1990 .

[13]  Les A. Piegl.  Less Data for Shapes , 1987, IEEE Computer Graphics and Applications.

[14]  Michael Goldapp,et al.  Approximation of circular arcs by cubic polynomials , 1991, Comput. Aided Geom. Des..

[15]  A. Ardeshir Goshtasby,et al.  Design and recovery of 2-D and 3-D shapes using rational Gaussian curves and surfaces , 1993, International Journal of Computer 11263on.

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

[17]  Gerald Farin,et al.  Geometric modeling : algorithms and new trends , 1987 .

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

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

[20]  Les A. Piegl,et al.  Infinite Control Points-A Method for Representing Surfaces of Revolution Using Boundary Data , 1987, IEEE Computer Graphics and Applications.

[21]  Rangachar Kasturi,et al.  Machine vision , 1995 .

[22]  E. F. Eisele.  Best Approximations of Some Parametric Surfaces by Bézier Surfaces , 1992, IMA Conference on the Mathematics of Surfaces.