A cyclic basis for closed curve and surface modeling

We define a cyclic basis for the vectorspace of truncated Fourier series. The basis has several nice properties, such as positivity, summing to 1, that are often required in computer aided design, and that are used by designers in order to control curves by manipulating control points. Our curves have cyclic symmetry, i.e. the control points can be cyclically arranged and the curve does not change when the control points are cyclically permuted. We provide an explicit formula for the elevation of the degree from n to n+r, r>=1 and prove that the control polygon of the degree elevated curve converges to the curve itself if r tends to infinity. Variation diminishing property of the curve is also verified. The proposed basis functions are suitable for the description of closed curves and surfaces with C^~ continuity at all of their points.

[1]  Juan Manuel Peña,et al.  Shape preserving representations and optimality of the Bernstein basis , 1993, Adv. Comput. Math..

[2]  J. Joseph,et al.  Fourier transforms , 2012 .

[3]  Juan Manuel Peña,et al.  Corner cutting algorithms associated with optimal shape preserving representations , 1999, Comput. Aided Geom. Des..

[4]  Qiaode Jeffrey Ge,et al.  Low-harmonic rational Bézier curves for trajectory generation of high-speed machinery , 1997, Comput. Aided Geom. Des..

[5]  Helmut Pottmann.  Zur Geometrie höherer Planetenumschwungbewegungen , 1984 .

[6]  Juan Manuel Peña,et al.  Shape preserving representations for trigonometric polynomial curves , 1997, Comput. Aided Geom. Des..

[7]  Juan Manuel Peña,et al.  Shape preserving alternatives to the rational Bézier model , 2001, Comput. Aided Geom. Des..

[8]  Wolfgang Schuster.  A closed algebraic interpolation curve , 2000, Comput. Aided Geom. Des..

[9]  Zhang Jiwen,et al.  C-curves: An extension of cubic curves , 1996, Comput. Aided Geom. Des..

[10]  Josef Schicho,et al.  Algorithms for Trigonometric Curves (Simplification, Implicitization, Parameterization) , 1998, J. Symb. Comput..

[11]  Javier Sánchez-Reyes,et al.  Harmonic rational Bézier curves, p-Bézier curves and trigonometric polynomials , 1998, Comput. Aided Geom. Des..

[12]  George Polya,et al.  Remarks on de la Vallée Poussin means and convex conformal maps of the circle. , 1958 .

[13]  Fujio Yamaguchi,et al.  Computer-Aided Geometric Design , 2002, Springer Japan.

[14]  Juan Manuel Peña,et al.  Totally positive bases for shape preserving curve design and optimality of B-splines , 1994, Comput. Aided Geom. Des..