Normalized C-Basis and C-Curves

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An alternative to the rational Bézier model is mixed algebraic and trigonometric curves named helix splines or C-curves (C for cycloidal). The basis of the C-curves space, named C-basis, is determined in order to retain some properties of the Bernstein bases.

Contributed by: Isabelle Cattiaux-Huillard (March 2015)
Open content licensed under CC BY-NC-SA


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An alternative to the Bézier model (see the related link) is the -order space of mixing algebraic and trigonometric polynomials . For example, the space allows exact representations of circles, ellipses, and cycloids. In [1], it is determined that the corresponding normalized C-basis for is defined by:

where

,

,

and

In this Demonstration, you can vary the value of . The corresponding curves are determined by the control points, which you can drag.

Reference

[1] E. Mainar, J. M. Peña, and J. Sánchez-Reyes, "Shape Preserving Alternatives to the Rational Bézier Model," Computer Aided Geometric Design, 18(1), 2001 pp. 37–60. doi:10.1016/S0167-8396(01)00011-5.



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