Normalized B-Bases for Trigonometric Polynomial Curves

B-curves are trigonometric polynomial curves; they are plots of linear combinations of a B-basis, which generalizes the polynomial Bernstein basis. B-curves can easily be controlled and modified by a control polygon, very much like the more common Bézier curves.

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DETAILS

A classical function space is the order- trigonometric polynomial space . The following is a normalized B-basis for such a space, for [1]:
,
with
.
This basis allows the definition of a B-curve by
,
where is the control polygon.
In this Demonstration, we plot such B-curves, with an adjustable (odd) number of control points. You can also vary the value of and see the corresponding normalized B-basis as well. The order of this basis corresponds to the number of control points, which has to be odd for the B-curve and its basis to exist.
Reference
[1] J. Sanchez–Reyes, "Harmonic Rational Bézier Curves, -Bézier Curves and Trigonometric Polynomials," Computer Aided Geometric Design, 15, 1998 pp. 909–923. doi.10.1016/S0167-8396(98)00031-4.
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