Periodic Bézier curves
2009 - J. Sánchez-Reyes
Computer Aided Geometric Design 26,989-1005, (2009)
We construct closed trigonometric curves in a Bézier-like fashion. A closed control polygon defines the curves, and the control points exert a push-pull effect on the curve. The representation of circles and derived curves turns out to be surprisingly simple. Fourier and Bézier coefficients of a curve relate via Discrete Fourier Transform (DFT). As a consequence, DFT also applies to several operations, including parameter shift, successive differentiation and degree-elevation. This Bézier model is a particular instance of a general periodic scheme, where radial basis functions are generated as translates of a symmetric function. In addition to Bézier-like approximation, such a periodic scheme subsumes trigonometric Lagrange interpolation. The change of basis between Bézier and Lagrange proceeds via DFT too, which can be applied to sample the curve at regularly spaced parameter values. The Bézier curve defined by certain control points is a low-pass filtered version of the Lagrange curve interpolating the same set of points.