Shape preserving representations for trigonometric polynomial curves
Computer Aided Geometric Design
The NURBS book (2nd ed.)
Cycles upon cycles: an anecdotal history of higher curves in science and engineering
Proceedings of the international conference on Mathematical methods for curves and surfaces II Lillehammer, 1997
A closed algebraic interpolation curve
Computer Aided Geometric Design
An Algebra of Geometric Shapes
IEEE Computer Graphics and Applications
Control Point Representations of Trigonometrically Specified Curves and Surfaces
Geometric Modelling, Dagstuhl, Germany, 1993
Quadratic trigonometric polynomial curves with a shape parameter
Computer Aided Geometric Design
Representing circles with five control points
Computer Aided Geometric Design
Cubic trigonometric polynomial curves with a shape parameter
Computer Aided Geometric Design
Quadratic trigonometric polynomial curves concerning local control
Applied Numerical Mathematics
Advancing front circle packing to approximate conformal strips
Computational Geometry: Theory and Applications
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We construct closed trigonometric curves in a Bezier-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 Bezier 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 Bezier 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 Bezier-like approximation, such a periodic scheme subsumes trigonometric Lagrange interpolation. The change of basis between Bezier and Lagrange proceeds via DFT too, which can be applied to sample the curve at regularly spaced parameter values. The Bezier curve defined by certain control points is a low-pass filtered version of the Lagrange curve interpolating the same set of points.