In this thesis, we construct closed planar periodic trigonometric curves in a manner similar to Bézier curves. The shape of such a curve is determined by a closed control polygon, whose vertices define the control points. Trigonometric Bézier curves allow for the exact representation of curves such as circles, ellipses, trochoids, and others. To achieve this, we employ trigonometric polynomials, which are linear combinations of sines and cosines. The curves are defined on the interval $[0, 2\pi)$ and are periodic. We prove that the basis of trigonometric polynomials can be expressed as a linear combination of shifted copies of a special function κn, which is analogous to the Bernstein basis polynomials. This enables us to represent trigonometric curves in a form similar to Bézier curves. Furthermore, we show that the coefficients in the curve representation are related to Fourier coefficients via the Discrete Fourier Transform (DFT). As a consequence, various operations on trigonometric curves-such as parameter shifting, differentiation, integration, and degree elevation-can be performed efficiently using the DFT. We also discuss trigonometric Lagrange interpolation, where the DFT is again applied to convert the basis from Bézier to Lagrange. Finally, we demonstrate that a Bézier curve defined by arbitrary control points can be viewed as a filtered version of the Lagrange curve interpolated on the same set of control points.
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