In this thesis we address the problem of parametric polynomial circle approximation, with a focus on optimal approximation. We define Bernstein polynomials and Bézier curves and describe their properties. We measure the error with a simplified radial and radial distance and show that the latter, with additional assumptions, implies the Hausdorff distance. We define geometric approximation of order k and consider when it is optimal. We examine Taylor's interpolant and some optimal geometric approximations, implement them, and compare them to each other. With the help of Taylor series and asymptotic analysis, we determine the order of the error.
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