In this PhD thesis several new methods for an approximation of the circular arc are presented. The first part represents asymptotically the best single-sided geometric approximation of order one (G1) according to the radial error. It is an approximation with the quartic Bézier curve. The control points are set so that in addition to the boundary points the approximative curve touches the circular arc with order one at two inner points. Due to the complexity of the system of equations, the proof of the existence of a solution is made using the homotopy. The generalization of the derived method is then used for the approximation of conic sections. The method is used in a subdivision process to determine new vertices until some additional conditions are met. Each individual part is then approximated using the derived method.
In the second part of the thesis curvature error is used instead of the radial one. Some simple low-degree polynomial methods and biarc methods are presented. It is confirmed that the derived methods still have the optimal approximation order according to the radial error, while the approximation order according to the curvature error is reduced by two as expected due to the second order derivatives.
As it turns out the optimal approximation of the circular arc is always achieved when the error equally oscillates. The last part of the thesis is thus dedicated to the methods of this type. These are low-degree approximations that have a geometric contact of some order with the circular arc at the boundary points, or they only approximate the same angle as the circular arc. In the latter case, we obtain a scaled Chebyshev polynomial. Similarly as in the first part, when a complicated system of equations appears, the proof of the existence of a solution is done using the homotopy.
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