Approximations of planar curves by quadratic rational B-splines평면곡선의 이차 유리 B-spline에 의한 근사

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We study on the geometric properties of the quadratic rational $Bézier$ curves and approximations using them. We find necessary and sufficient conditions for the curvature of a quadratic rational $Bézier$ curve to be monotone, to have a unique local minimum, to have a unique local maximum and to have both extrema, and we also visualize them in figures. We characterize the best approximation of a regular plane curve by a quadratic rational $Bézier$ curve with possible contact order at both end points and prove its uniqueness. We also present a Remes type algorithm to obtain the best approximation. We apply our characterization to the degree reduction of cubic rational $Bézier$ curve to quadratic one and also to the cubic offset approximation, and present the numerical results. For the circular arc of angle 0<α<π we present the explicit form of the best $GC^3$ quartic approximation and the best $GC^2$ quartic approximations of various types, and give the explicit form of the Hausdorff distance between the circular arc and the approximate $Bézier$ curves for each case. We also show the existence of the $GC^4$ quintic approximations to the arc, and find the explicit form of the best $GC^3$ quintic approximation in certain constraints and their distances from the arc. All approximations we construct in this thesis have the optimal order of approximation, twice of the degree of approximate $Bézier$ curves.
Kim, Hong-Ohresearcher김홍오researcher
한국과학기술원 : 수학과,
Issue Date
135103/325007 / 000955209

학위논문(박사) - 한국과학기술원 : 수학과, 1998.2, [ 70 p. ]


Curvatures; Geometric approximation; B{\``e}zier curves; Quadratic rational B-splines; Arc approximation; 원호 근사; 곡률; 기하적인 근사; 베지어 곡선; 이차 유리 비-스플라인

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