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Structure and approximation of real planar algebraic curves
Blažková, Eva ; Šír, Zbyněk (advisor)
Finding a topologically accurate approximation of a real planar algebraic curve is a classic problem in Computer Aided Geometric Design. Algorithms describing the topology search primarily the singular points and are usually based on algebraic techniques applied directly to the curve equation. In this thesis we propose a more geometric approach, taking into account the subsequent high-precision approximation. Our algorithm is primarily based on the identification and approximation of smooth monotonous curve segments, which can in certain cases cross the singularities of the curve. To find the characteristic points we use not only the primary algebraic equation of the curve but also, and more importantly, its implicit support function representation. Using the rational Puiseux series, we describe local properties of curve branches at the points of interest and exploit them to find their connectivity. The support function representation is also used for an approximation of the segments. In this way, we obtain an approximate graph of the entire curve with several nice properties. It approximates the curve within a given Hausdorff distance. The actual error can be measured efficiently. The ap- proximate curve and its offsets are piecewise rational. And the question of topological equivalence of the...
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Structure and approximation of real planar algebraic curves
Blažková, Eva ; Šír, Zbyněk (advisor)
Finding a topologically accurate approximation of a real planar algebraic curve is a classic problem in Computer Aided Geometric Design. Algorithms describing the topology search primarily the singular points and are usually based on algebraic techniques applied directly to the curve equation. In this thesis we propose a more geometric approach, taking into account the subsequent high-precision approximation. Our algorithm is primarily based on the identification and approximation of smooth monotonous curve segments, which can in certain cases cross the singularities of the curve. To find the characteristic points we use not only the primary algebraic equation of the curve but also, and more importantly, its implicit support function representation. Using the rational Puiseux series, we describe local properties of curve branches at the points of interest and exploit them to find their connectivity. The support function representation is also used for an approximation of the segments. In this way, we obtain an approximate graph of the entire curve with several nice properties. It approximates the curve within a given Hausdorff distance. The actual error can be measured efficiently. The ap- proximate curve and its offsets are piecewise rational. And the question of topological equivalence of the...
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Structure and approximation of real planar algebraic curves
Blažková, Eva ; Šír, Zbyněk (advisor) ; Lávička, Miroslav (referee) ; Surynková, Petra (referee)
Finding a topologically accurate approximation of a real planar algebraic curve is a classic problem in Computer Aided Geometric Design. Algorithms describing the topology search primarily the singular points and are usually based on algebraic techniques applied directly to the curve equation. In this thesis we propose a more geometric approach, taking into account the subsequent high-precision approximation. Our algorithm is primarily based on the identification and approximation of smooth monotonous curve segments, which can in certain cases cross the singularities of the curve. To find the characteristic points we use not only the primary algebraic equation of the curve but also, and more importantly, its implicit support function representation. Using the rational Puiseux series, we describe local properties of curve branches at the points of interest and exploit them to find their connectivity. The support function representation is also used for an approximation of the segments. In this way, we obtain an approximate graph of the entire curve with several nice properties. It approximates the curve within a given Hausdorff distance. The actual error can be measured efficiently. The ap- proximate curve and its offsets are piecewise rational. And the question of topological equivalence of the...
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