環形曲面與環形曲面之相交曲線與拓樸分析

Abstract

計算曲面交線是電腦輔助設計系統不可或缺的核心功能，例如布林運算牽涉曲面相交計算來表示分割後的區域。計算的效率和正確性，是曲面交線研究中重要的指標。 本研究探討兩環形曲面之相交曲線之演算法則，以及相交曲線之拓樸結構。提出的演算法也會經由一些較為極端的範例測試其強健性。此法則牽涉搜尋相交曲線的關鍵點，並根據不同情形加以分類，進而求得拓特徵。 本研究實作的平台是利用CAD系統（Spring Solid System），搭配開放原始碼的電腦代數系統（Maxima）作為外部函式庫處理代數式之化簡，展開等操作。 最後，本文測試數種相交範例，以驗證本方法的可行性和分析誤差。

Computing intersection is a core function of CAD systems; for instance, a Boolean operation involves calculating surface intersections to determine divided region. Efficiency and correctness are the key issues of surface intersection fields. This thesis is focusing on the algorithm of computation between two toroidal surfaces and the topological structure of intersection. Extreme cases will be tested for examining its robustness of the proposed algorithms. These algorithms involve searching the characteristic points and applying the classification to determine topological features. The thesis is implemented on a CAD platform (Spring Solid System) and an open source computer algebra system (Maxima) for processing algebraic symbols. Finally, the thesis takes several intersection samples for measuring the feasibility and errors.