Rational B-Spline曲面之交線技術研究

求取自由曲面(Free Form Surfaces)之交線(Intersection Curves)問題，是在計算幾何學、電腦圖學及CAD/CAM系統中欲設計或構建一複雜外型之物件(Object)時，一重要且不可或缺之步驟。因設計或構建一複雜外型之物件，必是用多個平面、曲面或任意曲面構建物件之幾何模組，所以必需先求出面與面之相交曲線後，再將多餘部份曲面消去，才能得到想要之幾何外型。本研究探討自由曲面相交之對象是RBS曲面相交之問題，RBS曲面(Rational B-Spline Surfaces)在計算幾何學，電腦圖學及CAD/CAM等系統中，能表示最廣泛之自由曲面及具有多種修改曲面圖形因子。本文主要是採用前進法求解此交線問題。先採用方盒法（或稱邊界箱），配合細分RBS曲面定義域之參數平面求各曲面點；然後建方盒以判斷RBS曲面是否相交；若相交則利用牛頓疊代法及三點共圓收斂法求取起始點；前進追蹤交線其他交點；最後再做交線精緻化，以求得精準之RBS曲面交線。

關鍵字

曲面交線； RBS曲面；幾何模型

並列摘要

Finding intersection curves of surfaces not only plays an important role in computational geometry, computer graphics, and modern CAD/CAM systems, but also has wide applications in industries. The geometric model of an complicated object is always constructed and assembled from multiple surfaces, including free form surfaces and planes. The excessive portions of the surfaces are removed or trimmed after locating the intersection curves/lines. Therefore, finding such intersection curves of the surfaces is a must step in rendering geometry of objects and for subsequent processing. This paper is concentrated on the intersection problems of rational B-Spline surfaces. Rational B-Spline surfaces are the unified and most generalized surfaces rendering techniques. They possess flexibility of local modification and are suitable for interactive computer graphics applications. The intersection curve solver of this paper is based on marching methods basically. A boxing method combined with a modified subdivision technique is used to evaluate whether two rational B-Spline surfaces are intersected. Then, a starting point is obtained by the Newton's method or a spatial three-point method. Subsequent nodes on the intersection curve of two surfaces are found by a marching algorithm. A curve refinement procedure is implemented to improve the accuracy of the intersection curves.

並列關鍵字

surface intersection ； rational B-spline surfaces ； geometric modeling

國際替代計量

Rational B-Spline曲面之交線技術研究

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