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 作一圓與已知三個圓相切是著名的阿波羅尼奧斯(problem of Apollonius)問題，Cyclographic maps是求解方法中可以求得全部解的其中一種。對平面上的曲線來說，其Cyclographic maps是一個直紋曲面，我們以γ-maps稱之。直線或圓的γ-maps分別是平面與圓錐，PH曲線(Pythagorean Hodograph curves)擁有參數形式(parametric form)的γ-maps源自於PH曲線擁有多項式形式(polynomial form)的切線向量(tangent)與垂直向量(normal)。此篇論文利用PH曲線的Cyclographic maps來求得PH曲線的r-offset，以及一PH曲線與另一直線或圓的中軸轉換(Medial Axis Transform)。我們將等距曲線以及中軸轉換問題轉化成直紋曲面與直紋曲面相交(Ruled Surface/ruled surface intersection)的問題，而其交線將以有理貝茲曲線(rational Bézier curve)的形式展現。
 Cyclographic maps is an important tool in Laguerre geometry to solve the problem of Apollonius. The cyclographic maps for a planar boundary curve is a ruled surface, and we call it γ-maps. The γ-maps for line or circle are plane and cone respectively. PH curve is the free form curve whose tangent and normal are polynomial, so that its cyclographic maps has the parametric form.This paper concerns the r-offset of a PH curve and the medial axis transform (MAT) of a PH curve with a line or a circle. We switch the offsets finding problem and the MAT finding problem into ruled surface/ruled surface intersection problem. The intersection curve is represented by rational Bézier form.
 誌謝 i中文摘要 iiAbstract iii目錄 iv表目錄 v圖目錄 vi1. 緒論 12. 定義與定理 22.1 直紋曲面 (ruled surface) 22.2 平面PH曲線 22.3 Cyclographic maps 42.4 中軸轉換(Medial Axis Transform) 73. PH曲線的��-maps 113.1 PH曲線的r-offset、PH曲線與射線的MAT 123.2 PH曲線與有向圓的MAT 224. 實作範例 295. 結論 376. 參考資料 38
 [1]Ching-Shoei Chiang, Sheng-Hsin Tsai, James Chen, “The Control Vector Scheme for Design of Planar Primitive PH curves” International Journal of Mathematics Sciences Volume 1 Number 4 2007[2]Ching-Shoei Chiang, “Exact representation for the intersection of ruling surface with a plane” The 3rd International Conference on Computer Science & Education KaiFeng, China, 2008[3]Ching-Shoei Chiang, S.-Y. Lin, “The Cyclographic Maps for Bezier Curve” The 35th International Conference on Computers and Industrial Engineering, Vol.1, pp.453-458, Istanbul, TURKEY.June 19-22, 2005[4]N. M. Patrikalakis, T.Maekawa, K.H.Ko and H.Mukundan “Surface to Surface Intersections” CAD’04 http://www.cadconferences.com/[5]Rida T. Farouki, T. Sakkalis, “Pythagorean Hodographs” IBM J. RES. DEVELOP. VOL 34, No. 5, September 1990.[6]Rida T. Farouki, V.T. Rajan, “Bernstein algorithms for polynomials in Bernstein form” Computer Aided Geometric Design 5 (1988) 1-26[7]Rida T. Farouki, “Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable” ISBN: 978-3-540-73397-3, Springer-Verlag Berlin Heidelberg New York, 2008.[8]Helmut Pottmann, Johannes Wallner, “Computational Line Geometry” ISBN:3-540-42058-4, Springer-Verlag Berlin Heidelberg New York, 2001.[9]Farin, Gerald E, “Curves and surfaces for computer aided geometric design : a practical guide” ISBN: 0122490525, Boston : Academic Press, c1993.[10]E. Muller and J. Krames. Die Zyklographie. Franz Deuticke, Leipzig und Wien, 1929.[11]Christoph M. Hoffmann and George Vanecek, Jr. “Fundamental techniques for geometric and solid modeling.” In C.T. Leondes, editor, Advances in Control and Dynamics. Academic Press, 1991.[12]Christoph M. Hoffmann “Computer Vision, Descriptive Geometry and Classical Mechanics” in Computer Graphics and Mathematics, B. Falcidieno, I. Hermann and C. Pienovi, eds, Springer Verlag, Eurographics Series, 1992, 229-244.[13]Ching-Shoei Chiang “The Euclidean Distance Transform” Thesis of Purdue University, 1992.[14]George B. Thomas, Jr. “Calculus”, Pearson Addison Wesley, ISBN 0-321-24335-8.[15]Weisstein, Eric W. "Director Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DirectorCurve.html.[16]Chritoph M. Hoffmann, Ching-Shoei Chiang, Robert E. Lynch “how to compute offsets without self-intersection” In Proceedings SPIE Conference on Curves and Surfaces in Computer Vision and Graphics, November 1991[17]Ching-Shoei Chiang, Sheng-Hsin Tsai “The Control Points Scheme for Design of Plannar Primitive Pythagorean Hodograph Curves”, Computer Graphics Workshop, Chiayi,Taiwan, R.O.C. 2008,[18]Ching-Shoei Chiang, Tung-Tse Wu “The Application of the PH curve’s Cyclographic maps” , Computer Graphics Workshop, Chiayi,Taiwan, R.O.C. 2008,
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