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研究生:郭裔銘
研究生(外文):Yi-Min Kuo
論文名稱:直線與球體之相切問題
論文名稱(外文):The Common Tangent Lines to Spheres
指導教授:江清水江清水引用關係
指導教授(外文):Ching-Shoei Chiang
學位類別:碩士
校院名稱:東吳大學
系所名稱:資訊科學系
學門:工程學門
學類:電資工程學類
論文種類:學術論文
論文出版年:2005
畢業學年度:93
語文別:中文
論文頁數:35
中文關鍵詞:座標轉換4P1L問題限制型電腦輔助軟體Geometric constraint問題
外文關鍵詞:computer-aided design4P1L problemgeometric constraint solver
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現今有許多的限制型電腦輔助軟體,藉由使用者提供幾何元件,加上給定適當的限制條件,經由電腦的計算得到可能的設計圖樣,供使用者參考使用。在二度空間的Geometric constraint問題因為發展的時間較為長久,因此相關方面的技術較為成熟。在三度空間中,當元件中加入直線之後,將使得整個系統變為複雜,在這方面的問題當中,著名的4P1L問題,因為較少的文獻提出此問題的解決方法,於是,在本文當中,利用數學定理的推導,如座標轉換、兩球體之間的共同切線等定理,設計出解決此問題的演算法,並加以實作,提供日後相關問題的一個解決方法。
Given a set of geometric entities with a set of constraint among them, the geometric constraint solver (GCS) solves the problem by finding construction steps to construct a cluster whose geometric entities satisfied the constraint. The 2D GCS software is well-developed but the 3D GCS software is still in develop because the space line complex the problem. There are many basic problems needed to solve before the software can be developed. These problems include 6P problem, 5P1L problem, 4P1L problem, 3P3L problem, etc. This paper proposed 2 ways to solve the 4P1L problem, that is, finding the lines tangent to 4 spheres in the 3D space
誌謝 i
摘要 ii
Abstract iii
表目錄 v
圖目錄 vi
1. 緒論 1
研究動機 1
研究目的及方法 2
論文架構 2
2. 文獻探討 3
2.1 限制型幾何問題6P(Octahedral Problem): 4
2.2 限制型幾何問題5P1L: 5
2.3 限制型幾何問題3P3L: 5
2.4 限制型幾何問題4P1L: 6
3. 限制幾何4P1L Problem Solved之研究方法 8
3.1 方法一Brute Force演算法及其基本定理 8
3.2 方法二Domain Constraint演算法及基本定理 12
4. 實作結果 17
方法一:Brute Force Algorithm實作 17
方法二:Domain Constraint Algorithm實作 19
5. 結論及後續研究 23
6. References 24
附錄 26
空間中點的平移以及旋轉 26
[1]Christoph M. Hoffmann and Ching-Shoei Chiang, “Variable-Radius Circles in Cluster Merging, Part I: Translational Clusters”, Computer-Aided Design 34(11), Pages 787-797, 2002. (SCI)。國科會補助: NSC-39201F
[2]Christoph M. Hoffmann and Ching-Shoei Chiang, “Variable-Radius Circles in Cluster Merging, Part II, Rotational Clusters”, Computer-Aided Design 34(11), Pages 799-805, 2002. (SCI)。國科會補助: NSC-39201F
[3]Ching-Shoei Chiang and Robert Joan-Arinyo, “Revisiting variable radius circles in Constructive Geometric Constraint Solving”, Computer Aided Geometric Design vol.21 Issue 4, Pages: 371 - 399, April 2004.
[4]Fudos 1., “Constraint Solving for Computer Aided Design”, Purdue University, Ph.D. thesis, 1995.
[5]Muirhead, F.R. , “On the number and nature of the solutions of the Apollonian contact problem”, Proceedings of the Edinburgh Mathematical Society 14, p. 135-147, 1896.
[6]Bruen, A., Fisher, J.C., Wilker, J.B., “Apollonius by Inversion”, Mathematics Magazine, Vol. 56, No.2, 1983.
[7]Bouma, W., Fudos, I., Hoffmann, C.M., Cai , J. and Paige, R., “A Geometric Constraint Solver”, Computer-Aided Design 27(6), 487-501, 1995.
[8]Allner, J., Krasauskas, R. and Pottmann, H. , “Error propagation in geometric constructions”, Computer-Aided Design, Vol 32, No. 11, p631-641, 2000.
[9]Hoffmann C.M. and Vermeer, P., “Geometric constraint solving in R2 and R3”, Computing in Euclidean Geometry, World Scientific Publishing, 1995.
[10]Cassiano Durand and C.M. Hoffmann, “A systematic framework for Solving Geometric Constraints Analytically”, Journal Symbolic Computation, Vol. 30, p493-519, 2000.
[11]Trott, M. “Apollonius Spheres”, Mathematica in Education and Research, Vol.6, No. 1, p.21-23, 1997.
[12]Sottile, F. and Theobald, T., “Lines tangent to 2n-2 spheres in R3”, Trans. Amer. Math. Soc. 354:4815-4829, 2002.
[13]Hoffmann C.M. and Yuan, Bo, “On Spatial Constraint Solving Approaches”, Workshop for Automatic Deductions in Geometry, ETH Zurich, 2000.
[14]C. Durand. “Symbolic and Numerical Techniques for Constraint Solving”, Purdue University, Dept. of Comp. SCI., PhD thesis, 1998.
[15]Ching-Shoei Chiang, “The Geometric Constraint Solving Problem Using Cyclographic Maps,” Technique Report Soochow University Taipei, Taiwan, ROC, Computer and Information Science, 2001.
[16]http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html
[17]Edward Angel, Interactive Computer Graphics. Page 156 ~ 160, third edition, Addison Wesley.
[18]Christoph M. Hoffmann and Bo Yuan, “There 12 Common Tangents to Four Spheres.” Computer Science Department Purdue University West Lafayette, in 47907 USA, October 18, 2000.
[19]I. G. Macdonald, J. Pach and T. Theobald, “Common Tangents to Four Unit Balls in R3”, Discrete Comput. Geom., 26:1-17, 2001.
[20]I. Fudos and C.M. Hoffmann. “A graph-constructive approach to solving systems of geometric constraints.”, ACM Trans on Graphics, 16:179-216, 1997.
[21]C.M. Hoffmann, A. Lomonosov, and M. Sitharam. “Geometric constraint decomposition. In Geometric Constraint Solving and applications”, pages 170-195, New York, Springer Verlag, 1998.
[22]Christoph M. Hoffmann, Ching-Shoei Chiang, Bo Yuan. “Elementary Constructions in Spatial Constraint Solving.” Computer Science, Purdue University.
[23]Christoph M. Hoffmann and Cassiano Durand. “Variational Constraints in 3D.” Proc. Intl Conf on Shape Modeling and Appl, Aizu, Japan, 1999; 90-97.
[24]Paul Bourke, http://astronomy.swin.edu.au/~pbourke/geometry/2circle/, 1997.
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