Solving Dynamic Geometric Constraints Involving Inequalities

• Hoon Hong
• Liyun Li
• Tielin Liang
• Dongming Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4120)

Abstract

This paper presents a specialized method for solving dynamic geometric constraints involving equalities and inequalities. The method works by decomposing the system of constraints into finitely many explicit solution representations in terms of parameters with radicals using triangular decomposition and real quantifier elimination. For any given values of the parameters, if they verify some set of computed relations, the values of the dependent variables may be easily computed by direct evaluation of the corresponding explicit expressions. The effectiveness of our method and its experimental implementation is illustrated by some examples of diagram generation.

Keywords

Inequality Constraint Real Solution Geometric Constraint Geometric Object Radical Expression
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. 1.
2. 2.
Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput. 12, 299–328 (1991)
3. 3.
Dolzmann, A., Sturm, T., Weispfenning, V.: A new approach for automatic theorem proving in real geometry. J. Automat. Reason. 21(3), 357–380 (1998)
4. 4.
González-Vega, L.: A combinatorial algorithm solving some quantifier elimination problems. In: Caviness, B., Johnson, J. (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition, pp. 300–316. Springer, Wien (1996)Google Scholar
5. 5.
Hong, H.: Quantifier elimination for formulas constrained by quadratic equations via slope resultants. The Computer J. 36(5), 440–449 (1993)Google Scholar
6. 6.
Joan-Arinyo, R., Hoffmann, C.M.: A brief on constraint solving (2005), http://www.cs.purdue.edu/homes/cmh/distribution/papers/Constraints/ThailandFull.pdf
7. 7.
Kim, D., Kim, D.-S., Sugihara, K.: Apollonius tenth problem via radius adjustment and Möbius transformations. Computer-Aided Design 38(1), 14–21 (2006)
8. 8.
Lewis, R.H., Bridgett, S.: Conic tangency equations and Apollonius problems in biochemistry and pharmacology. Math. Comput. Simul. 61(2), 101–114 (2003)
9. 9.
Wang, D.: Elimination Methods. Springer, Wien (2001)
10. 10.
Wang, D.: Automated generation of diagrams with Maple and Java. In: Joswig, M., Takayama, N. (eds.) Algebra, Geometry, and Software Systems, pp. 277–287. Springer, Heidelberg (2003)Google Scholar
11. 11.
Wang, D.: GEOTHER 1.1: Handling and proving geometric theorems automatically. In: Winkler, F. (ed.) ADG 2002. LNCS (LNAI), vol. 2930, pp. 194–215. Springer, Heidelberg (2004)
12. 12.
Wang, D.: Elimination Practice: Software Tools and Applications. Imperial College Press, London (2004)
13. 13.
Weispfenning, V.: Quantifier elimination for real algebra — the cubic case. In: Proceedings of the 1994 International Symposium on Symbolic and Algebraic Computation, Oxford, UK, July 20–22, 1994, pp. 258–263. ACM Press, New York (1994)

Authors and Affiliations

• Hoon Hong
• 1
• Liyun Li
• 2
• Tielin Liang
• 3
• Dongming Wang
• 2
• 4
1. 1.Department of MathematicsNorth Carolina State UniversityRaleighUSA
2. 2.LMIB – School of ScienceBeihang UniversityBeijingChina
3. 3.Department of MathematicsUniversity of Science and Technology of ChinaHefeiChina
4. 4.Laboratoire d’Informatique de Paris 6Université Pierre et Marie Curie – CNRSParisFrance