Automatic Verification of the Adequacy of Models for Families of Geometric Objects

  • Aless Lasaruk
  • Thomas Sturm
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6301)


We consider parametric families of semi-algebraic geometric objects, each implicitly defined by a first-order formula. Given an unambiguous description of such an object family and an intended alternative description we automatically construct a first-order formula which is true if and only if our alternative description uniquely describes geometric objects of the reference description. We can decide this formula by applying real quantifier elimination. In the positive case we furthermore derive the defining first-order formulas corresponding to our new description. In the negative case we can produce sample points establishing a counterexample for the uniqueness. We demonstrate our method by automatically proving uniqueness theorems for characterizations of several geometric primitives and simple complex objects. Finally, we focus on tori, characterizations of which can be applied in spline approximation theory with toric segments. Although we cannot yet practically solve the fundamental open questions in this area within reasonable time and space, we demonstrate that they can be formulated in our framework. In addition this points at an interesting and practically relevant challenge problem for automated deduction in geometry in general.


Real Geometry Unique Representation Automated Proving Real Quantifier Elimination 


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  1. 1.
    Chou, S.-C.: Mechanical Geometry Theorem Proving. Mathematics and its applications. D. Reidel Publishing Company, Dordrecht (1988)MATHGoogle Scholar
  2. 2.
    Davenport, J.H., Heintz, J.: Real quantifier elimination is doubly exponential. Journal of Symbolic Computation 5(1-2), 29–35 (1988)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Dolzmann, A., Seidl, A., Sturm, T.: Efficient projection orders for CAD. In: Gutierrez, J. (ed.) Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation (ISSAC 2004), pp. 111–118. ACM Press, New York (2004)CrossRefGoogle Scholar
  4. 4.
    Dolzmann, A., Sturm, T.: Redlog: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31(2), 2–9 (1997)CrossRefGoogle Scholar
  5. 5.
    Dolzmann, A., Sturm, T.: Simplification of quantifier-free formulae over ordered fields. Journal of Symbolic Computation 24(2), 209–231 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dolzmann, A., Sturm, T., Weispfenning, V.: A new approach for automatic theorem proving in real geometry. Journal of Automated Reasoning 21(3), 357–380 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dolzmann, A., Sturm, T., Weispfenning, V.: Real quantifier elimination in practice. In: Matzat, B.H., Greuel, G.-M., Hiss, G. (eds.) Algorithmic Algebra and Number Theory, pp. 221–247. Springer, Berlin (1998)Google Scholar
  8. 8.
    Jüttler, B., Sampoli, M.L.: Hermite interpolation by piecewise polynomial surfaces with rational offsets. Computer-Aided Geometric Design 17, 361–385 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Loos, R., Weispfenning, V.: Applying linear quantifier elimination. The Computer Journal 36(5), 450–462 (1993)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Martin, R.R.: Principal patches. a new class of surface patch based on differential geometry. In: Eurographics 1983, pp. 47–55. North Holland, Amsterdam (1984)Google Scholar
  11. 11.
    Mäurer, C., Krasauskas, R.: Joining cyclide patches along quartic boundary curves. In: Dæhlen, M., Lyche, T., Schumaker, L.L. (eds.) Proceedings of the International Conference on Mathematical Methods for Curves and Surfaces II, Lillehammer, pp. 359–366. Vanderbilt University, Nashville (1998)Google Scholar
  12. 12.
    Pratt, M.J.: Cyclides in computer-aided geometric design. Computer-Aided Geometric Design 7, 221–242 (1990)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Prautzsch, H., Boehm, W., Paluszny, M.: Bézier and B-Spline Techniques. Springer, Berlin (2002)CrossRefMATHGoogle Scholar
  14. 14.
    Schöne, R.: Torische Splines. Doctoral dissertation, Department of Computer Science and Mathematics. University of Passau, Germany, D-94030 Passau, Germany (2007)Google Scholar
  15. 15.
    Schöne, R., Hintermann, D., Hanning, T.: Approximation of shrinked aspheres. In: Gregory, G.G., Howard, J.M., Koshel, R.J. (eds.) International Optical Design Conference 2006 (Proceedings of SPIE-OSA). Proceedings of SPIE, vol. 6342. SPIE, Bellingham (2006)Google Scholar
  16. 16.
    Srinivas, Y.L., Kumar, V., Dutta, D.: Surface design using cyclide patches. Computer-Aided Design 28, 263–276 (1996)CrossRefGoogle Scholar
  17. 17.
    Sturm, T.: Real Quantifier Elimination in Geometry. Doctoral dissertation, Department of Mathematics and Computer Science. University of Passau, Germany, D-94030 Passau, Germany (December 1999)Google Scholar
  18. 18.
    Sturm, T.: Reasoning over networks by symbolic methods. Applicable Algebra in Engineering, Communication and Computing 10(1), 79–96 (1999)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Sturm, T., Weispfenning, V.: Computational geometry problems in Redlog. In: Wang, D. (ed.) ADG 1996. LNCS (LNAI), vol. 1360, pp. 58–86. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  20. 20.
    Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5(1-2), 3–27 (1988)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Weispfenning, V.: Quantifier elimination for real algebra—the cubic case. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, Oxford, England (ISSAC 1994), pp. 258–263. ACM Press, New York (1994)Google Scholar
  22. 22.
    Weispfenning, V.: Quantifier elimination for real algebra—the quadratic case and beyond. Applicable Algebra in Engineering Communication and Computing 8(2), 85–101 (1997)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Weispfenning, V.: Simulation and optimization by quantifier elimination. Journal of Symbolic Computation 24(2), 189–208 (1997)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Aless Lasaruk
    • 1
  • Thomas Sturm
    • 2
  1. 1.FORWISSUniversität PassauPassauGermany
  2. 2.Departamento de Matemáticas, Estadística y Computación, Facultad de CienciasUniversidad de CantabriaSantanderSpain

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