# Applications of Gröbner Bases in Non-Linear Computational Geometry

## Abstract

*Gröbner* *bases* are certain finite sets of multivariate polynomials. Many problems in polynomial ideal theory (algebraic geometry, non-linear computational geometry) can be solved by easy algorithms after transforming the polynomial sets involved in the specification of the problems into Gröbner basis form. In this paper we give some examples of applying the Gröbner bases method to problems in non-linear computational geometry (inverse kinematics in robot programming, collision detection for superellipsoids, implicitization of parametric representations of curves and surfaces, inversion problem for parametric representations, automated geometrical theorem proving, primary decomposition of implicitly defined geometrical objects). The paper starts with a brief summary of the Gröbner bases method.

## Keywords

Prime Ideal Parametric Representation Common Zero Implicit Equation Primary Decomposition## Preview

Unable to display preview. Download preview PDF.

## References

- D. S. Arnon , T. W. Sederberg, 1984.
*Implicit Equation for a Parametric Surface by Gröbner Bases*. In: Proceedings of the 1984 MACSYMA User’s Conference (V. E. Golden ed.), General Electric, Schenectady, New York, 431 - 436.Google Scholar - A. H. Barr, 1981.
*Superquadrics and Angle-Preserving Transformations*. IEEE Computer Graphics and Applications, 1/1, 11–23.CrossRefGoogle Scholar - B. Buchberger, 1965.
*An Algorithm for Finding a Basis for the Residue Class Ring of a Zero-Dimensional Polynomial Ideal (German)*. Ph. D. Thesis, Univ. of Innsbruck ( Austria ), Dept. of Mathematics.Google Scholar - B. Buchberger, 1970.
*An Algorithmic Criterion for the Solvability of Algebraic Systems of Equations (German)*. Aequationes Mathematicae**4**/3, 374–383.MathSciNetCrossRefMATHGoogle Scholar - B. Buchberger, G. E. Collins, R. Loos, 1982. “Computer Algebra: Symbolic and Algebraic Computation”. Springer-Verlag, Vienna - New York.MATHGoogle Scholar
- B. Buchberger, 1985.
*Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory*. In: Multidimensional Systems Theory (N. K. Bose ed.), D. Reidel Publishing Company, Dordrecht–Boston–Lancaster, 184–232.Google Scholar - G. E. Collins, 1975.
*Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition*. 2nd GI Conference on Automata Theory and Formal Languages, Lecture Notes in Computer Science**33**, 134–183.Google Scholar - P. Gianni, 1987.
*Properties of Gröbner Bases Under Specialization*. Proc. of the EUROCAL ‘87 Conference, Leipzig, 2–5 June 1987, to appear.Google Scholar - P. Gianni, B. Trager., G. Zacharias, 1985.
*Gröbner Bases and Primary Decomposition of Polynomial Ideals*. Submitted to J. of Symbolic Computation. Available as manuscript, IBM T. J. Watson Research Center, Yorktown Heights, New York.Google Scholar - C. Hofmann, 1987.
*Algebraic Curves*. This Volume. Institute for Mathematics and its Applications, U of Minneapolis.Google Scholar - Hofmann, 1987a. Personal Communication. Purdue University, West Lafayette, IN 47907, Computer Science Dept.Google Scholar
- M. Kalkbrener, 1987.
*Solving Systems of Algebraic Equations by Using Gröbner Bases*. Proc. of the EUROCAL ‘87 Conference, Leipzig, 2–5 June 1987, to appear.Google Scholar - Kapur, 1986.
*A Refutational Approach to Geometry Theorem Proving*. In: Proceedings of the Workshop on Geometric Reasoning, Oxford University, June 30 - July 3, 1986, to appear in*Artificial Intelligence*.Google Scholar - D. Kapur, 1987.
*Algebraic Reasoning for Object Construction from Ideal Images*. Lecture Notes, Summer Program on Robotics: Computational Issues in Geometry, August 24–28, Institute for Mathematics and its Applications, Univ. of Minneapolis.Google Scholar - Kandri-Rody, 1984.
*Effective Methods in the Theory of Polynomial Ideals*. Ph. D. Thesis, Rensselaer Polytechnic Institute, Troy, New York, Dept. of Computer Science.Google Scholar - H. Kredel, 1987.
*Primary Ideal Decomposition*. Proc of the EUROCAL ‘87 Conference, Leipzig, 2–5 June 1987, to appear.Google Scholar - Kutzler, 1987.
*Implementation of a Geometry Proving Package in SCRATCH-PAD II*. Proceedings of the EUROCAL ‘87 Conferenc, Leipzig, 2–5 June, 1987, to appear.Google Scholar - B. Kutzler, S. Stifter, 1986.
*On the Application of Buchberger’s Algorithm to Automated Geometry Theorem Proving*. J. of Symbolic Computation,**2**/4, 389–398.MathSciNetCrossRefMATHGoogle Scholar - D. Lazard, 1985.
*Ideal Bases and Primary Decomposition: Case of Two Variables*. J. of Symbolic Computation**1**/3, 261–270.MathSciNetCrossRefMATHGoogle Scholar - R. P. Paul, 1981. “ Robot Manipulators: Mathematics, Programming, and Control”. The MIT Press, Cambridge (Mass.), London.Google Scholar
- F. P. Preparata, M. I. Shamos, 1985. “Computational Geometry”. Springer-Verlag, New York, Berlin, Heidelberg.Google Scholar
- T. W. Sederberg, D. C. Anderson, 1984.
*Implicit Representation of Parametric Curves and Surfaces*. Computer Vision, Graphics, and Image Processing**28**, 72–84.CrossRefGoogle Scholar - D. Spear, 1977.
*A Constructive Approach to Ring Theory*. Proc. of the MACSYMA Users’ Conference, Berkeley, July 1977 (R. J. Fateman ed.), The MIT Press, 369–376.Google Scholar - B. Sturmfels, 1987. Private Communication. Institute for Mathematics and its Applications.Google Scholar
- W. Trinks, 1978.
*On B. Buchberger’s Method for Solving Systems of Algebraic Equations (German)*. J. of Number Theory**10**/4, 475–488.Google Scholar - Van Den Essen, 1986. A
*Criterion to Decide if a Polynomial Map is Invertible and to Compute the Inverse*. Report 8653, Catholic University Nijmegen (The Netherlands), Dept. of Mathematics.Google Scholar - L. Van Der Waerden, 1953. “Modern Algebra I, II”, Frederick Ungar Publ. Comp., New York.Google Scholar
- F. Winkler, 1986.
*Solution of Equations I: Polynomial Ideals and Gröbner Bases*. Proc. of the Conference on Computers and Mathematics, Stanford University, July 30 - August 1, 1986, to appear.Google Scholar - W. T. Wu, 1978.
*On the Decision Problem and the Mechanization of Theorem Proving in Elementary Geometry*. Scientia Sinica**21**, 150–172.Google Scholar