An adjacency algorithm for cylindrical algebraic decompositions of three-dimensional space

  • Dennis S. Arnon
  • George E. Collins
  • Scott McCallum
Algebraic Algorithms III
Part of the Lecture Notes in Computer Science book series (LNCS, volume 204)


Given a set of r-variate integral polynomials, a cylindrical algebraic decomposition (cad) of euclidean r-space Er is a certain partition of Er into connected subsets compatible with the zeros of the polynomials. Each subset is a cell. Two cells of a cad are adjacent if their union is connected. In applications of cad's, one often wishes to know the pairs of adjacent cells. In a previous paper we gave an algorithm which determines the adjacent cells as it constructs a cad of the plane. We give such an algorithm here for three-dimensional space.


Boundary Point Limit Point Primitive Element Boundary Property Positive Degree 
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  1. 1. [ACM84a]
    D. S. Arnon, G. E. Collins, S. McCallum ”Cylindrical algebraic decomposition I: the basic algorithm”, SIAM J. Comp, 13, (1984), pp. 865–877.Google Scholar
  2. 2. [ACM84b]
    D. S. Arnon, G. E. Collins, S. McCallum ”Cylindrical algebraic decomposition II: an adjacency algorithm for the plane”, SIAM J. Comp, 13, (1984), pp. 878–889.Google Scholar
  3. 3. [ARN81]
    D. S. Arnon, ”Algorithms for the geometry of semi-algebraic sets”, Ph.D. Dissertation, Technical Report #436, Computer Sciences Department, University of Wisconsin — Madison, 1981.Google Scholar
  4. 4. [COL71]
    G. E. Collins, ”The calculation of multivariate polynomial resultants”, J. Assoc. Comp. Mach. 18, 4, (1971), 515–532.Google Scholar
  5. 5. [MAS78]
    W. S. Massey, Homology and cohomology theory, Marcel Dekker, New York, 1978.Google Scholar
  6. 6. [SCH83b]
    J. Schwartz and M. Sharir, On the ‘piano movers’ problem II. General techniques for computing topological properties of real algebraic manifolds, Advances in Applied Mathematics, 4 (1983) pp. 298–351.CrossRefGoogle Scholar
  7. 7. [WAL51]
    R. J. Walker, Algebraic Curves, Princeton University Press, 1951.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Dennis S. Arnon
    • 1
  • George E. Collins
    • 2
  • Scott McCallum
    • 3
  1. 1.Xerox PARCPalo Alto
  2. 2.Department of Computer ScienceUniversity of Wisconsin — MadisonMadison
  3. 3.Department of Computer ScienceUniversity of TorontoTorontoCanada

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