An adjacency algorithm for cylindrical algebraic decompositions of three-dimensional space
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.
KeywordsBoundary Point Limit Point Primitive Element Boundary Property Positive Degree
Unable to display preview. Download preview PDF.
- 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. [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. [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. [COL71]G. E. Collins, ”The calculation of multivariate polynomial resultants”, J. Assoc. Comp. Mach. 18, 4, (1971), 515–532.Google Scholar
- 5. [MAS78]W. S. Massey, Homology and cohomology theory, Marcel Dekker, New York, 1978.Google Scholar
- 6. [SCH83b]
- 7. [WAL51]R. J. Walker, Algebraic Curves, Princeton University Press, 1951.Google Scholar