EUROCAL 1985: EUROCAL '85 pp 246-261

# 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)

## Abstract

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.

## Keywords

Boundary Point Limit Point Primitive Element Boundary Property Positive Degree

## References

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
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## Authors and Affiliations

• Dennis S. Arnon
• 1
• George E. Collins
• 2
• Scott McCallum
• 3
1. 1.Xerox PARCPalo Alto