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

  • Authors:

  • Dennis S. Arnon;George E. Collins;Scott McCallum

  • Affiliations:

  • Xerox PARC, 3333 Coyote Hill Road, Palo Alto, California 94304, U.S.A.;Department of Computer and Information Science, The Ohio State University, Columbus, Ohio 43210 U.S.A.;Research School of Physical Science, Australian National University, Canberra ACT 2601, Australia

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 1988

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Abstract

Let A @? Z [x"1, ..., x"r] be a finite set. An A-invariant cylindrical algebraic decomposition (cad) is a certain partition of r-dimenslonal euclidean space E^r into semi-algebraic cells such that the value of each A"i @? A has constant sign (positive, negative, or zero) throughout each cell. Two cells are adjacent if their union is connected. We give an algorithm that determines the adjacent pairs of cells as it constructs a cad of E^3. The general teehnlque employed for E^3 adjacency determination is ''projection'' into E^2, followed by application of an existing E^2 adjacency elgorlthm (Arnon, Collins, McCallum, 1984). Our algorithm has the following properties: (1) it requires no coordinate changes, end (2) in any cad of E^1, E^2, or E^3 that it builds, the boundary of each cell is a (disjoint) union of lower-dlmenaionel cells.