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Proximity in arrangements of algebraic sets

Rieger, Joachim

February 1996, 25 pages.

Status: available - back from printing

Let $X$ be an arrangement of $n$ algebraic sets $X_i$ in $d$-space, where the $X_i$ are either parameterized or zero-sets of dimension $0\le m_i\le d-1$. We study a number of decompositions of $d$-space into connected regions in which the distance-squared function to $X$ has certain invariances. These decompositions can be used in the following of proximity problems: given some point, find the $k$ nearest sets $X_i$ in the arrangement, find the nearest point in $X$ or (assuming that $X$ is compact) find the farthest point in $X$ and hence the smallest enclosing $(d-1)$-sphere. We give bounds on the complexity of the decompositions in terms of $n$, $d$, and the degrees and dimensions of the algebraic sets $X_i$.

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Hide details for BibTeXBibTeX
  AUTHOR = {Rieger, Joachim},
  TITLE = {Proximity in arrangements of algebraic sets},
  TYPE = {Research Report},
  INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},
  ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},
  NUMBER = {MPI-I-96-1-003},
  MONTH = {February},
  YEAR = {1996},
  ISSN = {0946-011X},