# Item

ITEM ACTIONSEXPORT

Released

Report

#### Proximity in arrangements of algebraic sets

##### Locator

There are no locators available

##### Fulltext (public)

1996-1-003

(Any fulltext), 10KB

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

Rieger, J.(1996). *Proximity in arrangements of algebraic sets*
(MPI-I-1996-1-003). Saarbrücken: Max-Planck-Institut für Informatik.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0014-A1A7-9

##### Abstract

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 $X_i$.