# MPI-I-96-1-003

## Proximity in arrangements of algebraic sets

### Rieger, Joachim

**MPI-I-96-1-003**. February** **1996, 25 pages. | Status:** **available - back from printing | Next --> Entry | Previous <-- Entry

Abstract in LaTeX format:

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 $X_i$.

Acknowledgement:** **

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**URL to this document: **http://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1996-1-003

**BibTeX**
`@TECHREPORT{``Rieger93``,`

` 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},`

`}`