Namespaces
Variants
Actions

Equi-distant

From Encyclopedia of Mathematics
Revision as of 17:04, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

of a set in a metric space

The boundary of the tubular neighbourhood of in consisting of the balls of the same radius with centres in . If is a differentiable submanifold in a Riemannian space , then the equi-distant of is given (in a more restricted sense) by the set of end-points of the segments of equal length measured from on the geodesics perpendicular to at the corresponding points. If is complete, then the equi-distant is the image under the exponential mapping of the vectors of constant length in the normal bundle of in . If is not complete, then the equi-distant exists only for sufficiently small values of .

Examples of equi-distants. 1) An equi-distant in the Lobachevskii plane (a hypercycle) is the orthogonal trajectory of the pencil of straight lines perpendicular to some straight line (to a basic line, or basis). The equi-distant consists of two branches situated on different sides from the basis line and concave towards the basis. The curvature of the equi-distant is constant. 2) An equi-distant in the Lobachevskii space is a surface of constant positive exterior curvature.

How to Cite This Entry:
Equi-distant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equi-distant&oldid=31688
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article