Equi-distant

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.