View source for Equi-distant

''of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e0359801.png" /> in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e0359802.png" />''

The boundary of the [[Tubular neighbourhood|tubular neighbourhood]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e0359803.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e0359804.png" /> consisting of the balls of the same radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e0359805.png" /> with centres in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e0359806.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e0359807.png" /> is a differentiable submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e0359808.png" /> in a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e0359809.png" />, then the equi-distant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e03598010.png" /> is given (in a more restricted sense) by the set of end-points of the segments of equal length measured from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e03598011.png" /> on the geodesics perpendicular to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e03598012.png" /> at the corresponding points. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e03598013.png" /> is complete, then the equi-distant is the image under the [[Exponential mapping|exponential mapping]] of the vectors of constant length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e03598014.png" /> in the normal bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e03598015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e03598016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e03598017.png" /> is not complete, then the equi-distant exists only for sufficiently small values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035980/e03598018.png" />.

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.