# Parallel surfaces

Diffeomorphic equi-oriented surfaces and having parallel tangent planes at corresponding points and such that the distance between corresponding points of and is constant and equal to that between the corresponding tangent planes. The position vectors and of two parallel surfaces and are connected by a relation , where is a unit normal vector that is the same for at and at .

Thus, one can define a one-parameter family of surfaces parallel to a given , where is regular for sufficiently small values of for which

To the values of the roots and of the equation there correspond two surfaces and that are evolutes of , so that parallel surfaces have a common evolute (cf. Evolute (surface)). The mean curvature and the Gaussian curvature of a surface parallel to are connected with the corresponding quantities and of by the relations

lines of curvature of parallel surfaces correspond to each other, so that between them there is a Combescour correspondence, which is a special case of a Peterson correspondence.

#### Comments

For a linear family of closed convex parallel surfaces (depending linearly on a parameter ) the Steiner formula holds: The volume of the point set bounded by them is a polynomial of degree 3 in . An analogous result holds for arbitrary dimensions. The Steiner formula is a special case of formulas for general polynomials in Minkowski's theory of mixed volumes, and, even more general, in the theory of valuations.

For references see Parallel lines.

**How to Cite This Entry:**

Parallel surfaces.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Parallel_surfaces&oldid=17715