Parallel surfaces

Diffeomorphic equi-oriented surfaces $F_{1}$ and $F_{2}$ having parallel tangent planes at corresponding points and such that the distance $h$ between corresponding points of $F_{1}$ and $F_{2}$ is constant and equal to that between the corresponding tangent planes. The position vectors $r_{1}$ and $r_{2}$ of two parallel surfaces $F_{1}$ and $F_{2}$ are connected by a relation $r_{2} - r_{1} = h n$ , where $n$ is a unit normal vector that is the same for $F_{1}$ at $r_{1}$ and $F_{2}$ at $r_{2}$ .

Thus, one can define a one-parameter family $F_{h}$ of surfaces parallel to a given $F = F_{0}$ , where $F_{h}$ is regular for sufficiently small values of $h$ for which

$w (h) = 1 - 2 H h + K h^{2} > 0.$

To the values of the roots $h_{1}$ and $h_{2}$ of the equation $w (h) = 0$ there correspond two surfaces $F_{h_{1}}$ and $F_{h_{2}}$ that are evolutes of $F$ , so that parallel surfaces have a common evolute (cf. Evolute (surface)). The mean curvature $H_{h}$ and the Gaussian curvature $K_{h}$ of a surface $F_{h}$ parallel to $F$ are connected with the corresponding quantities $H$ and $K$ of $F$ by the relations

$H_{h} = \frac{H - K h}{w} (h), K_{h} = \frac{K}{w} (h);$

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[edit]

For a linear family of closed convex parallel surfaces (depending linearly on a parameter $ϵ > 0$ ) the Steiner formula holds: The volume of the point set bounded by them is a polynomial of degree 3 in $ϵ$ . 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.