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 $\mathbf r_1$ and $\mathbf r_2$ of two parallel surfaces $F_1$ and $F_2$ are connected by a relation $\mathbf r_2-\mathbf r_1=h\mathbf n$, where $\mathbf n$ 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

$$w(h)=1-2Hh+Kh^2>0.$$

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

$$H_h=\frac{H-Kh}{w}(h),\quad K_h=\frac Kw(h);$$

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

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