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