View source for Parallel lines

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''parallel curves''

Diffeomorphic smooth curves in space having parallel tangents at corresponding points. Such are, for example, the smooth components of equi-distant lines on a plane (see [[Equi-distant|Equi-distant]]) — they are characterized by the fact that the distance between corresponding points is equal to that between corresponding tangents. An example of parallel curves in three-dimensional space: If two surfaces are in [[Peterson correspondence|Peterson correspondence]] and have a common conjugate net, then the lines of this net have parallel tangents. Parallel curves in $E^m$ having parallel normals up to order $m<n$ are situated in a certain subspace $E^{n-m}$.

For a linear family of planar convex parallel curves (that is, convex curves whose position vector depends linearly on a parameter $\epsilon$) the Brunn–Minkowski theorem holds: The square root of the area of the domain bounded by them is a concave function of $\epsilon$.

A generalization of the concept of parallelism to the case of lines situated in Lie groups is obtained by means of the concept of equi-pollent vectors.



====Comments====
For a linear family of planar convex parallel curves there holds the Steiner formula: The area of the domain bounded by them is a polynomial of degree 2 in $\epsilon$. From this follows as a special case the Brunn–Minkowski theorem mentioned above.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.P. Do Carmo,   "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Schneider,   "Valuations on convex bodies"  P.M. Gruber (ed.)  J.M. Wills (ed.) , ''Convexity and its applications'' , Birkhäuser  (1983)  pp. 170–247</TD></TR></table>