Parallel lines

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) — 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 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

[a1]	M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976)
[a2]	R. Schneider, "Valuations on convex bodies" P.M. Gruber (ed.) J.M. Wills (ed.) , Convexity and its applications , Birkhäuser (1983) pp. 170–247