View source for 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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071360/p0713601.png" /> having parallel normals up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071360/p0713602.png" /> are situated in a certain subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071360/p0713603.png" />.

For a linear family of planar convex parallel curves (that is, convex curves whose position vector depends linearly on a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071360/p0713604.png" />) the Brunn–Minkowski theorem holds: The square root of the area of the domain bounded by them is a concave function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071360/p0713605.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071360/p0713606.png" />. 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>