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Difference between revisions of "Parallel lines"

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''parallel curves''
 
''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" />.
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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 <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" />.
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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.
 
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.
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====Comments====
 
====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.
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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====
 
====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>
 
<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>

Latest revision as of 14:14, 29 April 2014

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
How to Cite This Entry:
Parallel lines. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parallel_lines&oldid=31985
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article