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Difference between revisions of "Conical net"

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A conjugate net on a surface in three-dimensional space formed by the conical lines of the surface, that is, by the tangent lines to the cones circumscribed about the surface. A surface carrying a conical net is called a [[Peterson surface|Peterson surface]]. A [[Transport net|transport net]] can be considered as a special case of a conical net. The vertices of the cones of a conical net are positioned on two curves. Conical nets depend on at most four parameters. Only non-degenerate surfaces of the second order can carry a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024970/c0249701.png" />-parameter family of conical nets. A surface carrying a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024970/c0249702.png" />-parameter family of conical nets is a ruled surface belonging to a linear congruence (that is, a congruence formed by the common secants of two straight lines).
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A [[conjugate net]] on a surface in three-dimensional space formed by the conical lines of the surface, that is, by the tangent lines to the cones circumscribed about the surface. A surface carrying a conical net is called a [[Peterson surface]]. A [[transport net]] can be considered as a special case of a conical net. The vertices of the cones of a conical net are positioned on two curves. Conical nets depend on at most four parameters. Only non-degenerate surfaces of the second order can carry a $4$-parameter family of conical nets. A surface carrying a $2$-parameter family of conical nets is a [[ruled surface]] belonging to a linear congruence (that is, a [[congruence of lines]] which are the common secants of two straight lines).
  
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Ya.P. Blank,  "A certain generalization of S. Lie's problem on translation surfaces"  ''Trudy Geom. Sem. Inst. Nauchn. Inform. Akad. Nauk SSSR'' , '''3'''  (1971)  pp. 5–27  (In Russian)  (English summary)</TD></TR></table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  Ya.P. Blank,  "A certain generalization of S. Lie's problem on translation surfaces"  ''Trudy Geom. Sem. Inst. Nauchn. Inform. Akad. Nauk SSSR'' , '''3'''  (1971)  pp. 5–27  (In Russian)  (English summary)</TD></TR>
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Latest revision as of 18:55, 16 April 2018

A conjugate net on a surface in three-dimensional space formed by the conical lines of the surface, that is, by the tangent lines to the cones circumscribed about the surface. A surface carrying a conical net is called a Peterson surface. A transport net can be considered as a special case of a conical net. The vertices of the cones of a conical net are positioned on two curves. Conical nets depend on at most four parameters. Only non-degenerate surfaces of the second order can carry a $4$-parameter family of conical nets. A surface carrying a $2$-parameter family of conical nets is a ruled surface belonging to a linear congruence (that is, a congruence of lines which are the common secants of two straight lines).

References

[1] Ya.P. Blank, "A certain generalization of S. Lie's problem on translation surfaces" Trudy Geom. Sem. Inst. Nauchn. Inform. Akad. Nauk SSSR , 3 (1971) pp. 5–27 (In Russian) (English summary)
How to Cite This Entry:
Conical net. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conical_net&oldid=16604
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article