Difference between revisions of "Conical net"
From Encyclopedia of Mathematics
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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 [[ | + | 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==== | ====References==== | ||
| − | <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> | + | <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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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
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