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. 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).


[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)
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Conical net. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article