Difference between revisions of "Demoulin theorem"
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| + | A helicoid has an infinite number (viz. $ \infty ^ {2} $) | ||
| + | of systems of conjugate nets of lines which are preserved under continuous deformation of this surface — its principal bases (cf. [[Deformation over a principal base|Deformation over a principal base]]). Established by A. Demoulin [[#References|[1]]]. It then turns out that these principal bases are Voss nets (cf. [[Voss net|Voss net]]). Conversely (Finikov's theorem), the only surface with an infinite number of principal bases is a right helicoid [[#References|[2]]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Demoulin, "Sur les systèmes conjugués persistants" ''C.R. Acad. Sci. Paris'' , '''133''' (1901) pp. 986–990</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S.P. Finikov, "Bending and related geometrical problems" , Moscow-Leningrad (1937) (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Demoulin, "Sur les systèmes conjugués persistants" ''C.R. Acad. Sci. Paris'' , '''133''' (1901) pp. 986–990</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S.P. Finikov, "Bending and related geometrical problems" , Moscow-Leningrad (1937) (In Russian)</TD></TR></table> | ||
Latest revision as of 17:32, 5 June 2020
A helicoid has an infinite number (viz. $ \infty ^ {2} $)
of systems of conjugate nets of lines which are preserved under continuous deformation of this surface — its principal bases (cf. Deformation over a principal base). Established by A. Demoulin [1]. It then turns out that these principal bases are Voss nets (cf. Voss net). Conversely (Finikov's theorem), the only surface with an infinite number of principal bases is a right helicoid [2].
References
| [1] | A. Demoulin, "Sur les systèmes conjugués persistants" C.R. Acad. Sci. Paris , 133 (1901) pp. 986–990 |
| [2] | S.P. Finikov, "Bending and related geometrical problems" , Moscow-Leningrad (1937) (In Russian) |
How to Cite This Entry:
Demoulin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Demoulin_theorem&oldid=13871
Demoulin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Demoulin_theorem&oldid=13871
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article