Difference between revisions of "Focal net of a congruence"
From Encyclopedia of Mathematics
(Importing text file) |
m (link) |
||
| Line 1: | Line 1: | ||
| − | A net that is cut out on the focal surface of a congruence of lines (a hyperbolic one) by enveloping surfaces of that congruence. A focal net of a congruence is a [[Conjugate net|conjugate net]]. One family of lines of it consists of the cusps of one family of enveloping surfaces of the congruence; the other family is formed by the lines where the enveloping surfaces of the other family are tangent to the focal surface. Every conjugate net on a two-dimensional surface is a focal net of the congruence of tangents to the lines of one of the families of that net. Every hyperbolic congruence of lines has two focal nets. | + | A net that is cut out on the focal surface of a [[congruence of lines]] (a hyperbolic one) by enveloping surfaces of that congruence. A focal net of a congruence is a [[Conjugate net|conjugate net]]. One family of lines of it consists of the cusps of one family of enveloping surfaces of the congruence; the other family is formed by the lines where the enveloping surfaces of the other family are tangent to the focal surface. Every conjugate net on a two-dimensional surface is a focal net of the congruence of tangents to the lines of one of the families of that net. Every hyperbolic congruence of lines has two focal nets. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.P. Finikov, "Theorie der Kongruenzen" , Akademie Verlag (1959) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.P. Finikov, "Theorie der Kongruenzen" , Akademie Verlag (1959) (Translated from Russian)</TD></TR></table> | ||
Latest revision as of 18:17, 24 December 2020
A net that is cut out on the focal surface of a congruence of lines (a hyperbolic one) by enveloping surfaces of that congruence. A focal net of a congruence is a conjugate net. One family of lines of it consists of the cusps of one family of enveloping surfaces of the congruence; the other family is formed by the lines where the enveloping surfaces of the other family are tangent to the focal surface. Every conjugate net on a two-dimensional surface is a focal net of the congruence of tangents to the lines of one of the families of that net. Every hyperbolic congruence of lines has two focal nets.
References
| [1] | S.P. Finikov, "Theorie der Kongruenzen" , Akademie Verlag (1959) (Translated from Russian) |
How to Cite This Entry:
Focal net of a congruence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Focal_net_of_a_congruence&oldid=51070
Focal net of a congruence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Focal_net_of_a_congruence&oldid=51070
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article