Difference between revisions of "Pseudo-sphere"
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− | A surface of constant negative curvature formed by rotating a [[Tractrix|tractrix]] ( | + | <!-- |
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+ | $#C+1 = 6 : ~/encyclopedia/old_files/data/P075/P.0705840 Pseudo\AAhsphere | ||
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+ | A surface of constant negative curvature formed by rotating a [[Tractrix|tractrix]] ( $ x = u - \mathop{\rm tanh} u $, | ||
+ | $ y = \mathop{\rm sech} u $) | ||
+ | around its asymptote ( $ y= 0 $; | ||
+ | see Fig.). | ||
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p075840a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p075840a.gif" /> | ||
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The line element in [[Semi-geodesic coordinates|semi-geodesic coordinates]] has the form: | The line element in [[Semi-geodesic coordinates|semi-geodesic coordinates]] has the form: | ||
− | + | $$ | |
+ | d s ^ {2} = d u ^ {2} + \cosh ^ {2} | ||
+ | \frac{u}{a} | ||
+ | \ | ||
+ | d v ^ {2} ,\ a = \textrm{ const } | ||
+ | $$ | ||
− | (the line | + | (the line $ u = 0 $ |
+ | is a geodesic); while in [[Isothermal coordinates|isothermal coordinates]] it has the form: | ||
− | + | $$ | |
+ | d s ^ {2} = a ^ {2} | ||
+ | |||
+ | \frac{d x ^ {2} + d y ^ {2} }{y ^ {2} } | ||
+ | ,\ \ | ||
+ | a = \textrm{ const } . | ||
+ | $$ | ||
Every surface of constant negative curvature can be locally imbedded in the pseudo-sphere. The intrinsic geometry of a pseudo-sphere coincides locally with hyperbolic geometry (see [[Beltrami interpretation|Beltrami interpretation]]). | Every surface of constant negative curvature can be locally imbedded in the pseudo-sphere. The intrinsic geometry of a pseudo-sphere coincides locally with hyperbolic geometry (see [[Beltrami interpretation|Beltrami interpretation]]). | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.Ya. Vygodskii, "Differential geometry" , Moscow-Leningrad (1949) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , '''2''' , Moscow-Leningrad (1949) (In Russian)</TD></TR | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.Ya. Vygodskii, "Differential geometry" , Moscow-Leningrad (1949) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , '''2''' , Moscow-Leningrad (1949) (In Russian)</TD></TR> |
− | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) pp. 320, 378</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1957)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M.J. Greenberg, "Euclidean and non-Euclidean geometries" , Freeman (1974)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)</TD></TR></table> | |
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Latest revision as of 16:50, 8 April 2023
A surface of constant negative curvature formed by rotating a tractrix ( $ x = u - \mathop{\rm tanh} u $,
$ y = \mathop{\rm sech} u $)
around its asymptote ( $ y= 0 $;
see Fig.).
Figure: p075840a
The line element in semi-geodesic coordinates has the form:
$$ d s ^ {2} = d u ^ {2} + \cosh ^ {2} \frac{u}{a} \ d v ^ {2} ,\ a = \textrm{ const } $$
(the line $ u = 0 $ is a geodesic); while in isothermal coordinates it has the form:
$$ d s ^ {2} = a ^ {2} \frac{d x ^ {2} + d y ^ {2} }{y ^ {2} } ,\ \ a = \textrm{ const } . $$
Every surface of constant negative curvature can be locally imbedded in the pseudo-sphere. The intrinsic geometry of a pseudo-sphere coincides locally with hyperbolic geometry (see Beltrami interpretation).
References
[1] | M.Ya. Vygodskii, "Differential geometry" , Moscow-Leningrad (1949) (In Russian) |
[2] | V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 2 , Moscow-Leningrad (1949) (In Russian) |
[a1] | M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French) |
[a2] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) pp. 320, 378 |
[a3] | H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1957) |
[a4] | M.J. Greenberg, "Euclidean and non-Euclidean geometries" , Freeman (1974) |
[a5] | B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian) |
Pseudo-sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-sphere&oldid=16805