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Hyperbolic Polar Sine


The hyperbolic polar sine is a function of an n-dimensional simplex in hyperbolic space. It is analogous to the polar sine of an n-dimensional simplex in elliptic or spherical space. If the edges between vertices V_i and V_j have length E_(ij), the value of the hyperbolic polar sine of the n-dimensional hyperbolic simplex S in space with Gaussian curvature K<0 is given by

 polsinh^2S= 
 (-1)^n×|1 coshE_(01)sqrt(-K) ... coshE_(0n)sqrt(-K); coshE_(10)sqrt(-K) 1 ... coshE_(1n)sqrt(-K); | | ... |; coshE_(n0)sqrt(-K) coshE_(n1)sqrt(-K) ... 1|.

The hyperbolic polar sine is used in the generalized law of sines for a hyperbolic simplex.

The limit of the hyperbolic polar sine of an n-dimensional hyperbolic simplex as the curvature K of the space approaches zero is n!S(-K)^(n/2), where S is the content of the Euclidean simplex with the same edge lengths.


See also

Polar Sine, Sine

This entry contributed by Robert A. Russell

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Cite this as:

Russell, Robert A. "Hyperbolic Polar Sine." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HyperbolicPolarSine.html

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