Talk:Unit hyperbola

This article is rated C-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects:

Mathematics Low‑priority This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics Low This article has been rated as Low-priority on the project's priority scale.

I recommend changing the exponential function, from ${\displaystyle (e^{t},\ e^{-t})}$ to ${\displaystyle (e^{-t},\ e^{t})}$, and the linear mapping, from

${\displaystyle A={\tfrac {1}{2}}{\begin{pmatrix}1&1\\1&-1\end{pmatrix}}\ }$

to

${\displaystyle A={\tfrac {1}{2}}{\begin{pmatrix}1&1\\-1&1\end{pmatrix}}={\tfrac {1}{\sqrt {2}}}{\begin{pmatrix}\cos(\pi /4)&\sin(\pi /4)\\-\sin(\pi /4)&\cos(\pi /4)\end{pmatrix}}\ }$, so that A will be a rotation matrix (and scaling) in standard form, unless there is some historical reason for the non-standard form. The linear mapping can thus be seen to rotate the hyperbola clockwise through an angle of π/4 and scale it by ${\displaystyle 1/{\sqrt {2}}}$. See rotation matrix. Thoughts? — Anita5192 (talk) 03:40, 2 March 2014 (UTC)Reply

Yes, Anita, the matrix reverses orientation so it is not a rotation. Yes again, it has something to do with the historical development of the subject. Look at hyperbolic sector where there is a standard position and positive hyperbolic angles evolve in the direction opposite to the direction of evolution of circular angles. Thus the reversal of orientation. Circular angles evolve consistent with the Sun rising over the Earth from the East. The hyperbolic sector evolves as used by Gregoire de Saint-Vincent in the discovery of logarithm. The unit hyperbola of this article not only normalizes the x y = 1 of Saint-Vincent, but also orients its evolution to agree with the familiar circular evolution. Your suggestion would do the normalization but require a revised format from the traditional basis of hyperbolic angle.Rgdboer (talk) 21:57, 3 March 2014 (UTC)Reply

Thank you! — Anita5192 (talk) 03:33, 4 March 2014 (UTC)Reply

Given that Wikipedia is written for a general audience it makes sense to provide wikilinks for things like tanh. Axiom Theory (talk) 22:10, 11 February 2026 (UTC)Reply

 Done—Anita5192 (talk) 00:26, 13 February 2026 (UTC)Reply

Thanks. Axiom Theory (talk) 17:21, 22 February 2026 (UTC)Reply