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Given an hyperbola, is there a mathematical name that describes the region/area bounded by one arm of the hyperbola? In this image the area is marked grey.

To clarify my question: I'm looking for a shorter name for this region, so that I don't have to call it the region bounded by one arm of the hyperbola.

Lix
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2 Answers2

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The inverse hyperbolic cosine is close to what you're looking for, see this image.

By the way, did you want a name or a calculation? I just realised your question can be read both ways. I don't know a name, but it shouldn't be too hard to calculate using inverse hyperbolic functions, so if that's what you want, I'll add that to this answer.

Arthur
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  • Thank you and sorry for the confusion. I edited my question. I'm looking for a name. – Lix Jul 13 '16 at 08:26
  • Very minor observation: given that @Lux gave the image as an example, rather than a particular, the hyperbola could as well have been rotated. For future readers, it might be important to note that the inverse hyperbolic sine might then be relevant (already implicit in your answer) -- barring rotational transform. – Charles Rockafellor Jul 15 '16 at 08:08
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Just simply write it mathematically as $$x>a\sqrt{1+\frac{y^2}{b^2}}$$ which is more comprehensible than other uncommonly used names.

For example, have you heard abscissa or ordinate?

They mean $x$ and $y$ coordinates respectively.

Ng Chung Tak
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  • Thank you, but my hyperbolas are arbirarily rotated. I have the focal points $f$ and $g$ and some length $c$, then I write it as ${ p \in \mathbb{R}^2 : |p-f| - |p-g| > c }$, but I sometimes want to paraphrase it with some name. – Lix Jul 13 '16 at 09:08