Talk:Hyperbola

Hyperbolic functions

I removed the reference to hyperbolic functions parametrizing hyperbolas. The most obvious way that is done is only valid in special cases. It can be done more generally, but any hyperbola can be parametrized by rational functions. In fact, any conic can; these are curves of genus zero and are therefore rationally parametrizable. The hyperbolic functions are analogous to the circular functions, and that could be further explained, I suppose. Gene Ward Smith 05:28, 20 July 2005 (UTC)

Does this article really need a section describing hyperbolic functions when there is already an extensive article specifically about the subject?—Anita5192 (talk) 17:36, 21 April 2019 (UTC)

Congruence of hyperbolae with "Standard-Form" Origin-Centered hyperbolae

Every hyperbola, regardless of the direction of its opening (determined by the slope if its transverse axis) and regardless of the distance between its vertices, is congruent with a convenient pair of standard form hyperbolae: one is the origin-centered East-West opening hyperbola having its same shape or eccentricity; the other is the origin-centered North-South opening hyperbola of the same shape or eccentricity. Most of the illustrations in the article depict an equilateral hyperbola, that is one whose asymptotes cross at right angles. All such equilateral hyperbolae can be rotated and translated until they match the illustrated standard form. But picture a hyperbola on a diet - its asymptotes form an acute angle, actually a pair of acute angles, into which the arms of the hyperbola nestle. Because this one has a different eccentricity, its shape is different. It can be rotated to N-S or to E-W, and can be translated to the origin. Once there, it is congruent only to its twin with the same "skinny" shape and matching eccentricity. I added the qualification (same eccentricity) to the discussion of congruence of hyperbolae. Please look it over to see if I've got it right.Bookerj (talk) 06:43, 5 September 2011 (UTC)

Looks good to me! Thanks for putting it in. Duoduoduo (talk) 17:11, 5 September 2011 (UTC)

I might be embarrassingly wrong here, but I believe that the condition of the same shape or eccentricity is still insufficient. Take the two hyperbolae $x^{2}-y^{2}=1$ and $x^{2}-y^{2}=2$. They share the same eccentricity, but are not congruent - they require scaling in addition to translation to match. Well, unless I'm embarrassingly wrong, that is :p. Arielbr (talk) 14:51, 22 October 2014 (UTC)

Thanks—I've (belatedly) put in a correction. Loraof (talk) 14:20, 19 November 2015 (UTC)

Consistency in notation

In some places the symbol "e" is used for eccentricity, and in other places the Greek "epsilon" is used. I think we should be consistent. — Preceding unsigned comment added by 207.74.68.150 (talk) 20:09, 31 July 2012 (UTC)

Thanks for the suggestion. I've now standardized the notation. Loraof (talk) 14:54, 19 November 2015 (UTC)

Spelling hyberbola

I see occasionally the spelling "hyberbola" and "hyberbolic" in otherwise English pages on the web. Would that come from non-English languages?

It's not actually something different from hyperbola, in any case, is it?

Would it be inappropriate to redirect those forms here?--SportWagon (talk) 15:42, 7 April 2014 (UTC)

It's just a typo. I googled it and clicked a couple places that spell it that way, and they spell it correctly the rest of the time in the same site. Loraof (talk) 15:05, 19 November 2015 (UTC)

Reflective property of the hyperbola

I'm surprised no one has mentioned the reflective property of the hyperbola: a ray that is directed at one focal point that comes from outside the curve is reflected to the other focal point. This picture probably better represents it: http://cs.bluecc.edu/conics/hyperbola/reflex.gif. This should be a section like the parabola's reflective property has its own section. — Preceding unsigned comment added by 66.244.81.55 (talk) 18:30, 14 April 2014 (UTC)

It's long been in there in the section "Reflections and tangent lines". Loraof (talk) 15:10, 19 November 2015 (UTC)

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Conjugate hyperbola equation

The 2nd eq. under "conjugate hyperbola" is wrong . . . — Preceding unsigned comment added by 206.174.115.204 (talk) 01:21, 7 January 2019 (UTC)

Thank you for pointing this out. It is now fixed.—Anita5192 (talk) 17:28, 7 January 2019 (UTC)

Parametric equation

Why does the 3rd parametric equation representation have plusminus btan(t)? Unlike the first two representations, the plusminus is not necessary to get both sides of the hyperbola. In [0,2pi), either btan(t) or -btan(t) works Seedwagon (talk) 15:37, 25 October 2022 (UTC)

Hyperbolic linear property

Many beginners or new learners of math often ask and get confused by looking at parabola and ellipse because both of them look alike just it has two cones and parabola has one. Shouldn't the fact :

" Parabola has noticable curvature away from origin and hyperbola behaves linearly away from the origin as it progresses more towards the asymtope. "

Be marked and told explicitly at first? It would remove confusion among people. Prince khan official (talk) 14:42, 20 September 2023 (UTC)

This is already explicitly discussed in the lead section and in the article body. –jacobolus (t) 15:19, 20 September 2023 (UTC)