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)

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)

Pin and string construction - missing step introducing point A?

The Pin and string construction refers to A ("string with length |AB| is prepared") while point A is not defined anywhere.

On the image, it seems to perhaps be on double-radius of c2 (which is defined as 2a, so is it 4a), but can't be sure. It would be great if someone with knowledge redacted the section so it's clear how A is constructed. Zlamma (talk) 11:56, 14 August 2024 (UTC)

After some more reading and thinking, I now understand that point A can be chosen arbitrarily: on the image it's the end of the ruler, but it can be any length, as choosing a further point (which may need a longer ruler) will merely extend our ability to draw more of the hyperbola (note that it also makes the string longer), so in my edit I redacted the step verbiage so that it's a bit clearer (A is a point that the user picks, and the length is consequence of that choice). Zlamma (talk) 20:44, 27 October 2024 (UTC)

Chaotic text flow

The article starts with the definition $H=\{P:||PF_{2}|-|PF_{1}||=2a\}$. That's fine. But then we try to relate the canonical form $x^{2}/a^{2}-y^{2}/b^{2}=1$ (for $b=a$) with the graph of $f(x)=A/x$, without introducing the canonical form at all. This is done much further under Hyperbola#Equation. It's not a good best practice to use concepts that have not been introduced yet, so I suggest to add a subsection between "As locus of points" and "Hyperbola with equation y = A/x" (under "Definitions"), introducing the canonical form $x^{2}/a^{2}-y^{2}/b^{2}=1$ and with visualization/picture on the right hand-side of what a and b are. The proof of the relation between the two could be deferred to the much later section "Equation", but the concept should already be defined here. ~2026-18206-92 (talk) 13:01, 3 April 2026 (UTC)

I think the fundamental problem is that there's an over-focus on formulas and technical details, and too many lists, with not enough narrative prose. I think leading off with a bunch of technical minutiae about more than 10 various "definitions" (including proofs and lots of notation) makes the topic more challenging and much less legible than it should be, and makes it hard for readers to figure out what is most important. The difficulty of course is that any significant restructure would be quite a big project, which I don't feel motivated to tackle by myself any time soon, but would be happy to chat about or collaborate on (e.g. finding sources, etc.).

If it were up to me I'd make the initial definitions section shorter and more narrative, cutting it down to just 3–5 entries, none exhaustively discussed; there are an infinite number of possible definitions or characterizations of a hyperbola, but we don't need to put them all at the start of the article on equal footing. I think the most important "definitions" are probably (1) a section of a cone, (2) the locus of points whose distances from the two foci have a constant difference, (3) the points whose coordinates satisfy ⁠$xy=1$⁠, ⁠$x^{2}-y^{2}=1$⁠, or affine transformations of those. Maybe another 1–2 definitions could also be included, but there are quickly diminishing returns. After a definition section, I'd put a section about the most important geometrical properties (which could include some of the entries currently in "definitions"), and then after that put a top-level section with a more detailed focus on coordinates.

@Ag2gaeh made all of the diagrams in this article, which are wonderful. I'm not sure which editors did most of the writing and organization of the article, but it would be worth pinging them about any major changes. –jacobolus (t) 17:09, 3 April 2026 (UTC)