# Talk:Hyperbola

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One of the 500 most frequently viewed mathematics articles.
To-do list for Hyperbola:
 Here are some tasks you can do:
• Add image of hyperbola as the reciprocation of a circle in a second circle.
• Add animation of tangent circle generation and/or image of trilateration.
• Add images of conformal maps, such as w = z2.
• Discuss related "hyperbolic" things, such as "hyperbolic partial differential equation" or "hyperbolic motion".
• Animation of hyperbolic Kepler motion. May be useful in other articles, such as scattering.
• Create a diagram that includes the angle theta
Priority 9

Is this image used by permission? --Brion

It ain't local either. Aren't there laws against this type image use? --mav

IANAL, but I have the impression that case law would not be on our side if we're not using it with permission. If we got permission long long ago, in the before time, then it should be acceptable... but there's nothing in the talk page, and the pre-January edit history comments are still missing. However even if it's by permission, it doesn't look that great and we should use a local image anyway. Maybe I'll break out the old POV-Ray and make some new ones. :) --Brion

Why are there four different cartesian equations? How do they differ, which one should I choose, what are a, b and c? Equations without explanations are worse than useless. AxelBoldt 05:56 Jan 4, 2003 (UTC)

In the first set of two, the main difference is which way the hyperbola opens up (which direction the transverse axis is). The first one (the x term being positive) opens up horizontally, like in the picture, and the other one (with the y term being positive) opens up vertically. In these equations, a, b, h, and k are all constants.
As for the other equations, I had not seen them before, partly because they are not equations for hyperbolas, but rather equations dealing with hyperbolas, and I have not studied conics very deeply. According to my text book, c is the distance between one focus and the center. The text I have lists b2 = c2 - a2, which I cannot seem to make equal to the formula given. The eccentricity is just what is stated in the link. Loggie 00:55, May 6, 2005 (UTC)

I assigned my class homework to add information to this page and show me what they added. I hope it doesn't lead to a bunch of vandalism, but I fear it may. Sorry in advance. I hope it turns out that my experiment is good for the wiki. Tjdw 21:21, 30 Apr 2004 (UTC)

Followup: It seems it was a terrible experiment. Only 5 out of 30 kids even did the homework, and of those, only 2 wrote anything useful or correct. One student plagiarized, one vandalized the page, and one wrote an incorrect process in the External Links section for isolating y on one side of the standard form equation. What do you guys think? Leave some messages on my talk page. Tjdw 22:33, 3 May 2004 (UTC)

Whoops! Make that 2 students who plagiarized. That leaves 1 who wrote correct and original content. Tjdw 22:37, 3 May 2004 (UTC)

### Is this right?

A special case of the hyperbola is the equilateral or rectangular hyperbola, in which the asymptotes intersect at right angles. The rectangular hyperbola with the coordinate axes as its asymptotes is given by the equation xy=c, where c is a constant.

According to the formula given, $(x-h)(y-k) = c \,$, wouldn't xy=c anytime the center of the hyperbola is at (0, 0), not necessarily when the asymptotes intersect each other at right angles? The intercepts of the asymptotes are defined by the center (h and k), but the actual slope is defined by a and b. Loggie 01:02, May 6, 2005 (UTC)

It is right, but I corrected the other thing.--Patrick 14:27, 6 May 2005 (UTC)
I'm still slightly confused- When you say that "coordinate axes parallel to their asymptotes", are we refering to the x axis and y axis? And aren't the asymptotes of a hyperbola slanted? Can the axes of a hyperbola ever be parallel to the x and y axis? I wouldn't think so- as you made the slope of the asymptotes closer and closer to zero the hyperbola would just keep getting shallower and shallower, or steeper. I don't believe the slope of the asymptotes ever could be zero. I wish they would go into more detail when teaching me these things at school. Loggie 21:55, May 6, 2005 (UTC)
Nevermind! I got it all sorted out. Loggie 02:15, May 9, 2005 (UTC)

## 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)

## "Exaggeration"?

The American Heritage Dictionary translates the Greek as "a throwing beyond, excess (from the relationship between the line joining the vertices of a conic and the line through its focus and parallel to its directrix)". This provides more insight. Compare to parabola, "comparison, application, parabola (from the relationship between the line joining the vertices of a conic and the line through its focus and parallel to its directrix)", and ellipse, "to fall short (from the relationship between the line joining the vertices of a conic and the line through the focus and parallel to the directrix of a conic)". All three definitions implicitly refer to the eccentricity of the conic. KSmrq 03:28, 2005 August 11 (UTC)

Isn't eccentricity simply e=c/a ? Also, I see no mention of vertices or latus rectum, I thought that was an element of a hyperbola... --69.113.7.18

## The simplest case

Shouldn't the simplest and most common case of a hyperbola be mentioned and shown, I mean the function f(x) = 1/x?

nl:Gebruiker:Sokpopje

Done. Geekdog 13:50, 29 January 2007 (UTC)

IN FACT, I SEE THIS MENTIONED, BUT I DO NOT SEE THIS SHOWN. AND I AGREE THAT THIS SHOULD BE DONE Athkalani 20:41, 27 May 2007 (UTC)

Can you include the proof of the relationship between a, b, and c in a hyperbola is c^2=a^2+b^2 ? 68.83.181.181 03:10, 8 January 2007 (UTC)

## Ambigenal hyperbola

Does anyone actually have a clue what the definition given means? I'm not claiming to be the smartest person in the world, but I'm doing a PhD in maths, and if I can't make head or tail of that description then I suspect other people may not be able to either. A picture would help no end. All the other results for "Ambigenal hyperbola" on Google give the same definition, so I have been unable to find any clarification. As a side note, why are all the diagrams of such simple hyperbolas? They all look symmetric, which surely isn't true in the general case. Geekdog 16:05, 28 January 2007 (UTC)

I removed the "definition". It was quite unclear, and the term isn't used elsewhere in the article, so it wasn't needed. On your side note, all hyperbolas are symmetric, just not necessarily with respect to the axes, but linear transformation can make them so. So, geometrically it's all the same. A graph of y=1/x wouldn't be bad to include for an example of a "rotated" hyperbola. Cheers, Doctormatt 18:00, 28 January 2007 (UTC)
Okay, thanks for that. I'm still confused though (admittedly, geometry is not one of strong points). I've made a couple of very rough diagrams that try to explain what the problem I'm having is. This image attempts to illustrate a case where the hyperbola generated is obviously symmetric. Here is a similar case where the hyperbola is not obviously symmetric. It is not necessarily rotated (depending on coordinate choice, of course). When considered on the plane that's being used to generate the hyperbola, the two sides of the hyperbola do not seem (to my mind) to have the same curvature. When projected onto a plane containing the axis of the cone the two sides of the hyperbola are not equidistant from the centre point of the cone. I can't intuit whether they have the same curvature but I'm guessing not - of course, if they do have the same curvature then a simple translation will make the hyperbola symmetric.
You are saying that a linear transformation of this second case can produce a coordinate system where both sides of the hyperbola are symmetric, right? If so, this is not at all obvious to me.
Apologies for the low quality sketches, I hope they're enough to get my point across. Geekdog 19:56, 28 January 2007 (UTC)
I see what you mean: with your sketches, it is hard to see the symmetry. Hyperbolas are always symmetric, even the "non-vertical plane through the cone ones". Try looking at the illustrations on conic section; that might help. (It would be great to have a short animation showing the plane through the cone, and then moving so that the plane is viewed perpendicularly so you can see the symmetry). Or, consider the definition of the conic section as point loci; hyperbolas are the locus of points with the property that the absolute difference of the distances from each point to two fixed points (foci) is constant. This definition makes the symmetry most clear to me. Cheers, Doctormatt 22:03, 28 January 2007 (UTC)
Thanks, the sketch on the conic sections page made things clearer. I added a graph of y=1/x. Geekdog 13:51, 29 January 2007 (UTC)

## The parametric equations are very wrong

it is written beneath the parametric equations that a is the semimajor axis and b is semiminor. This is true for a and b in the cartesian equations, but is not true for the parametric equations given. In the parametric equations, each instance of 'a' should be replaced with 'b' and the other way round to be consistent with the statement beneath. —The preceding unsigned comment was added by 144.136.38.19 (talk) 05:56, 21 March 2007 (UTC).

## Pluralisation

English uses many irregular plural forms, a greek root is a common cause; the germanic pluralisation of greek words has become more common, which does not mean it is correct. By derivation, the plurals of hyperbola, parabola and formula should be hyperbolae, parabolae and formulae and, as the page on english pluralisation remarks, "correctly formed Latin plurals are the most acceptable, and indeed are often required, in academic and scientific contexts.". Diysurgery 02:44, 1 October 2007 (UTC)

I checked Merriam-Webster online, and it indicates that both are acceptable. This claim at english pluralisation is uncited; it is quite possibly just one person's opinion. I note that Mathworld suggests hyperbolas is correct ([1]) and even gives a citation (although I imagine that may be simply citing that author's usage). My trusty and worn Webster's New Collegiate Dictionary, copyright 1949, indicates hyperbolas as the only correct pluralization. This article at encarta [2] gives both. I can't find any source, besides Wikipedia, that suggests that -ae is required for hyperbola. Doctormatt 01:30, 1 October 2007 (UTC)
I've decided to focus on "formula", since it is a far more common word and the issue is (and my relevant findings are) identical. The Online Etymology Dictionary led me to the Charles Kingsley quote "Men who try to speak what they believe, are naked men fighting men quilted sevenfold in formulae.", dated 1861.
The 1982 concise OED lists both, but only uses -ae, whereas the 1996 compact OED lists both and uses -as in the entry and -ae in the entry for formulary (the english form of a latin word with a greek root, just imagine the carnage) - demonstrating the fact that, in matters of etymological minutiae, general purpose reference books simply cannot be relied upon (though I'd bet you believe me when I tell you the plural of minutia is minutiae, which neither dictionary disputes, even the laughably fallible dictionary.com agrees). Diysurgery 02:44, 1 October 2007 (UTC)

## a and b used before definition

In the Definitions' section $a$ and $b$ are used before they are defined later in the Equations' section. Also the formulae they are used in are not general enough.

--Ts4079 (talk) 16:48, 28 November 2007 (UTC)

This observation by Ts4079 (Nov 28, 2007) remains in effect, and is a source of confusion, as terms a,b,c, and theta are used to give a geometric interpretation/analysis of the hyperbola without clearly identifying these terms first. Someone placed into the article the request: "a and b are not yet defined, please fix." This text was deleted from the article (appropriately) by Wwillow on Aug 21, 2008 as part of a reorganization of the ==Definitions== section, but without resolving the request for clarification of terms. The appropriate spot for the requested definitions now lies early in the ==Nomenclature== section, where hyperbolas are defined geometrically using the terms a, b, c, and theta. Of these terms, only a has been defined so far, in the second paragraph of ==Nomenclature==, directly above (a is the distance from center to one vertex). So far, b is identified only as the numerator of the asymptote's slope b/a (plus and minus); further down it is identified with the semi-minor axis, but still with no way to correlate that line with a magnitude or distance. I think the length of the semi-minor axis is taken to be the length of a parallel line segment running from one vertex to the asymptote, i.e. the minor axis 2b is the length of a rectangle with its 4 vertices on the asymptotes, and two sides passing through the vertices perpendicular to the major axis. Is that correct for b? c I think is the distance from center to one of the foci, yes? And it appears theta represents the angle formed by the major axis and one of the asymptotes. Bookerj (talk) 07:10, 31 May 2009 (UTC)

## Degenerate hyperbola?

The 'degenerate hyperbola' is worth mentioning in this article. WinterSpw (talk) 05:48, 29 May 2008 (UTC)

Indeed. One could then point out that the curve y = c/x as usually introduced in high school is better written as xy = c so that the case c = 0 allows the Y-axis to belong to the degenerate hyperbola without having to explain why 0/0 should be understood as the whole Y-axis. --Vaughan Pratt (talk) 13:37, 12 December 2008 (UTC)

## "Not to be confused with hyperbole"

Uh, not to be confused myself: "Hyperbolic" redirects here. And "Hyperbolic" is the adjective form for both Hyperbole and Hyperbola. Either the redirect should be changed to a disambig, or this article's disambig. needs to be changed to the, "This article discusses the mathematical term. For the rhetorical term, "hyperbolic," see "hyperbole." LaughingVulcan 12:54, 11 September 2008 (UTC)

Seeing no responses, I changed the {{distinguish}} template to {{otheruses4}}. This is only to avoid creating a dab at Hyperbolic for only two terms. If anyone wants to do that instead, great! LaughingVulcan 00:42, 13 September 2008 (UTC)

## The section "In polar coordinates"

In the present version it says:

East-west opening hyperbola:

$r^2 =a\sec 2\theta \,$

North-south opening hyperbola:

$r^2 =-a\sec 2\theta \,$

Northeast-southwest opening hyperbola:

$r^2 =a\csc 2\theta \,$

Northwest-southeast opening hyperbola:

$r^2 =-a\csc 2\theta \,$

And in old versions (2003) it is already there::

Equations (Polar):

$r^2 = a\,\sec 2t$
$r^2 = -a\,\sec 2t$
$r^2 = a\,\csc 2t$
$r^2 = -a\,\csc 2t$

But this is wrong?

The first observation is that the dimension is wrong, r and a have the same dimension while

$\sec 2t = \frac{1}{cos 2t }$

has no dimension!

Then there is no reference to any eccentricity (or any semi-minor axis) at all!

The correct version is

East-west opening hyperbola:

$r^2 =\frac{b^2}{e^2 \cos^2 t -1} \,$

In case

$e= \sqrt{2}$

this can be written

$r^2 =\frac{a^2}{\cos 2t} \,$

what is almost the present formula!

Stamcose (talk) 11:52, 25 November 2008 (UTC)

What is the right number of definitions of "hyperbola" for the lead? Leads should be a short clear introduction to the body that doesn't delve into the esoterica of the subject. Some of the definitions in the lead are pretty esoteric!

The function y = 1/x and the conic section where a plane cuts a cone at an angle steep enough to produce two curves ought to give enough insight into hyperbola-hood for the average reader. The remaining definitions aren't exactly absorbable on a first reading. I propose moving those into the body. If you feel the lead warrants a third definition please state it in language appropriate to the lead, preferably from a perspective that makes it intuitively clear, and propose it for retention/inclusion in the lead.

Actually I have a candidate. A hyperbola is the apparent shape of a circle to a viewer whose retina or image plane cuts the circle. I'll only put that in the lead however if there is some show of support for it in there, otherwise it should go in the body. --Vaughan Pratt (talk) 08:27, 9 December 2008 (UTC)

Even shorter: A hyperbola is the apparent shape of a circle to a viewer standing inside it. --Vaughan Pratt (talk) 08:29, 9 December 2008 (UTC)
Only one branch of the hyperbola is conventionally visible, the other branch arising only in principle, by projecting the part of the circle behind the viewer through their head onto the back of their retina via optics complementing those of the eye. (This clarification probably should be relegated to the body so as not to clutter up the lead.) --Vaughan Pratt (talk) 08:35, 9 December 2008 (UTC)
It occurred to me that the introductory definition of the hyperbola should aim to match that of the ellipse more closely, namely as "the apparent shape of a circle viewed obliquely." I made this definition the first one in this article's lead, as for the ellipse, and drew the outside-inside distinction in both, which was previously absent from the ellipse article, rather an oversight in view of the fact that hyperbolas arise in the same way.
A side effect has been to yet further lengthen this already very long lead. What do people think about truncating the lead after the third definition (conic section) and simply indicating that the body of the article gives several other definitions of in terms of foci, directrix, eccentricity, etc., obtainable from their counterparts for ellipses by a suitable change of sign? --Vaughan Pratt (talk) 13:22, 12 December 2008 (UTC)
So far no objections to the idea of shortening the lead, suggesting that its time has come. Meanwhile at the suggestion of User talk:Greensburger I lengthened the body with a section relating conic sections to the appearance of a circle in perspective view, which while obviously not a new concept for the ellipse seems to be new for some people in the case of the hyperbola. But while the novelty of it in the case of the hyperbola argues for inclusion in the hyperbola article, one could equally well argue that it should be moved to the conic section article, or in view of its length, to its own article, linked to appropriately from all related articles. I'll sleep on this for now. --Vaughan Pratt (talk) 11:47, 14 December 2008 (UTC)

Suiting the action to the word, I've proposed a wholesale replacement for the current lead at Candidate new lead for hyperbola article. Please look it over and raise any objections during the next few days. Barring any that can't be resolved I'll install it around the end of the week (unless there's a consensus that earlier would be fine). --Vaughan Pratt (talk) 00:10, 16 December 2008 (UTC)

Replacement made as promised. Please check it over for problems. --Vaughan Pratt (talk) 06:38, 19 December 2008 (UTC)

## How can one draw a parabola or a hyperbola?

We can draw a circle with a compass, an ellipse with a loop, a pen, and two tacks. Is there a similar way to precisely draw a parabola and a hyperbola? Majopius (talk) 22:55, 27 May 2009 (UTC)

I would think you would need a device which could release an equal amount of string in two sections. The trick with the ellipse is that the sum of the two segments is fixed. For a parabola, you want to the distance to the point and the line to be equal, but not a fixed value. Plastikspork (talk) 16:30, 28 May 2009 (UTC)

## What is the equation?

In the nomenclature section, there is a beautiful graph of hyperbola, the one on the right. Great! But what is the equation for the hyperbolic curve? Ctchou (talk) 03:56, 22 June 2010 (UTC)

The equations are in the graphic on the left margin. Greensburger (talk) 00:25, 26 December 2010 (UTC)

## How to characterize special case

How can we characterize the special case in which the plane that intersects the cone is parallel to the symmetry axis of the cone? Does the eccentricity have a special value? Does it have a special appearance in the Cartesian plane? This question is motivated by this discussion on the talk page of Talk:Conic section, in particular the entry by Mark Dominus on 06:14, 9 February 2011. Duoduoduo (talk) 18:12, 9 February 2011 (UTC)

More generally, can we express the eccentricity as a function of (a) the angle between the plane and the axis of the cone, and (b) the angle between the lines that form the cone? Duoduoduo (talk) 20:28, 9 February 2011 (UTC)

This is a parabola, i.e. a conical section with the eccentricity 1. Here eccentricity > 1 is the theme! Stamcose (talk) 21:06, 9 February 2011 (UTC)
No, a parabola comes from a plane that is parallel to one of the lines forming the cone, not parallel to the axis of symmetry of the cone. So my question remains: How can we characterize the special case in which the plane that intersects the cone is parallel to the symmetry axis of the cone? Duoduoduo (talk) 22:08, 9 February 2011 (UTC)
Right, Tri-Duo's special case is the hyperbola illustrated informally as the right-hand figure number [3] in the graphic file File:Conic Sections with Plane.svg where the plane runs parallel to the cone's axis, not its surface (the left figure, [1]). It is also illustrated by the right-most gray cutting plane in the graphic File:Conic Sections 2.png which appears within the Conic Sections section of this article. Since the cutting plane of this special-case hyperbola parallels the cone's axis, its transverse axis also parallels the cone's axis. Intuitively, that suggests this is the equilateral or rectangular hyperbola, the special case in which the asymptotes intersect at right angles. That special case hyperbola has been described in three sections of this article (Nomenclature, Derived Curves, and Special Cases). Since the resulting conic section is still a hyperbola, its eccentricity is by definition greater than 1.
Relating to Mark Dominus' comments cited in this discussion, it seems clear this special parallel-plane hyperbola is the only one whose center lies on the plane perpendicular to the cone's axis and containing the cone's vertex (the cone's base plane). It is also the particular hyperbola whose two foci are equi-distant from the cone's vertex. Slightly shifting the cutting plane angle, Mark makes clear, moves one hyperbolic focus closer to the cone's axis (closer to one nappe) and the other focus further from the cone's axis (and its other nappe). In doing so, this shift of the cutting plane also moves the resulting hyperbola's center off the cone's base plane. This "tilted" plane is the one illustrated by the left hand gray plane in the right-most cone (#3) on File:Conic Sections 2.png. Mark's key point in responding to his OP is, I think, that despite the slanted cutting plane skewing the apparent symmetry of the resulting hyperbola with relation to the cone's slope and its vertex, the two arms of the hyperbola remain symmetric to each other. That is, the slanted cutting plane forms not a separate, fifth kind of conic, just a more general (less specialized) hyperbola.Bookerj (talk) 17:18, 24 February 2011 (UTC)

I suspect the eccentricity of the hyperbola is determined geometrically by the slope of the cone's surface of generating lines relative to its axis of symmetry. Imagine a cone, viewing just one nappe - an ice cream cone shape. Now imagine stretching it by either: a) keeping its circular base constant and raising/lowering its vertex; or b) keeping its height constant while enlarging or diminishing its circular base. As you raise the vertex (or diminish the base) you make the cone's shape grow skinnier, the slope of its surface steeper. Now restore the cutting plane into your mental image: as you stretch the cone vertically, the curved arms of the generated hyperbola narrow, its two foci recede from each other and from its center point, and its eccentricity grows larger. Can someone with a little more formal analytic geometry comment whether this intuitive connection of cone-shape with eccentricity bears out?Bookerj (talk) 19:42, 24 February 2011 (UTC)

## True anomaly

I improved the section! For the right branch (in this case the "far" branch) one certainly has that

$r =e x + a\,\!$
$r = e (-ae+r \cos \theta) + a\,\!$
$r = \frac{a(e^2-1)}{-1+e\cos \theta}\,\!$

This sure is a correct expression in polar coordinates for the far branch but this is not the "true anomaly" $\theta$ of this point. For the right branch it is the right focal point that is the origin which defines the true anomaly. If really the Polar coordinates section duplicating the True anomaly section should be kept it could be put in there!

Stamcose (talk) 18:35, 9 February 2011 (UTC)

## In polar coordinates

Without mentioning the applicable formula

$r = \frac{a(e^2-1)}{-1+e\cos t}$

for the far branch it says:

For the right branch of the hyperbola the range of $t$ is:

$-\arccos {\left(\frac{1}{e}\right)} < t < \arccos {\left(\frac{1}{e}\right)}$

Makes no sense! Either ignore the "far branch" or treat it properly! See talk "True anomaly"

Stamcose (talk) 19:04, 9 February 2011 (UTC)

## Ellipse where a or b equals i

Should it be noted in either this article or the ellipse article that an ellipse where a or b is equal to the imaginary number i will result in a hyperbola? It's just an interesting fact that because i squared is -1, and with a -1 in the denominator the equation of the ellipse is, effectively, the equation of a hyperbola. I'm noting this in respect to the standard forms of ellipses and hyperbolas, in case I am unclear. 165.199.1.20 (talk) 13:13, 10 May 2011 (UTC) (School IP)

Interesting observation, but a and b are defined as real numbers. I checked the article, and I think the fact that they are real is made clear by the nomenclature section, where they are defined as lengths in the real (x, y) plane. Duoduoduo (talk) 14:01, 10 May 2011 (UTC)
There is a useful discussion at Conic Section#Generalizations of conics derived in an complex space, and the equivalence of ellipses and hyperbolae in that space, due to the non-distinction there between 1 and -1. Bookerj (talk) 21:53, 12 August 2011 (UTC)

## Wrong word used?

In the section "Conic section analysis of the hyperbolic appearance of circles", a paragraph begins "The tangents to the circle where it is cut by the lens plane constitute the asymptotes of the parabola." Shouldn't that be the asymptotes of the hyperbola? Parabolas don't have asymptotes. 130.155.97.238 (talk) 04:38, 18 August 2011 (UTC)

You're right -- I've corrected it. Duoduoduo (talk) 13:38, 18 August 2011 (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)

## Axis intersecting hyperbola formula

Every angular plane that crosses the axis of a cone and creates 2 vertices would have a focal point halfway between the 2 vertices. And that would be the point of origin of any controlling force related to the rest of the path within the plane. However when illustrating the path of a controlled particle, the hyperbola used for the illustration practically always is that of a plane parallel to the axis of the cone. And with regard to the path in the plane a point is established and the distance from the path to the point is called the impact parameter. So if the hyperbola is the path of a plane intersecting a cone, why don't we call that point the point of intersection, and what would be the formula of such a path?WFPM (talk) 17:19, 28 February 2012 (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)

## Apollonius and eccentricity

I can't find where Apollonius mentions eccentricity referring to hyperbola. As far as I know, it's all about "application of areas", like for ellipse and parabola, see e.g. [3] (bottom of page), I think this argument is clear enough. So, the third sentence in the "History" section should be rephrased to reflect that. 79.132.172.210 (talk) 00:18, 2 November 2012 (UTC)

Actually that whole section seems like it could use some decent sourcing -- MacTutor website on Menaechmus:
"Menaechmus is famed for his discovery of the conic sections and he was the first to show that ellipses, parabolas, and hyperbolas are obtained by cutting a cone in a plane not parallel to the base. It has generally been thought that Menaechmus did not invent the words 'parabola' and 'hyperbola', but that these were invented by Apollonius later. However recent evidence in Diocles' On burning mirrors discovered in Arabic translation in the 1970s, led G J Toomer to claim that both the names 'parabola' and 'hyperbola' are older than Apollonius."
The same person who pointed this out to me suggested A History of Mathematics by Jeff Suzuki or the same by Victor Katz as possible good sources for further information. --JBL (talk) 23:49, 27 November 2012 (UTC)

## Temporary rename as: Hyperbola (mathematics)

To reindex the link in Google Search, I am renaming the article "Hyperbola" as "Hyperbola (mathematics)" (in the style of "Function (mathematics)") for about a three-week 5-day period. The problem, in Google Search, has been a link with prefix "https:" for a secure-server SSL protocol, which prompts some users with a security warning, and might deter them to not read the article after searching with Google, perhaps at a public library which limits access with secure websites (see: wp:Google https links). Also, the https-prefix pageviews have been omitted from April/May counts with stats.grok.se (see: wp:Pageview statistics). The new name, as "Hyperbola (mathematics)" will be listed in Google (or Yahoo or Bing.com) with a regular "http:" prefix, which any user can view without a security certificate. After about three weeks 5 days, the article can be renamed back to "Hyperbola" for access by any person with any browser. -Wikid77 (talk) 13:47, 12 May, revised 14:43, 14 May 2013 (UTC)

• Status: Although Google created the http-prefix link to the new title, the article was unrenamed back, within 2.5 days, and did not have time to fully index into Google. The following steps have been made:
1. 15:50, 12 May 2013‎ User:Wikid77 moved "Hyperbola" to "Hyperbola (mathematics)" as title with "http" prefix.
2. 15:52, 12 May 2013 Google created cache of "Hyperbola_(mathematics)" as snapshot of page as it appeared on May 12, 2013 15:52:35 GMT.
3. 15:53, 12 May 2013‎ User:Wikid77 edited redirect "Hyperbola" to reset as soft redirect to "Hyperbola (mathematics)", with "http:" prefix, to avoid https-protocol links which are not counted for pageviews in stats.grok.se website.
4.  ??:?? 14 May 2013 Google reindexed "Hyperbola" with soft redirect as "See: Hyperbola (mathematics)".
5. 03:52, 14 May 2013‎ User:Jcgoble3 edited "Hyperbola" as Requesting speedy deletion (wp:CSD#G6), to rename article back to original title.
6. 11:55, 14 May 2013‎ User:Wikid77 edited "Hyperbola" from mobile site "en.m.wikipedia.org" (via https) to set as simple #redirect to "Hyperbola (mathematics)" and put "__NOINDEX__" to set meta robots as "no index, no follow" in search engines.
7. 12:40, 14 May 2013‎ Google searches of hyperbola phrases began listing "Hyperbola (mathematics)" in top 10, or top 30 results; however, mobile-site page en.m.wikipedia.org/wiki/Hyperbola was often listed (as "http") for some phrases.
8. 03:51, 15 May 2013‎ User:Keegan renamed back to "Hyperbola" leaving "Hyperbola (mathematics)" as redirect.
9. 05:00, 15 May 2013 Google still listed Mobile-site "en.m.wikipedia.org/Hyperbola" as a separate match for some searches.
Keeping "Hyperbola (mathematics)" as redirect allowed some access from Google via http protocol; however Google listed mostly "https:../Hyperbola". After 12 May, within 2 days, the new title (with simple "http" prefix) was being listed in some results of Google Search; however, the original https-protocol title ("Hyperbola") still ranked higher for most searches on the 2nd day after renaming, even though the contents were a see-link to "Hyperbola (mathematics)" in the Google results blurb (but the Google cache was still 24 April 2013 revision). -Wikid77 14:43, 14 May 2013 (UTC)
The proper solution is to wait for the developers to fix the actual bug, not invent a new title that completely ignores the policy on article titles. The "benefit" that this provides is purely technical and of use only to a small subset of experienced users, and as such is nowhere near sufficient to justify invoking WP:IAR to violate a policy. jcgoble3 (talk) 23:41, 14 May 2013 (UTC)
Agreed. --JBL (talk) 00:36, 15 May 2013 (UTC)
Agree with User:Jcgoble3; Secure https links are not a bug but a feature. If a web site has them then Google should return them as they're more secure, not least for editors who are most at risk from e.g. having their login details captured (this may seem unlikely until you realise that details captured from WP, such as IDs, real names, passwords, could be used to access other sites, ones with more valuable or private information. WP is one of the most popular web sites, and requires regular logins, so is an obvious target). If a user's browser or connection doesn't support https they need to switch to a better browser/find another way of connecting. But this can effect few people: you pretty much can't function on the web today without https support. WP doesn't require one but many sites do, including some of the most popular.
As for pageviews if there's a problem with the stats server then it should be fixed there, as I'm sure it will be quickly if it's important. I don't think it is: I last looked at pageview stats many months ago, and I'm probably exceptional in doing so; most editors probably don't know they exist. But that's how to deal with problems with the stats, on the stats server. Not moving pages in violation of the page name policy.--JohnBlackburnewordsdeeds 01:13, 15 May 2013 (UTC)
I've moved the page back. Keegan (talk) 03:52, 15 May 2013 (UTC)
• Partial explanation of complex problems: There were several problems, which required broader understanding to consider, so I can appreciate all the confusion. The pageview levels confirmed the problems affected 70% as the vast majority of readers, and the wp:TITLE name as "Hyperbola" remained a soft redirect during the whole 2.5 days. However, the software problems which excluded https/ipv6 pageviews in webstatscollector have been reversed (by 19:00, 14 May), and so newer pageviews can be analyzed to reassess impacts to readers. The access was not a dichotomy between http/https protocols, but included the Mobile-site as Google still listed "en.m.wikipedia.org/Hyperbola" as a separate match for some searches. Access was blocked from en.wikipedia.com (versus ".org") for many browsers, such as MSIE. Plus pageviews of clicked wikilinks were also affected, as inheriting the https-protocol, except in the soft redirect inside page "Hyperbola" where explicit "http:" prefix allowed non-secure access when clicked, as confirmed by related pageviews. Although the unrename within 2.5 days botched the attempt to allow clean access, for all users, the continual logging of pageview entries during that time period provided valuable data, for multi-year studies, to measure the impacts of the various problems in more than 300 major articles, and thousands of other pages or images wikilinked to those articles. -Wikid77 (talk) 14:31, 15 May 2013 (UTC)

Wikid77 seems you are still concerned about this problem and consider your proposal for fixing it still relevant. Am I right to sum up this as:

• Problem: https link hits in Google should be eliminated to support widest common access (i.e. MSIE, some mobile browsers, some locked down sites)
• Proposed Fix: move article, wait for reindex, move back
• Expected Outcome: http will be listed higher than https
• You've created wp:Google https links as presumably you want them all fixed?

I'm trying to understand and consider this needing discussion first. 12:09, 16 May 2013 (UTC)

• Many https problems remain but perhaps the fix can wait another month: I think the Google-https links are still a problem, but the "cure might be worse than the disease" (such as not operating to remove shrapnel near major blood vessels likely to rupture/clot as more dangerous to health). I was hoping a slight rename (such as to typical "Hyperbola (mathematics)"), with redirect, would be allowed to float through the system; however, there have been some intense debates (even anger), and then I did not want to elaborate on how Google performs its company-proprietary, trade-secret re-indexing for PageRank (as an invasion of their privacy). However, if we stick to basic Google actions, we can note that getting Google to "forget" the old https-prefix name takes several days (with meta robots set "no index, no follow") and re-checking for hidden copies in Google's replicated servers. As to the scope of problems, there are also other concerns with the Google-https links, beyond the scary warnings for secure-server access in some browsers (MSIE) and duplicate listings of Mobile-site en.m.wikipedia pages, such as:
• HTTPS pages have 2-day delay in re-indexing a page with https-prefix, compared to just minutes to Google-update an http-type link.
• Users linking from https ".com" alias, en.wikipedia.com, get mandatory warnings in some browsers.
• The https-protocol has to be re-re-re-explained to the next user who wonders about the differences.
Now, for math pages, the 2-day delay to show "news" in Google-https pages is not a concern for https "Hyperbola" or https "Calculus" or "Geometry", etc. (except perhaps for "Alan Turing" or some new math techniques). However, with other articles, timely updates are valuable; for https "Shakira" or "Lady Gaga" or "Cancer" (as perhaps "now contagious") or "Water" (as perhaps "underground lakes found on Mars"), then a 2-day delay to update the lede of those pages would make Wikipedia seem dull, stuck-on-stupified, while the whole world had current information "2 days ago". But then, perhaps Google might "change the rules" (tomorrow) and give high-priority updates to such major articles despite the https-protocol debacle. In terms of reader impact, the pageviews reveal Google users are often half of pageviews, as much as all wikilinks, navboxes, or other websites combined. However, for us as wiki-surgeons, to cure the Google-https links, we also need to consider all the "other patients" (WP problems to fix) which might be delayed if dwelling too much on trying to cure these https patients who can hobble through as is. Meanwhile, we can learn by documenting the suffering, "symptoms and treatment" of the Google-https pages/images. Anyway, I am hoping the developers will discover a system-wide fix soon. Any other questions? -Wikid77 (talk) 15:47, 16 May 2013 (UTC)
I don’t understand the technical issues, but regardless: why should working around bugs in Google’s spyoftware take precedence over our article-naming policies?—Odysseus1479 (talk) 20:49, 16 May 2013 (UTC)
• Renaming for Google-https links always retained name as redirect: Plus, almost half of WP users come through Google Search. Yet the false claim how re-indexing Google would "take precedence over our article-naming policies" (wp:TITLE) was an exaggeration, because even for "Hyperbola (mathematics)" then the original title "Hyperbola" still redirected to the "Hyperbola" page, with no distraction of considering any other pages. However, even the temporary title as "Hyperbola (mathematics)" was another common name for the page, as in "Function (mathematics)" or "e (mathematics)" or "Division (mathematics)" etc. There was never any real problem with the interim page name. In general, beware discussions which can promote a straw man fallacy, claiming that some other imagined problem has been created, with the implicit slant that rejecting the initial work will stop the imagined problem (such as stop "take precedence over our article-naming policies"). Perhaps a more common imagined problem is claiming how new changes would "warp our little minds" (aka South Park TV phrase), where several attempted major improvements have been fought by claiming other editors, or the readers, are just too dull to ever cope with slight changes to Wikipedia practices. In reality, the vast majority of people have vast powers to learn new techniques, but might require extra time to adjust, such as learning "sampling" techniques of pageview counts to judge acceptance, as compared to counting every single reader's vote before making a decision. -Wikid77 (talk) 15:33, 18 May 2013 (UTC)
Stop confusing the issue with WP:TLDR responses and answer the question in a simple and concise manner: why is a Google bug, that is the sole responsibility of Google to fix, justification for violating Wikipedia policy? jcgoble3 (talk) 17:51, 20 May 2013 (UTC)

## Virtual hyperbola

when delta=0, hyperbola degenerates. Similar to this, ellipse degenerates when delta=0, and when C*delta>0 the ellipse is imaginary, when C*delta<0 the ellipse is real. Do we have similar imaginary hyperbola here? Jackzhp (talk) 16:29, 14 November 2013 (UTC)

No. As you can see by considering the simple case $ax^2 + by^2 + c = 0$, the sign determines which pair of the four quadrants determined by the asymptotes the hyperbola lies in. --JBL (talk) 19:01, 16 November 2013 (UTC)
Compare $x^2+y^2=\pm5$ versus $x^2-y^2=\pm5.$ In the former, ellipse, case, if the right side is -5 then if you solve for y in terms of x, any real x gives a complex solution for y. But nothing comparable happens for the hyperbola: no matter whether the right side is 5 or -5, there are real values of x that give rise to real values of y. Duoduoduo (talk) 00:28, 17 November 2013 (UTC)

## Bad depiction of vanishing point

The section Hyperbola#Conic section analysis of the hyperbolic appearance of circles uses an image which appears to suggest that parallel lines are naturally truncated by reaching the horizon. Usually the ground is curved, so this will happen as a result of occlusion, but not because there's a small distance along the ground after which you must see clouds above. If you can't see far enough to where the lines should converge, you simply can't tell the colors coming from that direction apart from the color reaching the spaces around it in the eye, including the spaces left and right of the point of convergence. Unless you live on an asteroid, you would see indistinctness of straight lines in the distance before you'd see occlusion. Compare Visible convergence and Convergence occluded. ᛭ LokiClock (talk) 01:46, 5 January 2014 (UTC)

For someone standing directly above the center of a horizontal circle and looking straight ahead, say north, the lens plane will intersect the circle in an east-west diameter directly below the viewer. The tangents at the diameter's endpoints will then be parallel, and what you say will then be correct.
The point of "gazing down slightly" in the caption is to rotate the lens plane about a horizontal east-west axis so that it intersects the circle in an east-west chord behind the viewer whose tangents intersect at a finite distance to the south of the viewer. Everything above the horizon is the virtual image of the two half-tangents that are behind the viewer including their intersection. If the viewer gazes up slightly the intersection will move down, flip from south to north as it crosses the horizon, and thereafter be a real image on the retina. Vaughan Pratt (talk) 17:57, 20 January 2014 (UTC)
I added a sentence to the caption including a pointer to the text which describes all this in more detail. Vaughan Pratt (talk) 18:22, 20 January 2014 (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)

## 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)