Talk:Conic section

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Request for edits[edit]

I am of the opinion that perhaps the section on Apollonius should be edited to reflect a few facts. The name of the work is "The Conics", not "Conic Sections;" else wise we would see a form of τομη in the title given in various MSS (,204,204,191,206, and others). Moreover, it should be noted that book eight is entirely lost, and five, six, and seven only survived in the Arabic tradition, meaning that Europeans would not have had access to them until very late, probably even after Heiberg compiled his critical edition of the the commentary on The Conics. This is important, because there is, as aforementioned, a companion text by Eutocius of Ascalon to The Conics. This is a text that also deserves mention, because unlike books 5-8 of the Conics, most European mathematicians who were studying conic sections would have been aware of Eutocius's commentary. Eutocius, in fact, recommends that they be taken together (viz. typeset side-by-side). We also know that Eutocius was writing his commentary, at least in part, for Anthemius of Tralles, who designed the Hagia Sophia. As it stands, the commentary only exists today in the Greek and the Latin, but I do know that soon a French translation is forthcoming from the same team that did all seven surviving books of the The Conics. An American professor's dissertation takes the first book's commentary from the Greek into the English, which is slightly over half of the text. He also claims that books two, three, and four are forthcoming. — Preceding unsigned comment added by (talk) 06:52, 24 June 2014 (UTC)

"Directrix" is circularly defined[edit]

The Wikipedia entry for "Directrix" redirects to this page on conic sections; however this page does not define "directrix" other than in one vague graphical example. It does however use the term "directrix" to define other terms such as "latus rectum" and thus it's a frustrating Wikipedia experience trying to learn what either of these terms really mean.

I have an intuitive sense of what the directrix is but with my high school analytic geometry am not qualified to define it rigorously in this article. Can someone help? Cvkline (talk) 04:36, 26 May 2009 (UTC)

I myself came to this page specifically to find out what the term directrix means, or actually I was re-directed here. It seems that someone went to the bother of making sure any search for directrix on wiki would end up here, but then did not bother to define it here. If, as it appears, this is a term basal to the rest of the subject, then it is beyond sloppy to have left it in this state for more than a year, as I see the previous post was about one year ago. Please, someone fix it. I'd actually really like to know what it means. DB —Preceding unsigned comment added by (talk) 19:46, 3 June 2010 (UTC)

I have changed the lead of the article to introduce the concept, and I also changed the subsection header of "Eccentricity" to Conic section#Eccentricity, focus and directrix. DVdm (talk) 20:25, 3 June 2010 (UTC)

Discussion of irreducible conics[edit]

Should be added —Preceding unsigned comment added by (talk) 19:10, 3 June 2008 (UTC)

Is there a Cartesian coordinate equation for a hyperbola caused by a plane which intersects the axis of the cone and has a defined distance between the path of the points of the hyperbola and the point of intersection of the axis of the cone? WFPMWFPM (talk) 15:47, 26 August 2008 (UTC)And if there is, why wouldn't that be the appropriate equation for Rutherford to have used in his alpha particle deflection equations rather than an equation where the plane? is parallel with and doesn't cut the axis of the cone.WFPMWFPM (talk) 16:44, 26 August 2008 (UTC)

Also is there a possible conical orbit path such as that proposed for the "orbitals" of the electrons in the electron orbital theory, whereby the electron con move in an orbital that has a constant amount of "lost electrostatic energy potential" between changes from one level to another? WFPMWFPM (talk) 18:33, 26 August 2008 (UTC)

Please notice that in your picture of the relationship of the cone to the cutting plane you leave out the notion of the cone to a flat angled plane that cuts both lobes of the cone as well as the axis of the cone and I think is a kind of hyperbolic path. Please tell me if I'm wrong.WFPMWFPM (talk) 19:24, 26 August 2008 (UTC)WFPMWFPM (talk) 19:29, 26 August 2008 (UTC)

Polar coordinates[edit]

This section is a fragment, stub, etc. Thats the basic equation in polar format for a hyperbola, has no real substance concerning the actual equations or methods of fiding the equations for the others or calculating the others, in short someone should give it some loving. (talk) 02:58, 10 December 2007 (UTC)


I have removed the first picture because it is incorrect (it shows only one half of the cone), and redundant. I have moved the already existing correct representation of the three conics at the top. Ceroklis 00:23, 9 October 2007 (UTC)


Again this page really needs a visual and should be written in a way accessible to all readers. This is not complex material. And a revision should be fairly easy...before this whole topic becomes esoteric.

Uh, it already is pretty esoteric - Anon

I recommend the following link for graphics on conics. User:Dick Beldin

Very good visuals, thank you. RoseParks

Incorrect determinant[edit]

I moved this:

Finally, if the following determinant,

                 | a h g | 
                 | h b f |
                 | g h e |

equals 0, it represents a pair of straight lines, that may not coincide.

This is incorrect. There probably is a determinant like this, but it would be preferable to use the correct one. AxelBoldt 18:32 Oct 23, 2002 (UTC)

Actually, the determinant is incorrect due to a typo.
The correct version is:
                 | a h g | 
                 | h b f |
                 | g f c |
Also note that if it is zero, it does not nééd to represent two lines, it may alo represent a single point, this can be seen as 2 imaginary lines that cross eachother in this special point. —Preceding unsigned comment added by (talk) 13:55, 27 August 2005

Improve hyperbola image[edit]

I hate to complain when someone puts a lot of work into creating images, but it would be nice if the image of the hyperbola made it visually apparent that a hyperbola has two asymptotes. (A common error committed by students asked to draw graphs showing the asymptotes is to draw the lines in the right places and then draw a curve that does not at all appear to approach the lines; a good image could help them understand that that is an error.) Michael Hardy 21:23 17 May 2003 (UTC)

I came here to point out that the picture of the hyperbola does not appear to be very hyperbolic; then I saw that Michael Hardy had a similar complaint. Is that picture really a hyperbola? Dominus 02:44 12 Jun 2003 (UTC)

Dimensionality of conic sections[edit]

Surely conic sections are one-dimensional? Rvollmert 17:02, 19 Apr 2004 (UTC)

A conic section is one-dimensional in the sense of being locally homeomorphic to R1, but two-dimensional in the sense of being a subset of the plane that is not a subset of any line. This latter sense is much closer to the conventional meaning of "two-dimensional". Even topologists recognize that a statement like "the sphere is a two-dimensional manifold" requires additional explanation for a general audience. As a geometric object, the sphere is three-dimensional, because it is a subset of R3 but not of R2; the conic section, analogously, is a two-dimensional geometric figure, even though it is topologically a one-dimensional manifold. -- Dominus 20:30, 19 Apr 2004 (UTC)
Hmm. I see what you mean, sort of. This "conventional" meaning of dimension is not what the linked article on dimension specifies, though. When the term is used in an incompatible sense, that should at least be noted. Maybe dimension should be updated to cover this meaning, too? I'll remove the note on dimension for now, but feel free to readd it if you think it's not generally confusing. Rvollmert 13:40, 26 Jul 2004 (UTC)


The first image has been replaced, original didn't have a verified copyright. I also made an alternative at Image:Conic_sections_2.png, based on a suggestion at Wikipedia:Image recreation requests. Take your pick. Duk 20:37, 18 Feb 2005 (UTC)

Semi-latus rectum and polar coordinates[edit]

Can anyone explain the origin of the term ? It's clearly half of the "latus rectum"; and my dim memory of Latin says "latus" means "carried" and I suppose "rectum" is saying the line is at right angles to the major axis. Is each line parllel to the directix called a "rectum" ? Why is the one through a focus called "carried" ? comment from user: moved from article to here

  • latus can be the perfect passive participle of fero, ferre, but it could also be the noun latus, lateris, n., which means "side" as in lateral. So the phrase semi-latus rectum could break out to "half the side, having been made straight." HTH (Sorry, I'm not a mathematician, just a grammarian). --Fulminouscherub 22:50, 13 December 2005 (UTC)


I'm a precalc student interested in higher-dimention sections. To me, "higher dimention" has two meanings: first, more than two variables, and second, more than squares of those variables (cubes, quartics, quintics, whatever those are called...). This article links to higher dimentions in the first sense, but ignores this second sense. In paricular, I was wondering if this was the proper way to expand the equation (all expressions set equil to zero):

Various powers of two variables:

Single Number:

(x+y)^0 => a

2D Line:

(x+y)^0 + (x+y)^1 => a + bx + cy

2D Curve (Conic Section general equation):

(x+y)^0 + (x+y)^1 + (x+y)^2 => a + bx + cy + dxy + ex^2 + fy^2

2D Extracurve (the shape I'm interested in learning about):

(x+y)^0 + (x+y)^1 + (x+y)^2 + (x+y)^3 => a + bx + cy + dxy + ex^2 + fy^2 + g3(x^2)y + h3x(y^3) ix^3 + jy^3

And et cetera for higher degrees (that is, higher powers). Greater than two dimentions use, for example, (x+y+z)^n rather than (x+y)^n.

This system is a complete guess, but is supposed to represent 2D intersections with objects of higher dimentionality than a cone. A cone is a 3D object that describes ^2 polynomials. Therefore, I figured there would be a 4D object to cover ^3 polynomials, and so on. I can't seem to find any information on this.

A couple comments:
  • Dimension is spelled with an S in US English. I thought it was in British English as well, but please let me know if I'm wrong.
  • Another way to "add a dimension" is:
(x+y+z)^0 + (x+y+z)^1 + (x+y+z)^2
  • Consider the pattern:
  • A line is the intersection of two planes.
  • A conic section is the intersection of a plane and a cone (with both lobes included).
Perhaps the extracurve might be an intersection of two cones or of a plane and a 4th-dimensional cone. One form of a 4th-dimensional cone might be with time as the 4th dimension. That is, the shape of the cone (and thus the conic section), varies over time. StuRat 19:01, 1 October 2005 (UTC)
The higher dimension generalizations are generally considered to be quadratic in their many variables (so higher dimensional in the 'first sense' using the student's terminology) from what I gather. This is because these quadratic curves will retain much of the nice properties that make '2D' conic sections useful. I think this is probably why functions cubic in their variables aren't discussed. Small note: I don't think if we are to consider higher dimensional spaces that it is useful to consider the added dimension time since this implies (at least to some people), the use of the Minkowski metric, which will cause problems if we define conics using analytic geometry. Threepounds 04:30, 27 November 2005 (UTC)

Degenerate Conics[edit]

There are two more degenerate cases, not listed in the introductory section. They require the cone itself to be degenerate; where the angle generating the cone is either 90 or 0 degrees. When the angle is 90 degrees, the interior of the cone encompasses all of three-dimensional space and the exterior of the cone is the plane passing through the apex and orthogonal to the cone's axis. That same plane may be chosen as the intoersector, yeilding the plane, included in its entirety. On the other hand, when the cone's generating angle is zero and the plane is parallel to (but not encompassing) the cone's axis, the intersection is null.

Algebraically, those are obtained by setting all parameters equal to 0 (giving the entire plane) or setting c not equal to zero while all other parameters do equal zero, giving the empty set.

Though these cases seem trivial, I think that since they are obtainable both algebraically and geometrically, they are demonstrably conics and should be mentioned.

Also, I added one word to the page to state that in the degenerate case of two lines, those lines must intersect.

Please forgive my unfamiliarity with how to add a timestamp. Dvd_Avins edited on Feb. 16, 2006 Dvd_Avins

Use four tildes (~~~~) to add both your screen name and the date at the end. I think including those "degenerate cone" degenerate cases might be going a bit far, myself, as adding such material might make it harder for beginners to understand the basics. Can you show us how they are obtainable algebraically ? (This might sway people toward including those degenerate cases.) StuRat 20:03, 16 February 2006 (UTC)
With the form , suppose a, b, f, g, and h are all zero. If c also equals zero, the equation simplifies to 0 = 0, designating the entire plane. If c equals, say 5, than we get 5 = 0, designating the empty set. Dvd Avins 21:40, 17 February 2006 (UTC)
Ok, I see. What does everybody else think ? Is there any reason why these degenerate cases shouldn't be added ? StuRat 05:00, 18 February 2006 (UTC)

er, simplify?[edit]

Hi -- any chance of getting something added that just quickly reminds you how to tell whether a curve is parabolic or hyperbolic? For non mathematicians? --Jaibe 12:28, 3 January 2006 (UTC)

The only completely non-mathematical way to tell is that hyperbolas are "pointier" and parabolas are smoother. StuRat 19:54, 16 February 2006 (UTC)
Or, if you could see enough of the curve, you could see that a hyperbola approaches a 'v' shape while a parabola does not. Nat2 (talk) 02:21, 15 November 2011 (UTC)

Correct image[edit]

Is the illustration of a hyperbola correct? Mathworld ( )shows illustrates a hyperbola as consisting of two curves. NJS. —Preceding unsigned comment added by (talk) 10:48, 23 March 2006

I hope so

You're right, the image isn't correct. The cone that is shown in the first picture should actually be 2 cones, one upside down, and meeting at their tips. The hyperbola is the only one of the four conics that intersects the upside down cone, so it does have a double. That picture illustrates it better. ImMAW 00:11, 13 March 2007 (UTC)


Are conics quadratics? 03:39, 19 May 2006 (UTC)

yes. idiotoff 06:52, 23 May 2006 (UTC)

Eccentricity illustration[edit]

Isn't the illustration under a heading Eccentricity incorrect? The eccentricity is defined in text as a ratio of distances from a fixed point F and line M. But in order to have zero eccentricity, the point on a conic has to be an infinite distance away from line M or at zero distance from point F. This means that we can't draw a circle that satisfies those conditions, only a degenerate case: a point. Nevertheless, the illustration clearly shows a proper circle, not barely a point. Is this definition of eccentricity truly adequate for all conics, including circle or are there other definitions? --Riojajar 12:22, 4 June 2006 (UTC)


I'd like to remove the derivation as I believe that it breaks Wikipedia:What Wikipedia is not#Wikipedia is not an indiscriminate collection of information (see: Textbooks and annotated texts) and belongs on Wikibooks. As it happens, there is already a book on Wikibooks:Conic Sections that I've added a link to. --Swift 23:52, 1 August 2006 (UTC)

I agree the derivations is overkill. I've deleted the section. --Salix alba (talk) 19:02, 24 September 2006 (UTC)

Conic Sections in Polar Coordinates[edit]

I think the polar coordinate equations in Conic Sections in Polar Coordinates is sufficiently important to be merged into this article. Thoughts? --Salix alba (talk) 19:08, 24 September 2006 (UTC)

The article already mentions the Cartesian coordinate system, so talking about polar would fit in well. --Swift 05:16, 25 September 2006 (UTC)
Anybody willing to do the work? :) If not, one could just redirect that one to here.Oleg Alexandrov (talk) 02:49, 26 September 2006 (UTC)
Be bold :-) ... or you could wait till I get around to it on my over full to-do list. I don't think we should redirect without actually mentioning polar coordinates in the destination article! --Swift 07:47, 26 September 2006 (UTC)

Done, there was already a section which did have the polar equation, which I've expanded a bit. --Salix alba (talk) 11:04, 26 September 2006 (UTC)

Good job! --Swift 20:18, 26 September 2006 (UTC)

The anon has pointed out some problems with the quadratic equation. Not all of these equations will be conic sections at all, for instance the cases when A>0,B>0 and F>0 or A<0,B<0 and F<0. I think these sort of relate to complex solutions. I don't off hand know the general conditions to include these illegal cases. --Salix alba (talk) 08:02, 3 October 2006 (UTC)

emmendation on statements concerning the circle[edit]

the claim that a circle is generated from a cut perpendicular to the axis is erronious in cases of oblique cones (i.e. cones with axes that are not perpendicular to their bases.) the statement in this article is fine so long as the reader assumes the cone to be right, but is misleading if otherwise. also, in oblique cones two circles are generable, one parallel and another subcontrariwise to the base (c.f. Apollonius Conics I. 4 and I. 5) —The preceding unsigned comment was added by (talk) 08:07, 28 February 2007 (UTC).

Merger proposal[edit]

Conics intersection is a small article and should not be an article by itself. It is also directly related to conic sections, as it is just the intersection of two conic sections. --Pbroks13 01:21, 9 March 2008 (UTC)

Degenerate conics (again)[edit]

Can anybody explain what a rectangular hyperbola is, and how a conic can degenerate into a cylinder?

00:23, 14 April 2008 (Talk) (16,872 bytes) (→Types of conics) introduced the rectangular hyperbola comment.

03:36, 21 April 2008 (Talk) (17,156 bytes) (→Degenerate cases: cylindric section) added the comment about cylindrical sections.

I think these are related, but the IPs are different and there was a few days (and edits) in between...

Feyrauth (talk) 03:52, 13 June 2008 (UTC)

It would be good to add a section explaining that the equation of a right cone with vertex (0,0,0) is

|x|^2-(1+\lambda^2)|x\cdot n|^2=0,

where x=(x_1,x_2,x_3) is the position vector, n a unit vector parallel to the axis of the cone, and \lambda=tan(\theta/2), where \theta is the opening angle of the cone.

The degenerate case of a cylinder is then \lambda=0, so when the opening angle of the cone is zero.

Since the article is locked, I can't add this... Would someone with the appropriate credentials do this please? (Btw, the article shows nicely the strengths and limitations of Wikipedia...) —Preceding unsigned comment added by (talk) 23:13, 14 January 2009 (UTC)

Image is blocking text[edit]

On my screen, the first image (File:Conic sections 2.png) is covering several words in the lead paragraph. I am using Safari. —Preceding unsigned comment added by (talk) 14:35, 5 February 2009 (UTC)

Fine here. Suggest you file a bug-report or figure out a way to fix the page rendering without causing problems for others. --Swift (talk) 05:11, 5 March 2009 (UTC)

Illustration of an ellipse[edit]

The subject thumbnail shows a grossly incorrect illustration of the semi latus rectum as it pertains to an ellipse. For an ellipse, the semi latus rectum is equal in length to 1/2 the major axis, or alternatively, 1/2 the distance from one focal point to any given point on the ellipse to the other focal point. The illustration shows the semi latus rectum as a line parallel to the minor axis of the ellipse from one focal point to the ellipse that is far shorter than the semimajor axis.

I assume the original intent was to illustrate the latus rectum as the sum of the length of lines from both focal points. Don Seib Don Seib 19:53, 27 September 2009 (UTC)

The image is consistent with the article and with everything else I've read on conics. See for example the diagram at the "Derivations of Conic Sections" in the external links section. Can you cite a source that gives a different meaning to the term?--RDBury (talk) 08:54, 28 September 2009 (UTC)
The sum of the length of lines from both focal points to a point on an ellipse is nothing but the length of the major-axis, or . The latus rectum is a completely different notion. In fact if and only if the conic section is a circle or a rectangular hyperbola. Barsamin (talk) 23:31, 19 October 2009 (UTC)

A different problem with the illustration is that the formulas use the terms "a" and "b", obviously constants, but these are not shown on the illustration. Neither are they defined in the discussion. These constants should be defined, and shown on the illustration. — Preceding unsigned comment added by Saintonge235 (talkcontribs) 18:15, 21 January 2011 (UTC)

Eccentricity Formula[edit]

Instead of writing the eccentricity formula for an ellipse as , wouldn't it be more clear to write it as (and the same for the hyperbola using the + sign). It is then immediately clear that the eccentricity is between zero and one for the ellipse and larger than one for the hyperbola. (talk) 11:37, 7 October 2010 (UTC)

Yes, agree. Good catch. Go ahead. I went ahead, as the article seems to be semi-protected. DVdm (talk) 11:47, 7 October 2010 (UTC)

The eccentricity definition depends on a & b parameters. If a>b definition e=c/a is correct, but if b>a then e=c/b. So e= as it must be to have sense. A general definition of e comes from the "one focus definition" (I call it so...) Given one focus F and a directrix, let P be a point of the plane and AP the distance of P from the directix. If e=PA/PF is constant then P lays over a conic and the ratio is called eccentricity. — Preceding unsigned comment added by (talk) 11:01, 1 November 2011 (UTC) (Alessio, Math Teacher from Italy; 1/11/2011)

Edit request from Preetum, 21 October 2010[edit]

{{edit semi-protected}} Please add this modified form of the polar equation of an ellipse, which uses the same constants a,b as in Cartesian form: b^2/(a - Cos[t] Sqrt[a^2 - b^2])

Preetum (talk) 23:21, 21 October 2010 (UTC)

Not done: Please provide a source for this. Also, where did you want to add this? Celestra (talk) 22:55, 25 October 2010 (UTC)

Edit request to Conics Intersection paragraph, 29 November 2010[edit]

{{edit semi-protected}} Please add the reference to this MATLAB Central URL containing the code to detect conics intersection:

Pierluigi 8:52, 29 November 2010 (UTC)

Fifth type of conic?[edit]

What would happen if the plane intersecting the cones had a slope greater than that for a parabola, but less than that of an ellipse? One would have a form similar to a hyperbola, but the two forms would appear to be parabolas of different eccentricities. Could someone please either prove me wrong or explain this on the page? Thank you. (talk) 15:56, 7 February 2011 (UTC)

The cone can be considered as the union of a family of lines that all intersect in a single point. Every line in this family has the same slope s. The slope of the intersecting plane can be greater than, less than, or equal to s, in which cases the intersection is a hyperbola, an ellipse, or a parabola, respectively. Thus there are only three cases. In particular, there is no case "between" a parabola and an ellipse: if the intersecting plane has slope exactly s, the intersection is a parabola; if it is less, the intersection is an ellipse. Hope this helps. —Mark Dominus (talk) 16:18, 7 February 2011 (UTC)

I believe that I misstated the above. Sorry. I meant that the slope would be between that of the plane that produced a parabola and that of the plane that produced a hyperbola. I would think that such a plane would create a hyperboloid with one branch with a different eccentricity than the other. My various math books have always seemed to indicate that a hyperbola's conic plane must be vertical. My question is why do the schoolbooks never recognize the existence of this irregular hyperbola? (talk) 22:59, 8 February 2011 (UTC)

It might seem like it will be irregular, but it won't. The two branches come out the same. Offhand I can't think of any way of making this obvious, sorry. —Mark Dominus (talk) 00:34, 9 February 2011 (UTC)
Using a vertical plane is the simplest way of obtaining a hyperbola, so in this sense 'the schoolbooks" are right. I don't know what they say exactly, could you quote the exact passage? If they seem to imply that the plane must be vertical, there could be a misunderstanding here. If we tilt the plane away from the vertical position, an odd thing happens, namely that one of the two branches of the intersection curve will be closer to the origin in 3-space than the other branch. Is that what you are worried about? Tkuvho (talk) 05:55, 9 February 2011 (UTC)
I think the OP's concern is that the tilted plane seems to cut the the two nappes of the cone in very different ways, and so the two branches of the hyperbola would seem to have different shapes. For example, consider a plane that intersects the cone in a parabola. It will intersect only one nappe, say the lower one, and misses the other nappe entirely. Now perturb the plane slightly so that it intersects the cone in a hyperbola instead. The parabola has become the lower branch of the hyperbola, but it is still very nearly parabolic. The plane intersects the upper nappe very far away, and at a very small angle. It is not immediately clear that the upper branch is exactly the same shape as the lower branch.
Or consider a vertical plane. This plane cuts the upper and lower nappes at the same angle, by symmetry. Now rotate the plane slightly so that its upper half moves slightly closer to the cone and the lower half moves slightly farther away. The angle that the plane makes with the upper nappe increases, and the angle with the lower nappe decreases. In the limiting case, the intersection is a parabola and the plane misses the lower nappe entirely. But before this limiting case is reached, the intersection is a hyperbola. Since one angle of intersection increases and the other decreases, why should the two branches of the hyperbola be the same shape?
I hope this makes the question clearer rather than more obscure. —Mark Dominus (talk) 06:14, 9 February 2011 (UTC)

This discussion is continued at Talk:Hyperbola#How to characterize special case. Duoduoduo (talk) 20:41, 24 February 2011 (UTC)

Thanks for the pointer. —Mark Dominus (talk) 21:16, 24 February 2011 (UTC)

Edit request on 18 December 2011[edit]

Please, add the following entry to the external links section:

Interactive three-dimensional conic sections

Jjbuchholz (talk) 16:50, 18 December 2011 (UTC)

When I tried the link I got some javascript errors, external links should work on pretty much any machine.--RDBury (talk) 17:30, 18 December 2011 (UTC)

Edit Request February 12 2012[edit]

Aren't lines and points conic sections also? If so, please change the page to include them. (talk) 14:07, 12 February 2012 (UTC)

See Degenerate conic. Tkuvho (talk) 14:14, 12 February 2012 (UTC)

Parabola equation[edit]

Please add this equation for the parabola

The source is given in the article Parabola (talk) 11:26, 2 April 2012 (UTC)

Not done: please provide reliable sources that support the change you want to be made. The source needs to be given here. And please say where exactly this formula should go. Thanks, Celestra (talk) 14:58, 2 April 2012 (UTC) Here it is: (talk) 10:40, 3 April 2012 (UTC)

Ah, the equation , as well as the equation for the circle can be given in the Features section of this article. (talk) 10:45, 3 April 2012 (UTC)
Sorry, that was wrong. I tried it in GeoGebra. (talk) 12:14, 5 April 2012 (UTC)

Not done: please provide reliable sources that support the change you want to be made. That source would fall under self-published sources. Sorry, Celestra (talk) 20:02, 5 April 2012 (UTC)

Infinite eccentricity[edit]

What will happen if the plane is fully vertical to the cone (perpendicular to the base of the cone) and does not passing through the apex/vertex? (talk) 11:35, 2 April 2012 (UTC)

The editors at Wikipedia:Reference desk/Mathematics are experts at questions like this. Please consider posting there, and you'll likely get a faster response. Dru of Id (talk) 11:27, 3 April 2012 (UTC)

Edit request on 27 October 2012[edit]

Add as an external link:The Conics Generated by the Method of Application of Areas at Theodoros v (talk) 08:28, 27 October 2012 (UTC)

Not done, per WP:ELNO point 13. - MrOllie (talk) 16:42, 27 October 2012 (UTC)

sorry! the link directed to the abstract of the source, this is the link to the source itself! cheers   — Preceding unsigned comment added by Theodoros v (talkcontribs) 16:42, 4 November 2012 (UTC) 

Edit request on 29 October 2012[edit]

"Please change the 3rd paragraph below the title "The conic sections were named and studied at least since 200 BC, when Apollonius of Perga undertook a systematic study of their properties." to: "The conics were most probably discovered as plane curves by the Method of Application of Areas and took their names according to the type of application used for their construction (Add a Note:Proclus' Commentary on the First Book of Euclid's Elements,edited by G.Friedlein,Leipzig, 1891-3,p.419). Later on they were studied thoroughly as conic sections by Apollonius" because the former paragraph is (a) ambiguous,(b)it is contradicted by the next section on Early Works and (c) it is not verifiable". Sardelisdim (talk) 09:42, 29 October 2012 (UTC)

Not done: Personally, I feel that this book will notbe in widespread use, so it will be hard to verify - can you link to a webpage that we can see this information on? Thanks, Mdann52 (talk) 13:16, 1 November 2012 (UTC)

The link is found at the Proclus page of Wikipedia. It reads:

   Proclus' Commentary on Euclid, Book I. PDF scans of Friedlein's Greek edition, now in the public domain (Classical Greek)

The relevant pages are 419-421 of the Commentary.

I should stress that Proclus is one of the two most reliable sources on ancient greek Mathematics (The other one is Pappus) — Preceding unsigned comment added by Sardelisdim (talkcontribs) 16:20, 3 November 2012 (UTC)

The section "Intersecting two conics" needs work[edit]

  • Currently the section "Intersecting two conics" says
The best method of locating these solutions exploits the homogeneous matrix representation of conic sections, i.e. a 3x3 symmetric matrix which depends on six parameters.

ending the paragraph, and then the next paragraph begins

The procedure to locate the intersection points follows these steps:....

But the procedure given makes no explicit reference to the 3×3 matrix. Are these two paragraphs unrelated? Or is there supposed to be some flow from the one to the other?

  • The steps in the procedure include the sentence
This can be done by imposing that , which turns out to be the solution to a third degree equation.

which doesn't make sense since an equation cannot be the solution of an equation.

  • The first step in the procedure says
given the two conics and consider the pencil of conics given by their linear combination

Maybe this will be clear to someone who is more familiar with this than I am, but I can't understand what specific mathematical expressions we're taking a linear combination of.

  • One step says that we should
intersects each identified line with one of the two original conic

Which line should be intersected with which conic? If we intersect both lines with both conics, won't we get 8 intersection points, even though there are no more than 4 answers?

  • One step says
this step can be done efficiently using the dual conic representation of

What is meant by the dual conic representation?

  • This section is unreferenced. Is it OR?
  • Anyway, wouldn't it be more straightforward to find the intersection points of the conics by just solving one of the conic equations for two solutions for y in terms of x, and substituting each y expression separately into the other conic equation and solving to get two solutions for x, for a total of four solutions (some of which may be real)? Duoduoduo (talk) 20:34, 11 December 2012 (UTC)
While I certainly agree that the section is not clearly written, you are being a little harsh in your comments. This is all fairly straightforward using the properties of a pencil of conics. Two conics determine a pencil. All the conics of the pencil pass through the four points of intersection of the original two, including the always present degenerate conic of the pencil. This consists of two lines. If you can get your hands on their equations, each of them intersected with either of the original pair will give the same 2 of the four intersection points (so, 4 points in total), so you can get the solutions algebraically by solving quadratic equations rather than the implicit quartic equations you suggested. Everything I've said so far assumes that you are counting things with multiplicity and are working over the complex numbers. The representation of the pencil that is given in the article is pretty standard. In practice, the 's can be replaced either by the quadratic forms of the conics or by the 3x3 matrices which represent them. By using the determinant condition to find the degenerate conics, it is clear that the editor was using the matrix forms. The awkward sentence is supposed to mean that the variables λ and μ are solutions of a cubic (rather than a higher degree) polynomial. I can not tell whether or not this is OR. Modern texts do not seem to get into this kind of detail and I haven't been able to find a reference in any of my older texts. My gut feeling is that this is not OR, but it will be hard to get a reliable reference. I think that I can fix up the section in a week or so, and I'll keep looking for a reference. Bill Cherowitzo (talk) 05:19, 12 December 2012 (UTC)
Thanks! Duoduoduo (talk) 13:53, 12 December 2012 (UTC)

Questions on section on pencil of conics[edit]

I think the new edits about the pencil of conics are very helpful. I have these questions:

But the present section then goes on to say that we are thinking of C1 as the 3×3 symmetric matrix, which comes from a homogeneous form in x, y, and z. So should binary form be replaced by something like trinary form or whatever?

You are right, I am being a bit sloppy and am not making the distinctions between affine and projective versions of the same thing. Note that the rest of the article suffers from the same problem (at one point describing things affinely and at another point projectively). When I have a little more time I'll go back and clarify these differences. Bill Cherowitzo (talk) 18:26, 19 December 2012 (UTC)
  • The last sentence of the section says two concrete representations (either equations or matrices) give the same conic if they differ by a non-zero multiplicative constant. I don't understand the if they differ by ... part. Duoduoduo (talk) 14:18, 19 December 2012 (UTC)
Homogeneous coordinates are only determined up to a multiplicative constant, (x,y,z) and (ax,ay,az) represent the same point. This is sometimes indicated with the notation (x:y:z) for homogeneous coordinates. Consequently, when a 3×3 matrix is used to represent a conic, any scalar multiple of the matrix will represent the same conic (you can think of the scalar as being pulled out of the matrix and put into the vector representing a point ... changing the vector but not the point). As homogeneous coordinates are needed for this, I'll have to clarify that we are dealing with the projective version. Bill Cherowitzo (talk) 18:26, 19 December 2012 (UTC)
  • Also, you say This symbolic representation can be made concrete with a slight abuse of notation. What specifically is the "abuse"? It seems to me that if you define C1 in a certain way, as a quadratic form or as a 3×3 matrix, then the notation is accurate. Duoduoduo (talk) 14:26, 19 December 2012 (UTC)
The abuse is that we are using the symbol C1 to represent both the name of the conic and also the algebraic expression that is used to define the coordinates of the points of the conic. We do this type of thing so often that most people just gloss over the difference. Bill Cherowitzo (talk) 18:26, 19 December 2012 (UTC)
  • Also, you say
Furthermore, the four base points determine three line pairs (degenerate conics through the base points) and so each pencil of conics will contain at most three degenerate conics.[12]

I'm probably misunderstanding something here. With four points, I think we can draw three line pairs each of which intersects at a particular point; since there are four points each of which could be the intersection point, that gives 12 degenerate conics. No? Duoduoduo (talk) 14:37, 19 December 2012 (UTC) Striking that -- didn't make sense since the line pair representing the degenerate conic has to pass through all four points. I don't see how any line pair can pass through all four points unless three of them are collinear; but then I can only draw two line pairs each of which passes through all four points. Can you help me out here? Duoduoduo (talk) 15:43, 19 December 2012 (UTC)

If the four points (no three collinear) are A,B,C and D then the line pairs are AB,CD; AC,BD; and AD,BC. If three of the points are collinear then one line must contain the three and any line through the fourth point as the second line (in this case you lose uniqueness). If all four points are collinear you get either one line repeated or the line containing all four points and any other line (this again loses uniqueness).Bill Cherowitzo (talk) 18:26, 19 December 2012 (UTC)

Okay, I get it now -- I was forgetting that the lines extend past where they intersect, rather than just having a vertex. May I suggest this wording?: Instead of

Furthermore, the four base points determine three line pairs (degenerate conics through the base points) and so each pencil of conics will contain at most three degenerate conics.

how about

Furthermore, the four base points determine three line pairs (degenerate conics); in each pair of lines one line passes through two of the points and the other passes through the other two points. So each pencil of conics will contain at most three degenerate conics.

Duoduoduo (talk) 18:05, 19 December 2012 (UTC)

This is ok.
  • Also: the current wording of the section contains
A conic is uniquely determined by five points in general position (no three collinear)

To me this wording seems ambiguous, though I know what is intended. It means

Five given points (no three collinear) determine a unique conic.

I think this wording would be preferable, since someone might think the current wording means

A given conic is determined by a unique set of five points (no three collinear).
  • Also I think that in the above wording "a unique conic" should be replaced by "a unique non-degenerate conic", to prepare the reader for the later mention of degenerate conics through four of the points. Duoduoduo (talk) 18:18, 19 December 2012 (UTC)
I think that I would prefer "A non-degenerate conic is completely determined ..." Bill Cherowitzo (talk) 18:26, 19 December 2012 (UTC)

Example of intersecting two conics[edit]

I always like to see an example worked out, so I can try to infer anything that I did not understand in the text. I worked out a couple of examples here:

I think adding the second one (more general, the first one is unnecessary really) to the wiki page would be a huge help for this section.

daviddoria (talk) 20:45, 28 February 2013 (UTC)

standard form for the equation of a conic section[edit]

Is it not possible, inside Wikipedia, to agree about a standard form for the equation of a conic section? In each of the following articles the form used for the equation of a conic section is different.

Jhncls (talk) 16:32, 13 August 2013 (UTC)

The problem with talking about a standard form is that you have to agree on what geometry the conic section is supposed to live in. These articles do not agree and do not make their assumptions explicit. The first is the most general form for the equation of a conic in a 2-dimensional affine space over an arbitrary field. The second places also places the conic in an affine plane but over a field whose characteristic is not 2 (this is a common assumption for an algebraic geometer to make). The third article does not talk about equations but rather forms, and gives the same affine form as the second article, just recast in this terminology. There is also a form for the conic in a projective plane, but again restricted to fields whose characteristic is not 2. The most general form for a conic section equation in a projective plane over an arbitrary field is:
with all constants coming from the field used to construct the projective plane. Can this problem (if it is a problem) be fixed? I don't think so, or at least not without some major rewrites. We could, however, make some of the unwritten assumptions in these articles more explicit. Bill Cherowitzo (talk) 17:56, 13 August 2013 (UTC)

Error in "As slice of quadratic form"[edit]

The following statement is patently false.

Parabolas and hyperbolas can be realized by a horizontal plane (), while ellipses require that the plane be slanted.

Clearly the equation for an ellipse (non-circular even) can be extracted even with .

--Noldorin (talk) 02:43, 29 March 2014 (UTC)

You are quite right. The type of conic section is determined by the discriminant, B2 - 4AC, which is independent of the values of D and E. Bill Cherowitzo (talk) 04:27, 29 March 2014 (UTC)

Semi-protected edit request on 14 April 2014[edit]

The Polar Coordinates formula is incorrect. The sign should be negative.

Happily, this makes it consistent with the same formula some lines earlier.

David Geo Holmes (talk) 19:14, 14 April 2014 (UTC)

Red information icon with gradient background.svg Not done: please provide reliable sources that support the change you want to be made. Anupmehra -Let's talk! 19:31, 14 April 2014 (UTC)

'Directrix' Concept Inapplicable to the Circle[edit]

The previous version of the article used to read (prior to my edit):

Conic sections are exactly those curves that, for a point F, a line L not containing F and a non-negative number e, are the locus of points whose distance to F equals e times their distance to L

but this definition does not apply to the circle, even though the circle is a bona fide conic section. Attempts to save it by talking about a 'directrix at infinity' don't work because multiplying infinity by zero gives an indeterminate, not a constant, value. I've modified that section in question but would welcome further refinement by an experienced mathematician explaining simply but rigorously what, if anything, can be saved from this definition of a conic section as applied to the circle. — Preceding unsigned comment added by ChengduTeacher (talkcontribs) 03:04, 29 December 2014 (UTC)

Sorry, but I had to revert those edits as they are not correct. The directrix of any circle is the line at infinity. To see that this is so, you must use the actual definition of eccentricity and not a formula derived from it. The definition is a ratio, distance to focus/distance to directrix. For the circle, the distance to the focus is the length of a radius (some positive number) and in order to get a constant ratio, you must divide by something infinite (actually you are taking a limit as this distance grows without bound) to get zero. I have fixed the section in the article concerning eccentricity of the circle which also argued from an invalid formula. By the way, the tradition on these talk pages is to put new comments at the bottom of the page (as opposed to other places on the web where they go on top), so I took the liberty to move your comment. Bill Cherowitzo (talk) 05:40, 29 December 2014 (UTC)

Sorry, but what you say above makes absolutely no sense to me. Please give an actual mathematical definition of a circle using the 'directrix at infinity' concept. You CANNOT define as the circle the locus of points the ratio of whose distance from the focus to the distance from the 'line at infinity' equals zero! Why? Because that ratio is zero for ALL points in the plane, not merely the points on the circle! Please respond with some actual mathematics or I will re-revert.ChengduTeacher (talk) 12:02, 30 December 2014 (UTC)

My apologies, I reverted your edits too rashly and have replaced the essential qualification. The problem is that the article claimed that all conic sections could be defined by the focus-directrix property, but for the reason you point out, this is not true for circles. On the other hand, circles do have the focus-directrix property when the directrix is taken as the line at infinity (and so, the eccentricity is 0). This is a natural definition. Consider a right circular cone and a plane through its vertex orthogonal to the axis of the cone. An ellipse is determined by a second plane cutting the cone and the directrix of the ellipse is the intersection of the cutting plane and the original plane. As the inclination of the cutting plane to the original plane decreases, the directrix moves further away from the ellipse. Ultimately, the cutting plane is parallel to the original plane and the ellipse is a circle. The situation of parallel planes is often referred to as the planes meeting at the line at infinity (that statement can be made precise in the projective setting). Technically, you can say that the circle has no directrix in the plane, but this is rarely phrased that way and modern geometers look beyond the confines of the Euclidean plane to explain the consequences of unseen properties (for instance, all circles meet the line at infinity in the same two points which explains why only three non-collinear points are needed to define a circle while five are required for a general conic). Again, my apologies. Bill Cherowitzo (talk) 19:08, 30 December 2014 (UTC)
that statement can be made precise in the projective setting - Indeed, the idea of points and lines at infinity is a very useful one in projective geometry, and in that context can be made precise. However, we are not doing projective geometry here, but ordinary Cartesian algebraic geometry. If you want to add something referring to projective geometry, the types of transformations it involves, and its relation to conic sections, I would have no problem with that. However, as long as we're staying with Cartesian algebraic geometry, the 'directrix at infinity' of the circle is a useless and misleading idea. I'll take a look at your edits and maybe do some cleanup tomorrow. — Preceding unsigned comment added by ChengduTeacher (talkcontribs) 14:12, 31 December 2014 (UTC)

Semi-protected edit request on 5 May 2015[edit]

In the "Conic Section#In other areas of mathematics" section, under the "Quadratic forms" definition list entry, the bulleted list needs two changes to be formatted correctly:

  1. All of the bullets * need to have : prepended so they're indented correctly.
  2. The comma between "x2+y2" and "positive definite" needs to go away so the three lines are consistent.

I'd also consider promoting "Quadratic form classifications" and "Eccentricity classifications" to subsubsection headings, but I'd want to see how that looked. Thank you! (talk) 10:09, 5 May 2015 (UTC) (talk) 10:09, 5 May 2015 (UTC)

 Done Did not deal with the headings issue. Bill Cherowitzo (talk) 18:17, 5 May 2015 (UTC)

Elaboration on Polar Form of Conic Sections and Another Reference[edit]

(1) Eccentricity, directrix, focus, and semi-latus rectum have VERY visual connections with a 45-degree double-napped cone when using a polar form of the equation for conic sections. I have not seen these ideas presented anywhere on the Internet--just in books--but I think they tie some wonderful ideas together which would be perfect for Wikipedia. I do not have editing privileges for Conic Sections in Wikipedia. (I requested permission yesterday, but was denied and he suggested I try this route.) I uploaded a visual to Conic Sections in Wikimedia Commons (Polar_Conics.png) based on some graphs I created for my precalculus students this year. If someone with editing rights would like to incorporate these ideas into this article, I would be quite willing to work with you and modify the visuals to make them more suitable for Wikipedia. Thank you for your consideration.

(2) An excellent reference to add to the entire section is the following book: Kendig, Keith (2005), Conics, The Dolciani Mathematical Expositions #29, Washington, DC: The Mathematical Association of America, ISBN 0883853353.

Kent Thele (talk) 13:59, 30 June 2015 (UTC)

Semi-protected edit request on 17 December 2015[edit]

I suggest that this entry on conic sections be written in language easily comprehended by intelligent laymen. I can do real analysis -- yes, delta-epsilon proofs and more -- yet I struggle to get anything out of this entry. My son, a highly intelligent high-schooler, tried to read this entry because his teacher was unintelligible on the subject. But the entry was zero help. The goal of Wikipedia is to be helpful, right? (talk) 00:17, 17 December 2015 (UTC) (talk) 00:17, 17 December 2015 (UTC)

Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format. A request to essentially rewrite the entire article is a bit out of scope of a simple edit request Cannolis (talk) 04:10, 17 December 2015 (UTC)


In response to the above request, I've looked over the article as a whole and can definitely see some problems with the presentation. It is not well organized and mixes levels of sophistication in a hodge-podge fashion. As I see it, the article should concentrate on those themes that unify the conic sections, leaving particular properties to the individual pages of ellipse, hyperbola and parabola. I am suggesting a reorganization of the page along the following lines:

  • Scrap the current lead, a total rewrite will be necessary.
  • Material concerning the conic sections in Euclidean geometry should be presented first. This will include
    • A geometric definition
    • Focus, directrix, eccentricity properties
    • Cartesian coordinates
    • Polar coordinates
    • Common properties of the conics
  • The History section should follow the Euclidean presentation
  • Applications
  • Extension to the projective plane, including
    • Intersections with the line at infinity
    • Homogeneous coordinates
    • Focus-directrix property of the circle
    • Steiner generation of the conics - point by point construction
  • Extension to the complex plane
  • Degenerate conics
  • Advanced algebraic manipulations
    • Matrix form of equations (maybe earlier)
    • Pencil of conics
    • Intersection of two conics
  • Generalizations

In the above I've tried to include most topics on the current page, but there are some omissions (material that I think is either not germane or not interesting). Suggestions and modifications of this scheme are welcome. I'll start refactoring the page in a week (unless someone comes up with a good reason why I shouldn't before then). Bill Cherowitzo (talk) 19:14, 30 December 2015 (UTC)

I have finished the reorganization mentioned above with the exception of rewriting the lead. I have been adding material in this process but more expansion is still needed. In particular, the applications section is pretty anemic and something about duality and line conics needs to be written. The complex plane section needs some attention and general copy editing is in order. I might want to add something about conics over fields of characteristic two, but I need to be careful there lest I get too carried away. I will continue to work on this article, but probably not at the same pace as I have recently. Bill Cherowitzo (talk) 06:10, 4 February 2016 (UTC)

Redirect from Semi-Latus Rectum[edit]

I was looking for information on the semi-latus rectum and it redirected me to "conic section". However, since the last update any information on the semi-latus rectum or other features of conic sections has been removed. Please return these features to the article. — Preceding unsigned comment added by OneMeVz (talkcontribs) 06:54, 3 February 2016 (UTC)

Sorry about that. I haven't finished refactoring the article and I haven't decided where to put that piece of vocabulary yet. It will go back in soon. Bill Cherowitzo (talk) 17:27, 3 February 2016 (UTC)
 Done Bill Cherowitzo (talk) 20:06, 3 February 2016 (UTC)

Sub-section: As slice of a quadratic cone[edit]

It seems reasonable and I must be missing something obvious. Don't we normally call the generic graphs of z=Ax^2+Bxy+Cz^2 paraboloids (elliptic or hyperbolic) rather than quadratic cones? I am used to the form z^2=Ax^2+Bxy+Cy^2 for the sort of cones we think of slicing with planes to get conic sections. Can someone explain? (talk) 03:28, 16 February 2016 (UTC)J

You are right. I've hidden the section until a corrected version with a source can be found. Bill Cherowitzo (talk) 05:12, 16 February 2016 (UTC)


I reverted several edits involving constructions of conic sections primarily for two reasons. First of all, I don't think that the illustrations had any encyclopedic value. They did not illustrate anything that had not already been illustrated and required a considerable amount of time to decipher. Secondly, the comments were clearly draftsman instructions; they did not say anything about the conic sections, only how to draw these diagrams - violating WP:NOTHOWTO. Certainly more can be said (and illustrated) about constructing conics and there are several theorems that are used to find points on a conic that can be discussed in this section. Bill Cherowitzo (talk) 04:51, 21 February 2016 (UTC)

Assessment comment[edit]

The comment(s) below were originally left at Talk:Conic section/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

should the standard cartesian equation of a conic not be Ax^2 +2Bxy + Cy^2 ... to yield those necessary and sufficient conditions for it to be parabola/ellipse/hyperbola?

Last edited at 07:13, 13 May 2009 (UTC). Substituted at 01:54, 5 May 2016 (UTC)

Finding focus and directrix[edit]

I think this article should include a general formula to find out the focus and directrix of a parabola, hyperbola or ellipse from a general equation like — Preceding unsigned comment added by Sayan19ghosh99 (talkcontribs) 06:39, 1 August 2016 (UTC)

Since this is an article about all the conic sections, taken collectively, this specific type of information is more appropriate in the pages for the individual sections. However, to obtain the information you want, start with the general form of the equation and then algebraically turn it into a standard form. The information can be read off directly from the standard form. This is assuming that the conic section is in a standard position to start with. If it isn't, one has to translate and rotate to put it into a standard position before doing this manipulation. --Bill Cherowitzo (talk) 16:28, 1 August 2016 (UTC)

Edit Request[edit]

Under generalizations:"For example, the usual matrix representation of a quadratic form." is not a complete sentence (not a complete thought). - I have zero idea whether 'the usual matrix representation' is an example of 'conics in other fields' or 'some formulae' which cannot be used. (talk) 14:08, 16 May 2017 (UTC)

Better now? --JBL (talk) 14:19, 16 May 2017 (UTC)

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