Talk:Conic section

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Contents 1 Discussion of irreducible conics 2 Polar coordinates 3 picture 4 Visuals 5 Incorrect determinant 6 Improve hyperbola image 7 Dimensionality of conic sections 8 Image 9 Semi-latus rectum and polar coordinates 10 Dimentionality 11 Degenerate Conics 12 er, simplify? 13 Correct image 14 Quadratics 15 Eccentricity illustration 16 Derivation 17 Conic Sections in Polar Coordinates 18 emmendation on statements concerning the circle 19 Merger proposal 20 Degenerate conics (again)

[edit] Discussion of irreducible conics

Should be added —Preceding unsigned comment added by 128.186.159.121 (talk) 19:10, 3 June 2008 (UTC)

[edit] Polar coordinates

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. 24.63.157.84 (talk) 02:58, 10 December 2007 (UTC)

[edit] picture

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)

[edit] Visuals

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

[edit] Incorrect determinant

I moved this:

Finally, if the following determinant,

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

equals zero, 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 213.224.82.23 (talk) 13:55, 27 August 2005

[edit] Improve hyperbola image

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)

[edit] Dimensionality of conic sections

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)

[edit] Image

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)

[edit] Semi-latus rectum and polar coordinates

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:80.203.35.66 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)

[edit] Dimentionality

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)

[edit] Degenerate Conics

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 ax² + 2hxy + by² + 2gx + 2fy + c = 0, 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)

[edit] er, simplify?

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)

[edit] Correct image

Is the illustration of a hyperbola correct? Mathworld ( http://mathworld.wolfram.com/ConicSection.html )shows illustrates a hyperbola as consisting of two curves. NJS. —Preceding unsigned comment added by 82.147.19.154 (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. http://www.themathlab.com/dictionary/hwords/hyperbola.gif That picture illustrates it better. ImMAW 00:11, 13 March 2007 (UTC)

[edit] Quadratics

Are conics quadratics? 70.251.199.130 03:39, 19 May 2006 (UTC)

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

[edit] Eccentricity illustration

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)

[edit] Derivation

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)

[edit] Conic Sections in Polar Coordinates

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 Ax² + By² + F = 0 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)

[edit] emmendation on statements concerning the circle

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 63.167.237.65 (talk) 08:07, 28 February 2007 (UTC).

[edit] Merger proposal

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)

[edit] Degenerate conics (again)

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

00:23, 14 April 2008 64.7.77.226 (Talk) (16,872 bytes) (→Types of conics) introduced the rectangular hyperbola comment.

03:36, 21 April 2008 128.250.80.15 (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)