Talk:Conic section
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.
I recommend the following link for graphics on conics.
Very good visuals, thank you. RoseParks
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.
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)
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)
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)
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)
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)
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
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)