Talk:Ellipse

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Geometry guy 00:06, 17 June 2007 (UTC)

[edit] constant distance

so where is the equation that shows distance to foci is constant? (This is the principle definition, right?) —Preceding unsigned comment added by 71.31.146.220 (talk) 01:54, 13 March 2008 (UTC)

[edit] Clarification Please

For the ellipse equation: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

What are its a, b, c, e presented as A,B,C,D,E, &F?

where is its center, and where is its foci?

how much is the angle of its major axis and the x axis? Please put the answers here!

Furthermore, for in three dimensions, is the equation like this:

Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0

What is the condition for it to be an ellipse ball? And for that, what are the length of its axies? What are the those? and where is its center? How much is the angle of its major axis? Please. Jackzhp 22:20, 23 October 2006 (UTC)

You raise a good point, and this equation isn't much help if you don't know what the parameters are. Also, this equation is over generalized so it will create a lot of shapes besides just an ellipse. If you zero-out some terms you get the classic formula: Ax² + Cy² + F = 0 ...although, this equation is not general enough to represent all rotations of an ellipse about its center, and IMHO those are still ellipses, so a better formula is needed.

One solution is to use simple parametric equations with linear scaling to strech the ellipse and move it throughout the plane. First of all, a useful step is to normalize the sine function like this: (sin(t)+1)/2, so that the function varies from 0 to 1. This makes the linear scaling coefficients more intuitive.

The parametric equations can then be written as:

x = A_x (sin(t)+1)/2 + x₀

y = A_y (sin(t+phi)+1)/2 +y₀

where,

t = the parametric parameter, i.e. "time"
The scaling coefficients are:

A_x = ellipse width = x_max - x_min

x₀ = x-axis offset = x_min

A_y = height of ellipse = y_max - y_min

y₀ = y_min

(note if A_x is negative, then x_max < x_min, ditto for "y".)

phi = angle of ellipse.
On the interval: (0 < phi < pi/2) the major axis has slope = 1
On the interval: (pi/2 < phi < pi) the major axis has slope = -1
When phi = pi/2, it's a circle. Also, phi = n*pi is a line with the afore mentioned slope. The ability to switch from one of these intervals to the other is required to select a right-leaning or left-leaning ellipse (at least for positive scaling coefficients Ax and Ay). The variable "phi" offers a 5th degree of freedom, which may not be required.

A more sophisticated approach would be to add an orthogonal cosine term, or use the matrix notation described a few messages below. Mikiemike 02:50, 5 February 2007 (UTC)

[edit] Slight clean up?

Don't know if this is the place to point this out, but this article would benefit from consistent use of characters. For example, now 'e' is either the distance of focus from the center of the ellipse, or eccentrecity of the ellipse. Also 'a' is used in different pictures to denote two different things. Since the parameters also appear in formulas, it is easy to get confused with them. --80.222.17.246 18:36, 5 April 2006 (UTC)

I second that! 128.122.20.71 22:09, 30 August 2006 (UTC)

[edit] Image

I removed the image since it does not always add up to 10. It would be a nice image if it were fixed; as it stands, the picture is wrong. Silverleaftree 18:55, 21 January 2006 (UTC)

Something is wrong with this formula:

And the general equation of an ellipse in polar coordinates is

$r = \frac{1\left( 1 - e^2 \right)} {1 - e \cos \theta}$

What does "general equation" mean here?

The images look a bit fishy to me. Aren't the two foci too far apart? Also, one of them seems to be closer to the ellipse than the other one. AxelBoldt 07:48 Feb 9, 2003 (UTC)

The first 1 in the formula should be an a (so it's not a general equation at all - it's essentially the same as the polar coordinate equation that's already in the article).

I also thought the foci in the images were too far apart, but when I measured them they seemed about right (well, the one on the left anyway). But I may do some neater replacements somewhen. --Zundark 16:34 Feb 9, 2003 (UTC)

The "general equation" means that your vaules for a and b may vary. Therefore the eccentricity will then vary. So a may equal 1 and b may equal 2, therefore e = (sqrt(3))/2, however, a may equal 300 and b may equal 600, but e will still equal (sqrt(3))/2, since the ratio of a to b remains equal 1:2 . However, no ellipse in a polar equation my have a coordinate greater than 1 (unless there is a coefficent). Therefore the "general equation," is reffering to the simplest ratios of a to b. So the ellipse where a= 1/2 and b = 1 would be similiar to the ellipse a = 7 and b = 14. In fact, if both centers are at the origin, they will be cocentric ellipses if they are similiar or if a and b remain in the same proportion. So with the "general equation," it produces every ellipse of a different ratio from 1:1 (a circle where e=0) to 0:1 (where e=1 or a line whose length = b, or in the case of polar coordinates, it equals one as b always equals 1 and a decreases to zero).To produce an ellipse cocentric to another in polar coordinates, multiply by entire equation or f(θ) by a constant. Sorry if this was a vauge explanation, I don't have much time. So to generalize all that I've typed, e can represent the eccentricity of any ellipse which can be concentric to another ellipse with the same eccentricity, where the major and minor axes are in the same proportion. —Preceding unsigned comment added by 69.117.220.101 (talk • contribs) 22:56, 19 May 2006

Q: What tools are used for drawing the figures? Are the source files of the images avairable? I'm writing ja: page of ellipse (ja:楕円). --HarpyHumming 11:53, 27 Feb 2004 (UTC)

I don't know anything about the third image. The first two images were originally done by someone else, but they were somewhat messy, so I redid them, copying things like the arrows and the text from the originals. I used some experimental software of my own design to draw the ellipses themselves, and did the rest of the work in the GIMP. The PNG files were processed by pngrewrite and pngcrush (or maybe pngout) to reduce the filesize without affecting the image. So there are no source files, just the PNGs. --Zundark 13:47, 27 Feb 2004 (UTC)

Formula for case where minor/major axis are not parallel with x/y axis is missing. Tomislav

Yeah, someone needs to include that. I might be able to later, but I'm not sure. pie4all88 01:27, 20 Aug 2004 (UTC)

All right, I have it put in. Hopefully it helps. pie4all88 01:36, 20 Aug 2004 (UTC)

There's an electronic group called Oval. I wanted to write an article about them, but Oval redirects here. What I'd like to do is add a line to the top that says "Oval is an electronic group" and have the word Oval link to a page "Oval (band)". I really dislike automatic redirects... IF I don't get any negative feedback on this in a few weeks, I'll go ahead and do it.

Hwarwick 8:32, 07/06/04

The electronic group was already mentioned at the bottom. I've moved the disambiguation to Oval, which shouldn't have redirected here anyway. --Zundark 07:27, 7 Jul 2004 (UTC)

Excellent! Thanks much!

Hwarwick 10.04 7 JUL 04

[edit] How far off-topic is Newton?

Do we really want to tell the history of Kepler's laws of planetary motion including Newton's deduction of them from his law of universal gravitation here on the ellipse page? Or would that history be better told on one of those two other pages? Ben Kovitz 16:25, 4 Dec 2004 (UTC)

Yes, it's a bit off topic and it is closely explainded on other pages. But it's only one sentence and it provides the basic idea about the problem of planetary orbits which is closely related to the ellipse topic. I think it doesn't harm this topic and sometimes may lead the reader to further readings. That's my humble opinion.

-- Egg 19:13, 2004 Dec 4 (UTC)

I don't think that "Newton's deduction of them [Kepler's laws of planetary motion] from his law of universal gravitation," has been "closely explained on other pages." Many words have been generated to explain Newton's deduction. This deduction is supposed to be very simple and understandable. In his De Motu, Newton was supposed to have accomplished it in less than ten pages. But that book is generally unavailable. The explanation in the Principia is not immediately clear. 64.12.116.200 12:36, 9 September 2005 (UTC)Bruce Partington

[edit] Area

is a derivation of the area (πab) appropriate here? i find an explanation of a formula helps with remembering it, but i don't know whether it's too much detail for a wikipedia article. Xrchz 06:40, 17 September 2005 (UTC)

If it's too much detail for this article you could always make it a separate article and link it from here. --Zundark 08:08, 17 September 2005 (UTC)

Most people remember from high school geometry that the area A in a circle is given by:

$A=\pi r^2 \$ , where r = radius

If somebody understands that an ellipse is a general case of a circle stretched or compressed in the y (vertical) direction and understands rather elementary integral calculus, then it is rather easy to see how the area of an ellipse is obtained. For a circle having a center at the origin, set the radius r equal to a and then stretch it or compress it in the vertical direction until the desired ellipse is obtained with a semi-vertical axis length of b. The stretching (or compressing) factor in 1 direction (the y-direction) is then = b/a . Since the 2-dimensional geometric figure is stretched (or compressed) in only 1 dimension, the previous area of the stretched (or compressed) figure is multiplied by the stretching factor raised to the 1st power to get the new stretched area as follows:

$A=\pi a^2 (\frac b a) = \pi a b$

The statement about how the area of a stretched 2D geometric figure = area of the unstretched figure times the stretching factor can be easily demonstrated (in effect proven) using first year single-variable integral calculus. H Padleckas 09:07, 13 December 2005 (UTC)

[edit] A quick, short and exact formula for calculating the perimeter of an ellipse

An ellipse can be split into an infinite number of evenly spaced lines of varying length(height). These lines pass through the x-axis at 90 degrees. The first and last lines have a length of 0 and the middle line has a length equal to the height of the ellipse. The distance along the x-axis between each line is l/a. The x-axis starts from the left at 0, there are no negative x values.

To convert these x values into normal Cartesian values ( $x C$ ):

$x_C = x-\frac{l}{2}$

The distance between a specific line and the first line is:

$x = \frac{lb}{a}$

The height of a specific line is:

$y = \frac{2h\sqrt{ab-b^2}}{a}$

The area of the ellipse is:

$A = \frac{l}{a} \sum_{b=0}^a{\frac{2h\sqrt{ab-b^2}} {a}}$

The circumference of the ellipse is:

$C = 2\sum_{b=0}^{a-1}{\sqrt{(\frac{l}{a})^2+(\frac{h}{a}\sqrt{a(b+1)-(b+1)^2}-\sqrt{ab-b^2})^2}}$

l is the length of the ellipse.

h is the height of the ellipse.

$a$ is the number of points equally distributed along l.

b is a specific point along l.

x is the distance between the start of the ellipse and point b.

y is the perpendicular expansion of b, centred on l (the height of a line going through the x axis at 90 degrees).

A is the area of the ellipse.

C is the circumference of the ellipse.

For an ellipse of length=2 and height=1, the exact infinite series gives a value of 4.844224110291 (62 terms).

My equation gives a value of 4.84422411023477 ( $a = 1 * 10 9$ points).

-GoldenBoar

Your expression for A reduces to $\pi hl/4\,$ in the limit, which is the same as the $\pi ab\,$ given in the article. There's a typo in your expression for C. Now would probably be a good time to learn about integration - you are starting to reinvent it yourself, so you've already got the basic idea. --Zundark 15:59, 10 October 2005 (UTC)

My expression for A when using 1000 points does not give the exact same value as the standard $\pi ab\,$ equation, so they are not the same. Also, I specifically set out not to use

π

. I have just rechecked the equation for C, and there is no typo, the equation is correct. What made you think there was a typo?

Of course your expression for A isn't equal to $\pi ab\,$ when using 1000 points, I said in the limit. The limit of your A as the number of points tends to infinity is $2hl\int_0^1\!\sqrt{x-x^2}\,dx$ , which evaluates to $\pi hl/4\,$ . As for your expression for C, I meant that it should be

$C = 2\sum_{b=0}^{a-1}{\sqrt{\left(\frac{l}{a}\right)^2+\left(\frac{h}{a}\left(\sqrt{a(b+1)-(b+1)^2}-\sqrt{ab-b^2}\right)\right)^2}}$ .

--Zundark 18:18, 10 October 2005 (UTC)

Thanks, I was wondering how to do that.

[edit] Ellipse: Oblate vs. Prolate?

Spheroids can be either oblate (a>b) or prolate (b>a). Most (all?) ellipses are presented as oblate:

Is this just popular convention, or by actual definition—i.e., a "prolate" ellipse is just an oblate ellipse turned 90° (a is the vertical radius semi-major axis and b is the horizontal/semi-minor axis)? My point in asking is regarding the "modular angle", where sin{α} is the eccentricity, $\sin\{\alpha\}^2=\frac{a^2-b^2}{a^2}\,\!$ . A common question is "what is Earth's circumumference?", with the answer being it is found via the "elliptic integral of the 2nd kind". But what if you are dealing with a prolate spheroid, in which case it would be shaped like a watermellon standing on end—? Is it as simple as switching a and b in calculating the modular angle (thus $\sin\{\alpha\}^2=\frac{b^2-a^2}{b^2}\,\!$ ), as it would seem, or is the modular angle (and its applications—e.g., elliptic integrals) only meant for oblate cases? (Incidently, using the above image as reference, I still think the third, prolate vs. oblate, paragraph in spheroid is wrong!) ~Kaimbridge~ 15:06, 13 December 2005 (UTC)

Well in the oblate spheroid, elliptic integrals aren't nessecary to find the cirdumfrence at any given (in the case of the earth) "laditude," since at that point it is merely a circle. In the case of a prolate pheriod, on the other hand, it depends on its orientation. It its "standing on end," then the circumfrence at a given hieght is a circle. But if its on its side then ring is an ellipse and its circumfrence as simple as find. In the case of an ellipse, oblate and prolate are the same since on can merely be rotated to the other.-- He Who Is^{[ Talk ]} 16:13, 13 July 2006 (UTC)

[edit] More ellipse generalization...

Circle has one focus. Ellipse has two focus. I wonder if there is a generalization for k focuses.

--Yochai Twitto 22:50, 1 January 2006 (UTC)

First, of all, I only want to mention that the plural of focus is foci. An ellipse has two foci. Anyway, I suppose that the formula for such a shape would be: (For foci with coordinates {A,B,C}_{0,1} and the sum of the focal distances is d.)

$\sqrt{(x-A_0)^2+(y-A_1)}+\sqrt{(x-B_0)^2+(y-B_1)}+\sqrt{(x-C_0)^2+(y-C_1)} = D$

-- He Who Is^{[ Talk ]} 17:09, 23 June 2006 (UTC)

[edit] Quadratic Form, Quadrik

In the article the general expression for the ellipse : $A x 2 + B x y + C y 2 + D x + E y + F = 0$ with $B 2 < 4 A C$ is given. Maybe this looks better in the general form of

$\begin{pmatrix}x \\ y \end{pmatrix}\begin{pmatrix} \alpha & \beta\\ \gamma & \delta\end{pmatrix}\begin{pmatrix}x \\ y \end{pmatrix}+\begin{pmatrix}\epsilon \\ \mu \end{pmatrix}\begin{pmatrix}x \\ y \end{pmatrix}+\phi=0$

where $β = γ = B / 2$ . The solution is a an ellipse if the matrix is poitive definite i.e., all Eigenvalues are positive. So it is only $αδ - βγ = αδ - β 2 > 0$ and the 4, coming out of nowhere, is not required. In general the off diagonal elements can also be different, but I believe this gives only a tilt (maybe a distortion). Depending on the parameters also the hyperbola, parabola, lines, etc. are possible, which is (somehow) written in the Wiki article about the quadric. Unfortunately the 2D case is not written explicitely and the matrix form is not used, but a link could be helpful (also from Quadric to Ellipse). Mikuszefski 14:32, 4 May 2006 (UTC)

[edit] Do ellipses remain ellipses when scaled?

An anon added this to the article:

--- ~~The above statement is not correct.~~ A given anisotropic scaling in the plane (i.e. a scaling in a specified direction in the plane) may be considered to act around a scaling "centre line" L and have a scaling factor S. The scaling will act so that a point at distance D from the centre line will be transformed away from the line in a direction normal to that line so that the resulting distance from line to point is S*D. An important point about such anisotropic scalings is that they do NOT generally preserve angles, so that for an arbitrary anisotropic scaling the angle between two intersecting lines will generally not be the same after the scaling as before it.

~~This lack of preservation of angles has consequences for anisotropically scaled ellipses.~~
An ellipse may be defined by:

x(t) = c + a*cos(t)*d1 + b*sin(t)*d2

Where d1 and d2 are such that d1.d2 = 0 (for nonzero major and minor radii d1 and d2 will usually be normalised). ~~If a curve can be written in such a form as that above where d1.d2 != 0 then the curve is not an ellipse~~.

If an anisotopic scaling is applied to an ellipse, then the resulting transformed curve may be written in the above form, but d1 and d2 will not generally be orthogonal ~~due to the lack of angle preservation of isotropic scalings.~~ This can be true even if the scaling centre line passes through the centre of the ellipse. The only time that d1.d2 = 0 generally for such a scaling is when the scaling centre line is coincident to either the major or minor axis of the ellipse. MSM ---

This is correct up to this statement: If a curve can be written in such a form as that above where d1.d2 != 0 then the curve is not an ellipse. I think that's wrong. Just because it can be written in that form with d1 and d2 not orthogonal, doesn't mean it can't also be written with two different vectors that are orthogonal. —Keenan Pepper 15:38, 13 July 2006 (UTC)

Here's a simple proof: The image of a circle under any linear transformation is an ellipse, where the axes of the ellipse are eigenvectors of the matrix. The composition of any two linear transformations is another linear transformation, so whatever you do, it's still an ellipse. —Keenan Pepper 15:42, 13 July 2006 (UTC)

So every ellipse is a transformation of a circle? Just curious. --kris 21:16, 7 December 2006 (UTC)

[edit] Ellipse as intersection solutions

In Descriptive Geometry, an ellipse is also defined as an intersection between a cylinder and an oblique plane. But perhaps even more interesting is the fact that is also an intersection between a cone and an oblique plane. More interesting because in this case, rotating the plane can give us various results: if the plane is perpendicular to the cone's axis, it is a circle; if it is oblique and intersects all of the cone, is an ellipse; if it is oblique and doesn't intersect all of the cone, it draws an hyperbole, except in the one case when the plane is parallel to one of the defining lines of the cone (can't remember the specific name).

Shouldn't this be inside the ellipse thematic?

Also, are there correlations with other geometrical constructs? Hyperboloids and stuff? Thanks. —The preceding unsigned comment was added by 195.23.224.70 (talk) 12:31, 14 February 2007 (UTC).

[edit] Real World Example

I think that the the ellipse page and many other math shape related pages could use a real world example image such as a famous picture or sculpture showing an ellipse. It would go quite nicely under the top image. I tried to add one myself, but I am very new to this and have no idea how. —The preceding unsigned comment was added by 69.153.22.24 (talk) 22:43, 18 February 2007 (UTC).

[edit] fact check

From the article:

In 499, Indian astronomer Aryabhata discovered that the orbits of the planets around the sun are ellipses, and published his findings in his book, the Aryabhatiya [1].

I'd like to see direct quotes from the cited source that support the idea that Aryabhata "discovered that the orbits of the planets around the sun are ellipses". The Wikipedia article on Aryabhata does not make this claim. According to Ian Pearce, Mahavira (c850 AD) was the, "only Indian mathematician to refer to the ellipse". Aryabhata may have described planet motion using epicycles, but did he recognize that orbits are ellipses? --JWSchmidt 12:55, 19 February 2007 (UTC)

[edit] Errors:

Parameterisation section: I would prefer to see the drawing and section changed to conform with c as the hypotenuse.

The current drawing should use the form $b 2 = a 2 - c 2$ for $\frac {x^2}{b^2} + \frac {y^2}{a^2} = 1$

I believe the author has listed c as the base rather than the hypotenuse in the drawing (similar to some textbooks) but then incorrectly uses the standard Pythagorean Theorem.

The Pythagorean Theorem $a 2 + b 2 = c 2$ uses c as the hypotenuse.

I would question this section completely as it should be in harmony with the Alegbraic terms above rather than just noting the contradictions in the label.

Additionally this entry does not explain the use of Beta which seems spurious. [User:Hwgramm|Hwgramm]] 01:11, 24 February 2007 (UTC)

I redid some of the section, renamed it, removed the confusing/unhelpful pythagorean theorem figure. There's a bit more to do, but I think it's a little better now. Doctormatt 07:39, 24 February 2007 (UTC)

No, NO, NOOOoooo!!! P=)

A good, traditional explanation of the ellipse (and the x/y, a/b relationships) can be found here. My particular approach is based on the fact that, if you slice an oblate spheroid down the middle, lay the flat side down and trace its perimeter, you will have an ellipse: Thus, each point on the perimeter can be considered a "latitude". Now, look at the three most common types of latitude——the geographical (φ), reduced or parametric (β) and the geocentric (ψ)——and how they relate to each other. The parameterization is frequently expressed as a spherical coordinate, but the notation can be quite confusing!

I'm working on a major revamp of the ellipsoid article, a few snippets of which may best demonstrate my reasoning:

--------------------------------------

$o\!\varepsilon=\arccos\left(\frac{b}{a}\right)=2\arctan\left(\sqrt{\frac{a-b}{a+b}}\right);\,\!$

[edit] The core integrands

The most fundamental elliptic integrand is that of the elliptic integral of the second kind:

$\begin{matrix}{\color{white}.}E'(\theta)=C'(\frac{\pi}{2}-\theta)\!\!\!&=&\!\!\!\!{\color{white}\dot{{\color{black}\sqrt{\cos(\theta)^2+(\sin(\theta)\cos(o\!\varepsilon))^2}}}},\\&=&\!\!\!\!\sqrt{\cos(o\!\varepsilon)^2+(\cos(\theta)\sin(o\!\varepsilon))^2},\\&=&\!\!\!\!\sqrt{1-(\sin(\theta)\sin(o\!\varepsilon))^2};{\color{white}88888}\end{matrix}\,\!$

Its complement is simply the argument's sine and cosine positions reversed:

$\begin{matrix}{\color{white}.}C'(\theta)=E'(\frac{\pi}{2}-\theta)\!\!\!&=&\!\!\!\!{\color{white}\dot{{\color{black}\sqrt{\sin(\theta)^2+(\cos(\theta)\cos(o\!\varepsilon))^2}}}},\\&=&\!\!\!\!\sqrt{\cos(o\!\varepsilon)^2+(\sin(\theta)\sin(o\!\varepsilon))^2},\\&=&\!\!\!\!\sqrt{1-(\cos(\theta)\sin(o\!\varepsilon))^2};{\color{white}00000}\end{matrix}\,\!$

Its inverse has special meaning, too, as it is both the integrand to the elliptic integral of the first kind and the unit form of one of the principal radii of curvature:

$n'(\theta)=\frac{1}{E'(\theta)}.\,\!$

----------

[edit] Elliptic radii

The local elliptical radius can be found from all three latitude types:

$r'(\phi)=n'(\phi){\color{white}\dot{{\color{black}\sqrt{\cos(\phi)^2+\sin(\phi)^2\cos(o\!\varepsilon)^4}}}};\,\!$

$R=R(\phi)=a\,r'(\phi)=a E'(\beta)=\frac{b}{C'(\psi)};\,\!$

One of the principal radii of curvature is the perpendicular, or normal, radius of curvature (named as such, as it is perpendicular to its "meridional" counterpart):

$N=N(\phi)=a\,n'(\phi)=\frac{a}{E'(\phi)}=a\sec(o\!\varepsilon)C'(\beta)=a\sec(o\!\varepsilon)r'(\frac{{\color{white}\dot{{\color{black}\pi}}}}{2}-\psi).\,\!$

----------

[edit] Cartesian parameterization

There are different Cartesian equations for different applications, the most basic being the Cartesian parameterization.

[edit] Circle

With a circle, $\phi=\beta=\psi\,\!$ , which can all be generalized to ${}^{\color{white}.}\theta{}^{\color{white}1}\,\!$ .

Where r is the radius,

$x=r\cos(\theta);\quad\,y=r\sin(\theta);\,\!$

$\left(\frac{x}{r}\right)^2+\left(\frac{y}{r}\right)^2=\cos(\theta)^2+\sin(\theta)^2=1;\,\!$

$\theta=\arctan\left(\frac{y}{x}\right);\quad\,r=\sqrt{x^2+y^2}.\,\!$

[edit] Ellipse

In the case of an ellipse——as well as a circle——there technically are no such things as latitudes: Just points on a perimeter. However, as such points "behave" like latitude points (including reduced variations), $\phi\,\!$ , $\beta\,\!$ and $\psi\,\!$ (and all of their properties) can be applied to the ellipse, as can N and R. As should be apparent, Cartesian parameterization is based on the parametric latitude, $\beta\,\!$ , with equivalencies for $\phi\,\!$ and $\psi\,\!$ :

$x=a\cos(\beta)=R\cos(\psi)=N\cos(\phi);\,\!$

$y=b\,\sin(\beta)=R\,\sin(\psi)=\cos(o\!\varepsilon)^2N\sin(\phi);\,\!$

$\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=\cos(\beta)^2+\sin(\beta)^2=1;\,\!$

${}_{\color{white}.}\beta=\arctan\left(\sec(o\!\varepsilon)\frac{y}{x}\right);\quad\psi=\arctan\left(\frac{y}{x}\right);\quad\phi=\arctan\left(\sec(o\!\varepsilon)^2\frac{y}{x}\right);\,\!$

$\begin{matrix}{}_{\color{white}.}\\R&=&\sqrt{x^2+y^2}&=&\sqrt{(R\cos(\psi))^2+(R\sin(\psi))^2},\\ &=&a\,E'(\beta)&=&\sqrt{(a\cos(\beta))^2+(b\sin(\beta))^2},\\ &=&a\,r'(\phi)&=&\sqrt{(N\cos(\phi))^2+(N\sin(\phi)\cos(o\!\varepsilon)^2)^2}.\end{matrix}\,\!$

--------------------------------------

I would think that this is way too much for just this ellipse article——though feel free to adapt whatever parts you may feel will be helpful! P=) ~Kaimbridge~17:05, 24 February 2007 (UTC)

The parametric and cartesian equations as I've rewritten them are correct. Feel free to add information (such as your favorite developement of these equations), or correct bits in regards to which you feel the need to exclaim "No, NO, NOOOoooo!!! ". Cheers, Doctormatt 17:52, 24 February 2007 (UTC)

[edit] Ellipses in physics

Quote:

In 499, Indian astronomer Aryabhata discovered that the orbits of the planets around the sun are ellipses, and published his findings in his book, the Aryabhatiya.[1]

The reference summarizes:

The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It also contains continued fractions, quadratic equations, sums of power series and a table of sines.

So the reference does not support the sensational claim that the elliptical orbit of planets was found by Aryabhata. It should be removed until more convincing references are made. Bo Jacoby 16:09, 16 April 2007 (UTC).

The reference does in fact say "incredibly he believes that the orbits of the planets are ellipses". Of course, this in itself doesn't justify the word "discovered" in the text that you removed, since it may have been more a good guess than a real discovery. It would be interesting to see what Aryabhata actually wrote. --Zundark 16:39, 16 April 2007 (UTC)

Thank you. You are right and I am wrong. The reference http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aryabhata_I.html does indeed contain the above quote. The article on Aryabhata does not contain the word "ellipse", which it should if the claim is correct. The reference list http://www-groups.dcs.st-and.ac.uk/~history/References/Aryabhata_I.html contains no title containing the word "ellipse", but the reference "K S Shukla, Use of hypotenuse in the computation of the equation of the centre under the epicyclic theory in the school of Aryabhata I, Indian J. History Sci. 8 (1973), 43-57" seems to indicate an epicyclic theory rather than an ellipse theory. I agree on the word "incredibly". If you choose to revert my edit, I suggest that you include a request for a sufficient citation. Bo Jacoby 22:25, 16 April 2007 (UTC).

[edit] equation in matrix form

please put equation in matrix form. which might be centered in the origin, or a point not the origin. 70.52.48.32 21:17, 8 July 2007 (UTC)

[edit] Gauss map

I plugged in the Gauss map function from the article and found that the gauss-mapped point is not orthogonal to the original point (i.e., the vector connecting the two is not normal to the ellipse at the original point), as the Gauss map article says it must be. Rather, the vector passes through both points as well as the origin. Also, the normal is not (cosβ,sinβ). Is there a reason for this? SharkD (talk) 21:39, 16 February 2008 (UTC)

[edit] Inaccuracy

It is written: "With one focus at the origin, the ellipse's polar equation is

$r = \frac{ a\cdot(1-\varepsilon^{2})}{1 - \varepsilon\cdot\cos\theta}$ . "

While this equation is only correct for left focus point at the origin. For right one it would be:

$r = \frac{ a\cdot(1-\varepsilon^{2})}{1 + \varepsilon\cdot\cos\theta}$ .

Also a drawing showing theta would be usefull. 11.06, 26 February 2008 (UTC)

[edit] Picture Suggestion

There is a fine animated version of the ellipse in the Spanish version of this article. Would someone care to include it? It is a very graphical depiction of the definition of the ellipse. Juanmejgom (talk) 11:02, 1 April 2008 (UTC)

I see no reason why not.

[edit] article overhaul

I'd like to suggest a larger overhaul of the article, in particular in include some of the properties and graphics you can find in the german article (as well as from other languages), which are still missing here. And in the process of adding all that additional information it might make sense to reorganize the current content/information along with it. Also that could be uuse to add some more references. Is anybody of the original authors still actively working on the article right now? Any suggestions/comments regarding the possible overhaul?--Kmhkmh (talk) 10:47, 2 April 2008 (UTC)

I agree that the article could be organized better. For one thing, the facts that an ellipse is a conic section and a circle is a special case should be mentioned early on. Second, there should be a completely nontechnical paragraph at the beginning to explain to a layperson generally what an ellipse is and where they occur in real life. Starting a top 500 article with a phrase like 'locus of points' seems a bit stodgy. The more pictures and the fewer formulas the better in the introductory sections. Third, some of the material does not fit under the section it's been put under. There needs to be some cleanup to fix this. Fourth, there must be dozens of geometrical theorems involving ellipses that could be included. Perhaps this should be a separate page such as 'Theorems involving ellipses' or 'Theorems involving conic sections'. Fifth, some of the material, the proofs section in particular, could stand a rewrite to avoid copyright issues. Sixth, there could be more on anatomical features such as vertex & curvature, and related curves such as the evolute. I don't mean to be nitpicky, there is good material here, but some improvements could be made. —Preceding unsigned comment added by RDBury (talk • contribs) 17:14, 26 April 2008 (UTC)

[edit] Is this a mistake?

The distance c is known as the linear eccentricity of the ellipse. The distance between the foci is 2pac or 2aε.

2pac? —Preceding unsigned comment added by 58.172.146.121 (talk) 05:01, 31 May 2008 (UTC)

This was some joker making a rap reference; it's been fixed.--RDBury (talk) 14:32, 31 May 2008 (UTC)