Talk:Ellipse/Archive 1

Archive 1

Archive 2

First messages

aaaaaaaaaaaarg.

I can't save the tex equation, I get a huge error

${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$

(add a math tag at the front and put { for < to see the problem, i can't save otherwise

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This still? I thought taw fixed that. Guess not... grr... --Brion 20:45 Jan 29, 2003 (UTC)

Okay, taw recommended a simple fix that seems to have got it. If this recurs, let me know. --Brion 07:09 Jan 30, 2003 (UTC)

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)

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)

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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)

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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)

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)

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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

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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)

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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)

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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

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)

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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

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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:

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

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)

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 $\pi$. 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.

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)

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)

Quadratic Form, Quadrik

In the article the general expression for the ellipse :$Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$ with $B^{2}<4AC$ 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 $\beta =\gamma =B/2$. The solution is a an ellipse if the matrix is poitive definite i.e., all Eigenvalues are positive. So it is only $\alpha \delta -\beta \gamma =\alpha \delta -\beta ^{2}>0$ 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)

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)

If by "transformation" you mean linear transformation, or even more generally affine transformation (i.e. translation is allowed as well), then yes. More generally yet, ellipses are even preserved by projective transformation (as when a scene is projected onto your retina or an imaging device, generically the screen), with the caveat that the plane of the screen may not cut the ellipse. If it does cut it, the ellipse projects to a hyperbola whose two branches show up on opposite sides of the screen.

The converse of this is the case when the screen cleanly separates (and therefore does not intersect) the two branches of a hyperbola, which then projects onto opposite sides of the screen as an ellipse, with the two branches neatly connecting at the horizon (the projection onto the screen of the line at infinity). So if you draw both branches of a hyperbola on the ground and follow the two diverging arms of one branch out to the horizon with your eye, they will appear to converge. Unlike railroad lines however they will not meet at the horizon (except in the degenerate case of a parabola, which projects to an ellipse with the horizon as a tangent), instead they will form part of an ellipse cut off by the horizon. The other branch behind the screen projects onto (the back of) the screen in exactly the right place to complete that ellipse! --Vaughan Pratt (talk) 00:02, 6 December 2008 (UTC)

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: It is actually a fallacy(that has existed for centuries!)that an ellipse is formed by oblique section of a cone - what is formed this way is an egg shape. An ellipse can only be obtained by an oblique section of a cylinder. —Preceding unsigned comment added by 124.181.74.101 (talk) 12:19, 24 January 2009 (UTC)

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).

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)

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:

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

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

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:

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

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Elliptic radii

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

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

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

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):

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

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Cartesian parameterization

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

Circle

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

Where r is the radius,

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

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

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

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 $\psi \,\!$:

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

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

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

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

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

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

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)

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)

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)

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.

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)

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)

oval

Re the last reversion: While I agree that "an ellipse is not an oval" is wrong, there's room for a statement that "oval" is a broader and less well-defined term. —Tamfang (talk) 09:47, 4 October 2008 (UTC)

Improve introduction

The overview should be improved:

1)

Present text:

Algebraically, an ellipse is a curve in the Cartesian plane defined by an equation of the form

$Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\,$

such that $B^{2}<4AC$, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists.

Comment:

There are several ways to define an ellipse but the definition above is certainly not very intuitive! A more natural definition understandable for the typical reader should have been selected!

2)

Present text:

The line segment AB, that passes through the foci and terminates on the ellipse, is called the major axis.

and then

If the two foci coincide, then the ellipse is a circle; in other words, a circle is a special case of an ellipse, one where the eccentricity is zero.

Comment:

How can one talk about "foci" before they have been defined?

Present text:

An ellipse centered at the origin can be viewed as the image of the unit circle under a linear map associated with a symmetric matrix $A=PDP^{T}$, $D$ being a diagonal matrix with the eigenvalues of $A$, both of which are real positive, along the main diagonal, and $P$ being a real unitary matrix having as columns the eigenvectors of $A$. Then the axes of the ellipse will lie along the eigenvectors of $A$, and the (1 over the square root of the) eigenvalues are the lengths of the semimajor and semiminor axes, which are one-half of the lengths of the major and minor axes respectively.

Comment:

Why make it so complicated. Just use the "canonical" coordinate system and the equation

${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$

A minimal re-write maintaining as much of the present text as possible would in my opinion be:

Stamcose (talk) 16:35, 8 November 2008 (UTC)

Overview

Algebraically, an ellipse is a curve in the Cartesian plane defined by an equation of the form

$Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\,$

such that $B^{2}<4AC$, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists. It can be proved that for any such set of coefficients $A,B,C,D,E,F$ there exists a "canonical" coordinate system (translation and rotation) such that the equation above takes the form

${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$

for some values

$0<b<a$

The line segment between the points (-a,0) and (a,0) is called is called the major axis. The major axis is the longest segment that can be obtained by joining two points on the ellipse. The line segment between the points (0,-b) and (0,b) perpendicular to the major axis is called the minor axis.

An ellipse is a type of conic section: if a conical surface is cut by a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse.

For a short elementary proof of this, see Dandelin spheres.

And also:

Before the section "Drawing" it should be proven that the sum of the distances to the foci is constant!

If no (sensible!) protests are recieved I plan to update the text as proposed above in about a week and then also add a proof that that the sum of the distances to the foci is constant what presently is missing!

Stamcose (talk) 16:35, 8 November 2008 (UTC)

The fundamental property of an ellipse not demonstrated before

I think this should be section "eccentricity"

Stamcose (talk) 17:46, 8 November 2008 (UTC)

Eccentricity

The eccentricity of the ellipse with the equation

${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$

is defined as

$e={\sqrt {1-{\left({\frac {b}{a}}\right)}^{2}}}$

As

$0<b<=a$

the eccentricity is a value between 0 and 1

$0<=e<1$

The eccentricity is zero if and only if $b=a$ in which case the ellipse is a circle.

The points with the coordinates (-a e , 0) and (a e , 0) on the major axis are called the focal points of the ellipse. The square of the distance from a point (x,y) on the ellipse to the left focal point is

$r_{1}^{2}=(x+ae)^{2}+y^{2}=x^{2}+2xae+a^{2}e^{2}+(a^{2}-x^{2})(1-e^{2})=(a+ex)^{2}$

As

$0<=ex<a$

it follows that

$r_{1}={\sqrt {(x+ae)^{2}+y^{2}}}=a+ex$

The square of the distance from the same point to the right focal point is in the same way

$r_{2}^{2}=(x-ae)^{2}+y^{2}=x^{2}-2xae+a^{2}e^{2}+(a^{2}-x^{2})(1-e^{2})=(a-ex)^{2}$

and as before

$r_{2}={\sqrt {(x-ae)^{2}+y^{2}}}=a-ex$

Adding these equations one gets

$r_{1}+r_{2}=2a$

This property of the ellipse, that the sum of the distances to two given points is taking a given value, is often used as the definition of an ellipse. The method to draw an ellipse of the next section is an application of this.

Stamcose (talk) 17:46, 8 November 2008 (UTC)

I disagree that the eccentricity is defined as above. The equation is clearly not even true, let alone a definition, unless x and y are orthogonal coordinates at the same scale (which is customary but not always necessary) and a>b, which you did not state as part of the definition. —Tamfang (talk) 04:25, 14 November 2008 (UTC)

As annonced above I now release a re-write of "Introduction", "Overview" and "Eccentricity" re-organising the present content into what I think is a more logical way with the sections "Introduction", "Eccentricity" and "Reduction to canonical form". The proof of the fundamental property

$r_{1}+r_{2}=2a$

that was missing before is also inserted!

Stamcose (talk) 13:03, 13 November 2008 (UTC)

Well, the first obvious thing you did wrong was to remove any definition of ellipse from the first sentence. The etymology is not the most important thing! —Tamfang (talk) 04:21, 14 November 2008 (UTC)

As it is said in the introduction, there are many equivalent way to define an ellipse. In this case I assume the definition is:

The locus of points satifying an equation

${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$

relative an orthogonal Cartesian coordinate system

with

$0<b<=a$

If this definition is used the the eccentricity is certainly defined as:

$e={\sqrt {1-{\left({\frac {b}{a}}\right)}^{2}}}$

Stamcose (talk) 10:23, 26 November 2008 (UTC)

Aken or Pitteway as the generalizer to ellipses of Bresenham's method?

Pitteway generalized Bresenham's line drawing algorithm to conic sections in 1967. Pitteway catered for both ellipses and hyperbolas with the one method, which therefore subsumes Bresenham-type ellipse drawers. Is there some reason why the article credits Aken as the inventor of this generalization of Bresenham's method rather than Pitteway?

It should be mentioned that Pitteway's method fails when the ellipse is so thin as to be invisible to Bresenham-type or forward difference sampling of the quadratic function f(x,y) whose roots constitute the ellipse, a problem that Pitteway had been aware of early on but for which no solution had been found prior to 1985, occasional publicity by Pitteway about the problem notwithstanding. As one of a number of techniques in my SIGGRAPH'85 paper "Techniques for conic splines" I described a low-overhead method for making thin ellipses visible to Pitteway's method, namely by reorganizing the method's addition operations (without adding any further arithmetic operations) so as to spin off the partial differences f_x and f_y of the function as a side effect of updating f itself. The only additional work needed is to monitor the signs of these partials, one of which must change whenever Pitteway's method jumps over an ellipse too thin to be detected by f alone. I haven't followed the literature closely since then (too many other interests), but if some other comparably efficient modification to Pitteway's method solving that problem has been invented in the intervening 23 years I'd love to hear about it. --Vaughan Pratt (talk) 23:34, 5 December 2008 (UTC)

Math parser is out -- repair template ref? —Preceding unsigned comment added by 128.61.67.129 (talk) 01:45, 9 December 2008 (UTC)

Ellipse properties as projected circle properties?

First of all many thanks and congratulations to all who have worked on this page. I have, at best, a sketchy knowledge of advanced mathematics, and yet I have been able to make sense of presentation here, in corroboration with related wikipedia articles (such as "arclength," "elliptical curves," "kepler's laws," etc.).

The one question I have yet to answer, however, is as follows: How do you calculate the ratio between a given true anomaly and its corresponding arclength?

And can this ratio (and, for that matter, other values such as the ellipse's circumference or polar coordinates thereupon) be derived from corresponding values on the ellipse's corresponding auxiliary circle--through the relation between ellipse and auxiliary circle stated in the (somewhat misplaced) paragraph at the end of the "Semi-latus rectum and polar coordinates" section?

viz.--"An ellipse can also be thought of as a projection of a circle: a circle on a plane at angle φ to the horizontal projected vertically onto a horizontal plane gives an ellipse of eccentricity sin φ, provided φ is not 90°." —Preceding unsigned comment added by 190.84.245.19 (talk) 19:23, 10 December 2008 (UTC)

Area and circumference

This section points to elliptic integral, which, however, does not give any geometrical meaning to the parameters in integrals such as $\int _{0}^{\phi }{\frac {d\theta }{\sqrt {1-(\sin \theta \sin o\!\varepsilon )^{2}}}}.\,\!$. Albmont (talk) 17:12, 27 February 2009 (UTC)

Major and minor radius

An anonymous editor has been removing (1:) "major radius" and "minor radius" as alternatives to (2:) "semimajor axis" and "semiminor axis", claiming that the (1)-names are "wrong". Now, it is not the role of wikipedia to legislate on nomenclature; we should only record the names that people use. While the (2)-names may be those used in most (?) geometry textbooxs, there seem to be many technical people who use the (1)-names. The claim that "ellipses do not have radii" is groundless: "radius" is a general term for a line segment from the "center" of a figure to any point on its boundary. In some contexts the radius is the longest such segment, but in other contexts it sis any such line. Check polar coordinates or the radius of a polygon, for example. An engineer may say that "the radius of that hole varies between 4.99 and5.02 inches", for example. So ellipses do have radii, and the (2)-names are precisely the longest and shortest radii -- which is probably why people often use the (1)-names for them. All the best, --Jorge Stolfi (talk) 02:29, 9 April 2009 (UTC)

A Google search for ellipse and "semimajor axis" produces 124,000 hits, while a search for ellipse and "major radius" produces less than 4,000 hits. Given the magnitude of this discrepancy, I think there is a good argument for not including "major radius" as an alternative to "semimajor axis".

On the other hand, it certainly helps with the explanation to use the word "radius" to describe the semimajor and semiminor axes. Perhaps the best compromise would be to have a sentence describing the semimajor axis as the largest radius of the ellipse, and the semiminor axis as the smallest radius. Jim (talk) 03:33, 9 April 2009 (UTC)

Well, if you browse through those 4000 hits, you will find some authoritative sources, and many technical articles. On what authority can we tell those people that they are "wrong"? (Besides being more cumbersome, the classical nomenclature is nonsensical, and looks like a botched-up transtation from Latin or some Romance language, e.g. Italian "semiasse maggiore" which properly translates to "major semi-axis". The blame may lie with the French who write "demi-grand axis" instead of "grand demi-axis"...) By the way, Google gives some 4000 references to "major semi-axis" + "ellipse", including several in Wikipedia. Just because it is unusual, it does not mean that it is "wrong", either.
So here is another compromise proposal:

The semimajor axis (denoted by a in the figure) and the semiminor axis (denoted by b in the figure) are one half of the major and minor diameters, respectively. These are sometimes called the major and minor semi-axes, or (especially in technical fields) major radius and minor radius.

Fair enough? All the best,--Jorge Stolfi (talk) 04:22, 9 April 2009 (UTC)

I have made this change, and added some book and journal references that attest the alternate names. --Jorge Stolfi (talk) 23:57, 14 April 2009 (UTC)

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).

Currently under planetary orbits it says "Keplerian elliptical orbits are the result of any radially-directed attraction force whose strength is inversely proportional to distance." Presumably that should be "distance squared". Fathead99 (talk) 15:02, 9 June 2009 (UTC)

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)

This comment I believe is correct. However the article appears wrong against the diagram. The diagram shows origin on left focus point but the commentry has a positive sign in the denominator. I believe it should be negative. With theta zero, r should be a*(1+e). Please confirm or correct me or modify the article. I will modify with enough confirmations. Thanks --MereEngineer (talk) 23:14, 4 May 2009 (UTC)

Third definition of ellipse using directrix

The "Euclidean geometry" section gave a third definition of ellipse (besides "conic section" and "constant-sum-of-focal-distances"), namely "constant-ratio-of-dists-to-focus-and-directrix". However this property was already given in the section "Directrix", and it seems to be much less common than the other two. So I merged that third definition into the "Directrix" section. --Jorge Stolfi (talk) 00:02, 15 April 2009 (UTC)

How to draw an ellipse with pen and string given the semi-major and semi-minor axes

To draw an ellipse with semi-major axis A and semi-minor axis B, the separation of the two foci S is given by the equation:

S = 2 x A x Sin ( Cos-1 ( B / A ) )

And the length of the string used L is given by:

L = 2 x A —Preceding unsigned comment added by 163.166.8.14 (talk) 08:28, 17 April 2009 (UTC)

Isn't this information already given in the article? --Jorge Stolfi (talk) 22:46, 17 April 2009 (UTC)

Polar form relative to center

This looks wrong. (or maybe I'm misreading it and some clarification would help.) With the formulas as given, X and Y can never be negative. I would guess that there needs to be some way to select the negative versus positive square roots in the appropriate quadrant. —Preceding unsigned comment added by 199.43.48.129 (talk) 13:33, 14 July 2009 (UTC)

I changed this to a straight polar form instead of parametric. Since there's no harm in r always being positive it should fix your issue.--RDBury (talk) 23:12, 16 July 2009 (UTC)

Elliptical trainer

I was curious about the elliptical trainer used in gyms. My search for elliptical led me here with a "redirected from elliptical." I propose one of the following changes: 1. Change redirect from "ellipse" to "elliptical trainer" 2. Add "elliptical trainer" to ellipse disambiguation page 3. Add "for the gymnastic training machine, see elliptical trainer" to the top of the ellipse article.

What do y'all think would be the best idea? I think referring to the machine as an "elliptical" is common enough that some mention is warranted.74.10.227.130 (talk) 15:46, 15 July 2009 (UTC)

• The most important meaning of "ellipse" is the geometrical term, so ellipse should remain as it is now. I have added elliptical trainer to the ellipse (disambiguation page). All the best, --Jorge Stolfi (talk) 16:06, 15 July 2009 (UTC)

"General polar form" is not polar:

I deleted the following section:

General polar form

From the implicit general polar equation of an ellipse centered at $(x_{0},y_{0})$ and rotated by $\phi$

${\frac {\left(x_{0}-r\cos(\theta -\phi )\right){}^{2}}{a^{2}}}+{\frac {\left(y_{0}+r\sin(\theta -\phi )\right){}^{2}}{b^{2}}}=1$

follows the general polar form

$r(\theta )={\frac {ab{\sqrt {\left(a^{2}-x_{0}^{2}\right)\sin ^{2}(\theta -\phi )+\left(b^{2}-y_{0}^{2}\right)\cos ^{2}(\theta -\phi )-x_{0}y_{0}\sin(2(\theta -\phi ))}}+a^{2}y_{0}(-\sin(\theta -\phi ))+b^{2}x_{0}\cos(\theta -\phi )}{a^{2}\sin ^{2}(\theta -\phi )+b^{2}\cos ^{2}(\theta -\phi )}}$

The first equation above is not a "polar form" since the angle θ is not the angular coordinate of the polar coordinate system. That formula is merely a parametric equation of the ellipse in general position, already given in a previous section of the article.
I haven't checked the second formula, but if the angle θ there is the same as in the first formula, then it is not a "polar form" either. If the θ in the second formula is indeed the polar angle coordinate, then the second formula alone can be restored to the article. (In WP articles one does not usually provide proofs or derivations for the formulas.) All the best, --Jorge Stolfi (talk) 00:47, 4 December 2009 (UTC)

• The angle in the first formula indeed is the angular coordinate, but the equation is "implicit polar". Have restored a corrected version of the second formula. Wikilupos (talk) 11:43, 4 December 2009 (UTC)Robert Nowak

• I see, sorry for my mistake. However, the formula seems unnecessarily complicated, and is partly polar, partly Cartesian (since the center is given in Cartesian coordinates). What about:

General ellipse in polar coordinates

The following implicit equation on the polar coordinates (r,θ) describes an ellipse centered at a point with polar coordinates (r₀,0), with its major axis rotated by φ relative to the θ=0 axis:

${\frac {\left(r_{0}-r\cos(\theta -\varphi )\right){}^{2}}{a^{2}}}+{\frac {\left(r\sin(\theta -\varphi )\right){}^{2}}{b^{2}}}=1$

The radial coordinate r can be expressed as a function of the polar angle θ:

$r(\theta )={\frac {{\sqrt {2}}abM(\theta )-(a^{2}-b^{2})P(\theta )+(a^{2}+b^{2})Q(\theta )}{2N(\theta )}}$

where

$M(\theta )={\sqrt {(b^{2}-a^{2})\cos(2\theta -2\varphi )+a^{2}+b^{2}+r_{0}^{2}\cos 2\theta -r_{0}^{2}}}$

$P(\theta )=r_{0}\cos(\theta -2\varphi )$

$Q(\theta )=r_{0}\cos \theta$

$N(\theta )=a^{2}\sin ^{2}(\theta -\varphi )+b^{2}\cos ^{2}(\theta -\varphi )$

For a geral ellipse whose center has arbitrary polar coordinates (r₀,θ₀), the same formulas hold, with θ−θ₀ and φ−θ₀ substituted for θ and φ in the right-hand side of each equation.

What do you think? All the best, --Jorge Stolfi (talk) 18:03, 6 December 2009 (UTC)

Hello Jorge, as I posted before the equation was not corret (it erronenously rotated the ceneter (X0,Y0) by phi). changed everything to polar as sugested with center $(R_{0},\theta _{0})$.

$r(\theta )={\frac {P(\theta )+Q(\theta )}{R(\theta )}}$

where

$P(\theta )=R_{0}\left(\left(b^{2}-a^{2}\right)\cos \left(\theta +\theta _{0}-2\phi \right)+\left(a^{2}+b^{2}\right)\cos \left(\theta -\theta _{0}\right)\right)$

$Q(\theta )={\sqrt {2}}ab{\sqrt {R(\theta )-2R_{0}^{2}\sin ^{2}\left(\theta -\theta _{0}\right)}}$

$R(\theta )=\left(b^{2}-a^{2}\right)\cos(2\theta -2\phi )+a^{2}+b^{2}$

hopefuly we are finished

regards Robert Wikilupos (talk) 19:50, 8 December 2009 (UTC)

• Great! The formulas are now fairly nice. I only changed R₀ to r₀ for consistency, and phi to varphi because it is the only kind of "phi" in the WP Greek alphabet tool. (I try to avoid math embedded in the text; it looks terrible on my browser, because the font sizes do not match, and presumably that is the case for many other readers too.)
  Usually one assumes that a≥b. Does that matter for your formulas? If so we must add a note to that effect. All the best, --Jorge Stolfi (talk) 22:06, 8 December 2009 (UTC)

Merge with Ellipse/Proofs

It should be possible to simplify and summarize the material in Ellipse/Proofs so that it can be merged into the main article. In addition, the article name 'Ellipse/Proofs' does not conform to Wikipedia naming conventions. Therefore I propose that the other article be merged with this article.--RDBury (talk) 12:19, 14 December 2009 (UTC)

As a general rule, Wikipedia math articles, or encyclopedia articles in general, do not include proofs. Those belong in textbooks or journal papers, which can be cited in WP articles. It is of course assumed that every math statement in Wikipedia either can be easily checked by mathematicians or has a proof published somewhere. If we add proofs to this article, we must add them to every other math article; and that is just unfeasible, not merely inappropriate. There are exceptions, such as articles on famous theorems where specific proofs have historical interest (e.g. Euler's proof of the Königsberg Bridge problem, which started graph theory). But this is not the case here, and anyway the article is too long already.
As for the name, the material was taken out from the article some time in the past, and it was placed in that subpage (rather than being simply deleted) out of respect for the author's work. If the material cannot be turned into a separate stand-alone encyclopedic article (and I cannot see how it could), then we should just delete it. Perhaps in Wikibooks there is a good place for it. All the best, --Jorge Stolfi (talk) 14:57, 15 December 2009 (UTC)

General form

The second equation given under the topic of "In Analytic Geometry" -> "General Ellipse" is not a general equation for an ellipse at all. It would not be valid for an ellipse which passes through the origin! —Preceding unsigned comment added by Quadruplecheck (talk • contribs) 22:14, 1 February 2010 (UTC)

Reworked lead section

Recently the lead section was rewritten to say:

---

In mathematics, an ellipse (from Greek ἔλλειψις elleipsis, a "falling short") is one of the four conic sections. (The other three are the hyperbola, the parabola, and the circle (which is a special case of the ellipse). An ellipse is a smooth curve lying in a plane, and differing from hyperbolas and parabolas in that it is bounded, i.e. it can be contained inside some circle. An ellipse can be defined by its geometric properties or by the kinds of equations for which it is the solution set. Geometrically, an ellipse is formed by the intersection of a cone and a plane. If the plane is perpendicular to the axis of the cone, then the ellipse is a circle. An ellipse can also be defined as the locus of all points such that the sum of the distances to two fixed points (called focii) is constant. When an equation is used to define an ellipse, the equation has the property that it has two variables, is quadratic in both variables, and the coefficients of the two quadratic terms have the same sign.

Ellipses also arise as images of a circle or a sphere under parallel projection (and some cases under perspective projection). Ellipses are also the simplest Lissajous figures (figures formed when horizontal and vertical motions are sinusoids with the same frequency).

---

I understand that the change had good intentions, but the new text had some problems:

• Some of the wording (e.g. "An ellipse can be defined ... solution set"; "When an equation ... same sign.") is more appropriate for a textbook than for an encyclopedia. Wikipedia is not a textbook.
• In any article, the first paragraph, and preferably the first sentence, must define the article's topic as precisely as possible and distinguish it from other article's topics. Merely saying "is one of the conics" does not do that.
• If circles are a special case of ellipses (as everybody seems to agree), then there are only three cases of conics, not four.
• The concept of "bounded" is probably easier for a lay reader to understand than "is contained in a circle" (which may be misunderstood, e.g as "is inscribed in a circle"). Indeed the equivalence of the two concepts is a non-trivial theorem, and arguably "bounded" is the more "primitive" or "fundamental" of the two. (Consider that one could equally say "contained in a square" or "contained in a regular heptagon")
• Likewise the verbal paraphrase of the implicit algebraic equation will be hardly understandable to lay readers, and irritating for slightly more sophisticated readers. In any case it is imprecise (which "two variables" are that? Angle and radius? Slope and arc length? Does X²+ Y² = 0 define an ellipse? How about X²+ Y² + 1 = 0? What if both coeffs are zero?) Methinks that it is better give only the "sophisticated" version ("bounded case of an algebraic curve of degree 2") and leave explanations for the article body, where the equation can be given in algebraic notation.
• The revision eliminated the statement that ellipses are also (parametric) rational curves of degree 2, which is at least as important as the implicit formula (e.g. in computer graphics).
• The name "foci" and other ellipse-related terms (center, axes, semidiameters, etc.) are already given together in the article body. There is no need to name the foci (alone) in the lead section.
• The new text seems to say that *all* Lissajous figure have vertical and horizontal motions with the same frequency. Actually the frequencies are different in general, and the figure is an ellipse only when they are equal.

I have re-rewritten the lead to fix these problems. All the best,--Jorge Stolfi (talk) 03:48, 13 January 2010 (UTC)

I appreciate your attempt at a lede, but in correcting some mistakes you have introduced others. In particular, not every conic section in which the plane is not parallel to the axis of the cone is an ellipse. Ah, well, we can both keep trying. Nobody ever said it was going to be easy.

Addressing your bullet points. 1) Wikipedia is not a textbook, but it is one of the most widely used student resources, and it doesn't hurt to mention that the ellipse is a solution set to particular kind of equation. 2) I agree, but we need to get the definition right. 3) Most books say four kinds of conic sections. Probably best we avoid mentioning a number. 4) I agree. 5) I've tried to address your objections. 6) I think parametric equations can wait for further down in the article, because students are not introduced to them in elementary courses, but if you want to put them back, I will not object. After all Lissajous figures are an even more advanced topic. 7) I can live with that. 8) Good call. Rick Norwood (talk) 14:37, 13 January 2010 (UTC)

Rick, I hope we can converge to a middle ground. However:

1. Note that the first sentence says intersection with a *cylinder*, not a *cone*. So the condition is indeed "not parallel to the axis".
2. We should say "conical surface" instead of "cone" because the latter is usually understood to be (1) a solid (2) only one branch (3) with a bounding base. Any of these three features would break the definition. Idem "cylindrical surface" instead of "cylinder".
3. One would expect students to read something *beyond* the lead section...
4. The fixes to the "verbalized" equation only made more awkward. It reads like trying to describe the shape of the US by words instead of giving a map. Even readers who already know the algebra will have to pause and think hard to understand what it is trying to say. Presumably anyone who knows what is an "equation" and "degree" knows algebra, so the proper way to convey that information is by using algebra.
5. Moreover, the solution set is an ellipse only if the variables are *Cartesian* coordinates.
6. The cone does not have to be "right"; sections of oblique circular cones are conics too. In fact "right" is meaningless for a conical surface, it only applies to finite solid cones. "Circular" is necessary because conical surfaces can have arbitrary sections. (The section perpendicular to the axis could also be elliptical instead of circular, but that still gives the same conics).
7. Wikipedia is not written *solely* for elementary students; many readers are engineers, scientists, mathematicians, etc. So the introduction should be good for them too. Parametric curves are important in many applied areas; in computing, for instance, they are much more common than implicit ones. An encyclopedia entry (unlike a textbook) does not have to be *entirely* uderstandable to high-school students.

All the best, --Jorge Stolfi (talk) 19:42, 13 January 2010 (UTC)

I'm sure we can work toward common ground.

1) While it is true that every ellipse can be formed by the intersection of a plane and a cylinder, most sources use the cone, and we should follow the sources.

2) The word "cone" suggests an ice-cream cone, which is hollow. I think cone gets the idea. And mathematically, a cone is the set of all points on any line connecting a point on a plane curve to a fixed point not in the plane of the curve, so we're technically correct as well. We might want to say "right circular cone", and of course, in the hyperbola article, we need to say "double cone". "Conical surface" seems to me pedantic, like saying "the sum of the number represented by the numeral 2 and the number represented by the numeral 2 is the number represented by the numeral 4" instead of "2 plus 2 equals 4".

3) One would hope...

4) I have no objection to using algebra here.

5) Again, it seems pedantic to say Cartesian graph. The word "graph" suggests Cartesian if no other adjective is used.

6) True, but again most sources say "cone" or "right circular cone". To take the example nearest to hand, I'm getting ready to teach a class out of Thomas' Calculus. Let's see what Thomas says. "It is an example of a conic section, which are the curves formed by cutting a double cone with a plane." I've got a desk copy of Tan's Calculus that I haven't even opened yet. "...the intersection of a plane and a double-napped cone." Moving to a slightly higher level, let's see what Geltner and Peterson, Geometry, says, "Let C be a circular region and P a point not in the plane containing the circular region. A circular cone is the union C, P, and all points between P and each point on C." I'm picking these books haphazardly, not to prove my point.

7) I have no objection to mentioning parametric curves.

Rick Norwood (talk) 14:52, 14 January 2010 (UTC)

Rick,

1. Please note that the Geltner and Peterson are defining a solid, bounded, one-sided cone, not a hollow cone or conical surface. The Wikipedia article for cone (geometry) also gives (or gave until recently) the solid as the primary meaning. Thomas uses "double cone" and Tan uses "double-napped cone". So if anything the citations show that one canot just pick a definition from a book; each book has its nomenclature and definitions for the prior concepts. In Wikipedia one must pick a definition that makes sense when the article is read on its own, without assuming that the reader has read other articles first.
2. I concede that defining ellipse by using a cylinder instead of a cone is unusual, and some editors would call that "original research". However, textbooks use cones because they first define conic sections in general and then get to ellipses as a special case. This article will probably be acessed directly, not through conic sections; so defining ellipse as a special case of conic section is actually rather weird. Ellipses are special cases of many families (Lissajous figures, Lamé curves, rational curves, ...), so why pick conics? If ellipses can be defined directly in a more succint and rather elementary way, why should we use a roundabout road that only peple who have studied geometry will understand? As for "original research", IMHO that concept applies to most areas, but does not make sense in mathematics (except for history, notation and nomenclature). Any mathematically statement or definition that is correct and easy to check is just a paraphrase of "God's claims". However, while I am willing to fight for this view against the bureaucrats, I cannot ask others to go with it. I guess I can live with a cone section.

All the best, --Jorge Stolfi (talk) 18:13, 14 January 2010 (UTC)

Lead paragraph

I restored an older version of the lead paragraph because the current version is incorrect. First, a cylindrical surface may have any cross section, as article which the term links to. If you mean the surface of a circular cylinder then the word circular must be included to avoid ambiguity. Second, the definition as a section of a circular cylinder is one way to characterize an ellipse, but I've never seen it given as the definition in the literature. If it does have a reliable reference then it is by no means a standard that should used in an encyclopedia. Most sources define the ellipse either by the equation, the pins and string construction, or using focus and directrix, with most giving the conic section definition for historical reasons. The old definition has issues, but these can be repaired.--RDBury (talk) 17:56, 14 January 2010 (UTC)

As for cone versus cylinder and following textbooks, see above. You are right that it would be necessary to say "circular cylindrical surface". Come to think of it, perhaps the best defintion is indeed the the sum-of-distances one: not a special case of a more complicated thing, entirely two-dimensional, relevant to many applications, etc.. All the best, --Jorge Stolfi (talk) 18:18, 14 January 2010 (UTC)

Another good reason for having the plane intersecting a cone definition is that there is a picture of that just to the right. And while I have no objection to saying "double cone", of course a plane cannot intersect both parts of a double cone if it produces an ellipse. Rick Norwood (talk) 14:40, 15 January 2010 (UTC)

True but the phrase was defining general conic sections, not ellipses. I guess that could be considered an argument for using cylinder (which can be argued to be a special case of cone). But nolo contendere... All the best, --Jorge Stolfi (talk) 17:57, 15 January 2010 (UTC)

Polar form relative to center

I see that my deletion of the second version of this polar form has been reverted. I still believe that it is incorrect. For example, at $\theta =0$, the second form gives $r(0)=a\left({\frac {1+e}{1-e}}\right)$ rather than the correct value $r(0)=a$ as given by the first form. I was unable to see how the second form could be simply adjusted to make it a correct alternative. 64.118.100.243 (talk) 22:59, 18 January 2010 (UTC)

Derivation of area formula

The following section was recently added to the article. While we appreciate the effort that went into this edit, it had to be deleted because "Wikipedia is not a textbook". In particular, articles are not supposed to include mathematical proofs (unless the proofs are of hisorical interest on their own, which is not the case here).

---

The area for ellipse πab can be derived using vector calculus and Green's theorem.

Suppose an ellipse of the general equation ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$, the parametric equation for this ellipse is given by

$x=a\cos t$

$y=b\sin t$

Where t ranges from 0 to $\pi$.

Therefore, using Green's theorem, the area for an ellipse is given by:

$A={\frac {1}{2}}\oint _{C}x\,\mathrm {d} y-y\,\mathrm {d} y$

$={\frac {1}{2}}\int _{0}^{2\pi }[(a\cos t)(b\cos t)\,\mathrm {d} t-(b\sin t)(-a\sin t)\,\mathrm {d} t]$

$={\frac {1}{2}}\int _{0}^{2\pi }ab\,(\cos ^{2}t+\sin ^{2}t)\,\mathrm {d} t$

$={\frac {1}{2}}\,ab\,\int _{0}^{2\pi }\mathrm {d} t$

$=\pi ab$

---

This material would be quite appropriate for a textbook, and perhaps it can be used in a WikiBook. All the best, --Jorge Stolfi (talk) 07:51, 2 February 2010 (UTC)

Picture needed

The main page of this article has a section on "elliptical gears" but I am having trouble visualizing it. Could someone please post a couple of pictures on elliptical gears? That would really help. Dexter Nextnumber (talk) 00:01, 4 April 2010 (UTC)

Unreadable formulas

Some formulas are unreadable if the Windows user changes the color scheme of his computer from the default to another one. I use Vista with "High Contrast#1" color scheme (I want black bsckground to not strain my eyes). Some formulas appear just as a random collection of meaningless pixels.

Also, I searched the text but found none of the remarcable properties of an ellipse, like the bisectrice of the angle made by the foci with a point on the ellipse is perpendicular to the tangent in that point, how to compute where the foci are, formulas for tangent, normal, intersection with a line and so on. I came here to refresh my knowledge about ellipses but frankly learned nothing more than I already knew.

Sorry for my awful terminology, I learned mathematics in another language but I hope you still can understand what I mean. —Preceding unsigned comment added by Srelu (talk • contribs) 17:54, 11 May 2010 (UTC)

Formula for circumference

The circumference formula is not coherent with the definition of the E-function found at the Wikipedia page refered to, since the argument there is squared. Wouldnt it therefore be better to change the formula to E(epsilon) ? —Preceding unsigned comment added by 89.236.1.222 (talk) 22:53, 21 May 2010 (UTC)

I changed it.--Patrick (talk) 07:05, 22 May 2010 (UTC)

Code examples

I added an R function to draw an ellipse. Slightly concerned about clogging the page with code, but I don't think javascript is a special class especially deserving of code examples. Ken K (talk) 02:59, 18 July 2010 (UTC)

Errors

Subsection "General ellipse" of Section "In analytic geometry" contains a number of errors in formulas. For example: according to it, the equation $XY+1=0$ corresponds to an ellipse, since it is $AX^{2}+BXY+CY^{2}+DX+EY+F=0$ for A=0, B=1, C=0, D=0, E=0, F=1, thus $F(B^{2}-4AC)=1$ is positive as required. But in fact, the equation $XY+1=0$ corresponds to a hyperbola. For correct formulas see for instance Citizendium. Boris Tsirelson (talk) 20:47, 25 July 2010 (UTC)

Yes, that should be B^2 − 4AC < 0. For a ref to the correct formula, see for instance

DVdm (talk) 13:23, 26 July 2010 (UTC)

If Citizendium is correct B^2 − 4AC < 0 is necessary but not sufficient. In addition (A+C)f(t1,t2) <0 is required. Cynthia Y. Young is too sloppy. 131.174.224.199 (talk) 15:44, 26 July 2010 (UTC)

I had checked Citizendium but it doesn't give a source. I found a few sources for Young's version and picked one at random. Some more here, here, here, etc... If we can find a reliable source for Citizendium's version we can add a note. DVdm (talk) 18:15, 26 July 2010 (UTC)

$X^{2}+2Y^{2}-1=0$ is an ellipse, but $X^{2}+2Y^{2}+0=0$ is a single point, and $X^{2}+2Y^{2}+1=0$ is an empty set. This is why F matters, not only A, B, C. But of course, "The threshold for inclusion in Wikipedia is verifiability, not truth", I know. Thus, sooner or later "2+2=5" will probably appear in WP, and I cannot prevent it. A pity. Boris Tsirelson (talk) 19:26, 26 July 2010 (UTC)

Yes, perhaps it is indeed a good idea to explicitly exclude degenerate cases like a single point and the empty set. Since the first source from my short list does that, and since Young also adds a remark, I have added that source, and made a slight change to the text. Good point, no need for pity :-) DVdm (talk) 20:12, 26 July 2010 (UTC)

[unindent]

It seems to me that reference 13 is incomplete. Consider as a counterexample the equation

$-5x^{2}-2{\sqrt {3}}\;xy-7y^{2}+4x+12y-7=0\qquad \qquad \qquad \qquad (1)$

Clearly

$B^{2}-4AC=(2{\sqrt {3}})^{2}-4\times (-5)\times (-7)=-128<0$

hence reference 13 claims that equation (1) represents an ellipse (or circle). Yet, upon rotation and translation equation (1) becomes

$-8x^{2}-4y^{2}-1.799=0$

which clearly has no solution for real x and y (or stated pedantically: has only the empty set as solution). Citizendium states as extra condition

$(A+C)f_{t}<0\quad \Rightarrow \quad (-5-7)\times (-1.799)<0$

Clearly, the second condition is not satisfied and the quadric (1) is not an ellipse.--P.wormer (talk) 16:02, 28 July 2010 (UTC)

Sure, Boris already gave a similar empty set example: x^2+2y^2+1=0. So we say that the empty set is an ellipse, no big deal. Otherwise, let's find a source for Citizendium's version, or for something similar. DVdm (talk) 16:41, 28 July 2010 (UTC)

I beg to differ, I do find it a big deal, not to say very confusing, to call an empty set an ellipse (or a parabola, or a hyperbole, or a sphere, or a torus, and what have you). I agree that my example is essentially the same as Boris's, but my example requires some non-trivial calculations. Or did you see straightaway that Eq. (1) couldn't possibly be an ellipse? If that's so, I take off my hat to you and make a deep bow.--P.wormer (talk) 17:15, 28 July 2010 (UTC)

Don't bow too deep. You could hurt yourself :-)

Anyway, in geometry the ellipse is defined as the locus of points with some constant sum S of distances to two fixed points. So a defining value S smaller than the distance between the defining points produces the (not an) empty set, aka the empty ellipse. DVdm (talk) 17:30, 28 July 2010 (UTC)

In subsection "Parametric form in canonical position", there's a statement: "Note that the parameter t (called the eccentric anomaly in astronomy) is not the angle of (X(t),Y(t)) with the X-axis." Given the equation the statement refers to, that doesn't seem correct. Bobertmurphy (talk) 02:27, 29 November 2010 (UTC)

The "much faster series" for the circumference taken from reference 11 must be incorrect. Although it gives nice values for large to moderate epsilon, its limiting value for epsilon->1 is 0 instead of 4a. (wseewald@gmx.ch) 160.62.4.10 (talk) 12:47, 25 February 2011 (UTC)

Saturn's rings?

Circles only appear to be ellipses under parallel projection. The photograph of Saturn's rings uses perspective projection (because it's a photograph). Circles do not become ellipses under perspective projection. 157.193.175.207 (talk) 11:00, 22 June 2010 (UTC)

It's close enough to an ellipse that we can't tell. This image was taken from Earth, and the distance between the near and far edges of the rings is dwarfed by the distance from Earth -- so much so that you're probably getting more distortion of the ellipse from the tolerances of your monitor, never mind the relative distances between your eye and the near and far edges of the ellipse in the photograph. 155.43.165.127 (talk) 23:22, 8 November 2010 (UTC)

This relation is as small as the angel of view in which the rings appear from earth. --Helium4 (talk) 18:57, 16 March 2011 (UTC)

But circles ARE ellipses under perspective projections!! --99.235.122.40 (talk) 00:46, 1 May 2011 (UTC)

The picture was taken from almost a billion miles away, so it's so close to being parallel that the difference is less than a pixel. In any case, it's a basic fact of projective geometry that conic sections remain conic sections under perspective. So unless you're so close that it appears as a hyperbola, it will appear as an ellipse.--RDBury (talk) 12:31, 16 May 2011 (UTC)

I'm confused as to what you are saying. "...the difference is less than a pixel" means it's a close approximation to an ellipse. "...conic sections remain conic sections under perspective" means it is exactly an ellipse. And I don't understand "...unless you're so close that it appears as a hyperbola" -- as you get closer and closer are you saying there is some point at which it suddenly changes appearance from an ellipse to a hyperbola?!

As for the wording of the caption, the restored wording "they appear to be ellipses" is ambiguous: does it mean they appear to be but are not (except as close approximations), or does it mean they are ellipses in the plane of the photograph? The caption should either say "they appear as close approximations to ellipses" or "they are ellipses in the plane of the photograph". (BTW, at the time I made the change to the wording (March 28), there appeared to be a consensus that it was approximate. The first assertion that they are ellipses didn't come till May 1.) Duoduoduo (talk) 15:35, 16 May 2011 (UTC)

Please clarify here and in the caption -- Which is it -- "they appear as close approximations to ellipses" or "they are ellipses in the plane of the photograph"? We shouldn't just leave the ambiguous wording as it is now in the caption. Duoduoduo (talk) 14:54, 24 May 2011 (UTC)

General polar form

Tried using the equations listed under the section General Polar Form to obtain a list of ellipse radii corresponding to thetas. There is an issue. What's the admissible range of thetas, which can be used without making the square root in Q(theta) negative? It's very easy to get a negative square root there, so we should really specify the range (well, domain) of admissible thetas. PaulBilokon (talk) 16:04, 8 April 2011 (UTC)

Oval-shaped

Can the terms oval-shaped and egg-round be added as alternatives in the definition ? see http://www.thefreedictionary.com/elliptical

Oval is in the "see also" section. That includes Egg shape. --George100 (talk) 08:28, 18 August 2011 (UTC)

Simple formula for the focus position f

I think, the article would benefit from an easy to find and simple formula for the position of the focus as a function of the radii, especially for drawing ellipses with office programs, where the width (=2a) and the height (=2b) are usually given. If I read correctly, then the focus is at $f=ae$ and the eccentricity is $e={\sqrt {\frac {a^{2}-b^{2}}{a^{2}}}}$. Taking the square of both equations results in a very simple formula:

$f^{2}=a^{2}-b^{2}$ or after taking the square root the square $f={\sqrt {a^{2}-b^{2}}}$

I found this equation also in the eccentricity subsection. So I'll split this off and make a new subsection named Focus. — Preceding unsigned comment added by Zeptomoon (talk • contribs) 08:52, 14 April 2011 (UTC)

analytical: derivative at some coordinate?

I tried to find out, how to compute the slope at a certain point [x,y]; it could be of an ellipse centered at the origin, where the formula of the ellipse may be given in the form x^2/a+y^2/b=1. Unfortunately I don't see, where I could start in that material-rich article. My final goal is to find a formula for the distance of some point P=[x_p,y_p] to the ellipse (and also that nearest point E=[x_e,y_e] on the circumference of the ellipse) Gotti 13:33, 17 August 2011 (UTC) — Preceding unsigned comment added by Druseltal2005 (talk • contribs)

Very interesting questions. (1) To find the slope, totally differentiate the equation for the ellipse to get an expression that is linear in dx and dy, and solve for dy/dx. (2) To find the distance to the ellipse and the nearest point on the ellipse: The nearest point on the ellipse has a tangent line that is perpendicular to the line from E to P. So find the equation of a line going through the known point [x_p,y_p] and the unknown point [x_e,y_e] and having a slope equal to -1 / (slope of tangent line to ellipse at [x_e,y_e] ). Plug into this the solution of the ellipse equation for y (or maybe y^2) (to force point E to be on the ellipse). You end up with a fourth degree equation in x_e having either two or four real solutions, one of which is your desired nearest point's x-value. The article quartic function gives the very complicated formula for finding the roots. It's a mess -- I've tried it and gave up when I saw the quartic equation -- it didn't look like I'd find any useful insight. Duoduoduo (talk) 14:36, 17 August 2011 (UTC)

Also, see Azad, H., and Laradji, A., "Some impossible constructions in elementary geometry", Mathematical Gazette 88, November 2004, 548-551.

for a trigonometric proof that the nearest point on the ellipse is not classically constructible. This means algebraically that the x and y coordinates of the nearest point on the ellipse cannot in general be written using only square roots. Duoduoduo (talk) 21:20, 10 November 2011 (UTC)

My bad on last edit changing the polar formula

I forgot that mathematical convention measures counterclockwise starting from the positive x-axis -- in geophysical convention, it's measured clockwise from positive y or z (i.e., relative to true north or the local zenith).

Polar form relative to center Polar coordinates centered at the center.

In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate θ measured from the major axis, the ellipse's equation is

$r(\theta )={\frac {ab}{\sqrt {(b\cos \theta )^{2}+(a\sin \theta )^{2}}}}$

Don't think that the sqrt function is valid in the denominator. Can someone check this? — Preceding unsigned comment added by Offcut (talk • contribs) 01:53, 16 January 2012 (UTC)

Looks right to me. At theta=0 it correctly gives r=a if you include the sqrt, and at theta=pi/2 it correctly gives r=b if you include the sqrt. Duoduoduo (talk) 02:14, 16 January 2012 (UTC)

is there a difference in shape between a section of a cone and a section of a cylinder?

When I understood how to create an elipse, I thought it would be a section of a cylinder. All the definitions say it is a section of a cone, which is much harder to visualize. I checked with a mathematician friend who sent this definition, confirming that an ellipse is a section of a cylinder:

If a plane intersects a cylinder in such a way that it intersects the axis of the cylinder (but does not contain the axis), the intersection is an ellipse. If the cylinder has radius 1, the "short radius" of the ellipse has length 1 and the "long radius" has length secant(90degrees - theta), where theta is the angle between the axis and the plane. If theta = 90 degrees, the major semi-axis = secant(0) = 1. As theta approaches 0 degrees, the major semi-axis = secant(90 degrees - theta) approaches secant(90 degrees} = infinity and the intersection becomes two parallel lines in the limit.

This being the case, why do most definitions use the cone rather than the cylinder to describe an ellipse? To me, as a lay person, the section of a cylinder is much easier to visualize than that of a cone. (Or is a cylinder defined as a type of cone, and the definition is just being inclusive?) — Preceding unsigned comment added by 12.170.93.196 (talk) 17:15, 9 November 2011 (UTC)

Interesting question. Since the parabola and the hyperbola cannot be defined in terms of a cylinder, whereas all three can be defined in terms of a cone, focusing on the cone allows comparisons of different types of curves -- that's why it's always done in terms of cones. But I agree with you that a cylinder is easier to visualize. The article cylinder (geometry) mentions this. I'll put a mention of it into this article, too. Duoduoduo (talk) 17:49, 9 November 2011 (UTC)

A number of definitions have been used over the millennia, the focus-directrix definition is also common as well as the pins-and-string method. In the end it's somewhat arbitrary which, of several methods of construction you take as the definition since you still need to prove they produce the same curve. The definition in terms of a cone has the advantage that it makes it obvious why the curve is called a conic section.--RDBury (talk) 18:39, 9 November 2011 (UTC)

History of Pin and String Method

Currently we have an uncited claim that Maxwell invented this. But it seems weird that that the pin-and-string method should be preceded by the trammel (by 14 centuries or so!), which is much harder to construct. Anyone have a citation? Ethan Mitchell (talk) 22:56, 5 February 2012 (UTC)

Maxwell did have a paper called "On the Description of Oval Curves and Those Having a Plurality of Foci" presented to the Royal Society of Edinburgh when he was 14 (too young to present the paper himself). The Maxwell article (q.v.) cites Martin Gardner on this. It's more probable that Maxwell discovered a similar pin and string for drawing Cartesian ovals which is described in Gardner's article, but Gardner does not state it explicitly. I believe Apollonius of Perga would have at least known that the sum of the distances to the foci is constant, from which it's a small step to get the pin and string method. In general, one has to be very careful about this kind of anecdote, they tend to be repeated without being verified with original sources so exaggerations and distortions often appear in otherwise reliable sources. Actually there's other material in that section which is dodgy as well so I'll go ahead and remove the questionable bits.--RDBury (talk) 13:21, 6 February 2012 (UTC)

Thanks, RDBury. I'm also culling this:

Besides the well known ratio e=f/a, it is also true that e=a/d.

I think the point here is that the ratio of fa to fd is e, but (1) I'm not totally sure that's true, and (2) the above formula makes no sense. Ethan Mitchell (talk) 13:32, 6 February 2012 (UTC)

Oops, never mind. Ethan Mitchell (talk) 16:38, 6 February 2012 (UTC)

Pointing a mistake at Ellipse

My name is George Nemes, it is first time I am trying to comment on a mistake in a Wikipedia article. In this page "Ellipse", under the subsection "Parametric form in canonical position", the next to last formula is wrong. If you want to express tanΦ' with respect to tant using the focal distance f, instead of the currently wrong formula tanΦ' = (1 - f)tant, the following right one should be used: tanΦ' = [(1 - f^2/a^2)^1/2]tant. This equality is obtained from the last equation tant= (a/b)tanΦ' by solving for tanΦ' and replacing b/a with (1 - f^2/a^2)^1/2. Please make the appropriate correction. Thanks. George 78.96.220.246 (talk) 07:54, 24 June 2012 (UTC)

Done. Thanks for pointing it out! Duoduoduo (talk) 16:23, 24 June 2012 (UTC)

Parametric form in canonical position, missing angles in drawing.

In the section "Parametric form in canonical position" it mentions angles t, phi, phi' but there is no drawing showing the angles. Although there are even two drawings, one of which is animating. This is impossible Mathematics, doing drawings without labels.

phi' is called theta in the immediately following section titled "Polar form relative to center", and should if possible be theta in this section too.

Besides this I'm curious why a German Ellipse is such a different thing from an English Ellipse from a Russian Ellipse. In a Russian Ellipse, the focal parameter p is not needed altogether. The focal distance is called e, f or c, depending on language.

 — Preceding unsigned comment added by 84.75.9.8 (talk) 22:20, 17 July 2012 (UTC) 

Mistake concerning figure related with polar form of ellipse?

hi,

concerning formula : $r(\theta )={\frac {a(1-e^{2})}{1+e\cos \theta }}$ I think that the figure related on the right is wrong, $\theta$ comes from perihelion and not aphelion. r is minimum when $\theta =2\pi$ or =0 and maximum when $\theta =\pi$. Here, r is minimum when $\theta =\pi$. — Preceding unsigned comment added by 84.99.70.139 (talk) 17:04, 1 June 2012 (UTC)

Hi,

I don't know if this is actually your problem, but when I tried to use this formula, I thought that 'e' referred to the constant e, when it actually refers to eccentricity. Once I corrected that, the formula started producing ellipses instead of hyperbolas. Also, it would be worth mentioning that vertical ellipses should swap sine for cosine.130.85.56.84 (talk) 05:36, 28 October 2012 (UTC)