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The section General polar form gives without citation a general formula. See this math reference desk discussion of 1 Oct. 2012 about whether the formula could be correct. In particular, it seems to me that the radical should have a plus minus sign before it. Otherwise, when the origin is outside the ellipse and the angle theta is such that the ray cuts through the ellipse twice (entering and exiting), the formula fails to give two solutions. Duoduoduo ( talk) 15:16, 1 October 2012 (UTC)
Several pages on Wikipedia including this page and the page Conic Sections claim that ellipses are one type of conic section. My idea of such a conic section would result in a curve like an ellipse but with one end with a smaller radius, or 'pointier' than the other. This is backed up by an edit in this talk pages archives which suggests that an ellipse being a conic section is fallacious and the true conic section would be an egg-shape. If this is true, could this be clarified? Otherwise, am I (and the commenter in the archives) just wrong? 124.177.190.63 ( talk) 12:57, 19 November 2012 (UTC)
Being Ben BernankGrinch of the Federal Reserve stole yet another Christmas from me, I had time to improve upon the Blankenhorn-Ramanujan ellipse circumference formula. This formula provides exact end-points and with a maximum relative error of several magnitudes better than the Ramanujan formula. Some day, the Wiki editors will recognize the improvement, but I am not holding my breath for it.
C~pi(a+b)*(1+3h/(10+sqrt(4-3h)) +(1.5*h^6-0.5*h^12)/((11pi/(44-14pi))+24100(1-h)))
Where a and b are the half length axes and h = (a-b)^2/(a+b)^2
Referencing http://i39.photobucket.com/albums/e191/toomers/ell1.jpg, replace the h^5.20114 with 1.5*h^6-0.5*h^12, and the 19176 with 24100.
See: http://ellipse-circumference.blogspot.com
BEING THIS IS FACTUAL, as anyone can check out and I can provide a spreadsheet to anyone interested this should be able to be showing on wiki's site for people to have easy access to. When there is no doubt about the veracity, there should not be power mongers preventing publication. Numbertruth ( talk) 00:41, 26 December 2012 (UTC)
It would be nice for the general public to see an equivalent form of the Ramanujan II formula without the Hoelder mean:
C~pi(a+b)[1+((x-1)^2(10(x+1)-sqrt(x^2+14x+1)))/((x+1)(33x^2+62x+33))], where x=b/a
Anyone here should be able to verify this. If the power structure here still doesn't want this shown to the public, well, I made my appeal. Numbertruth ( talk) 08:41, 9 January 2013 (UTC)
I simply didn't understand what exactly was meant here. Is the approximate ellipse which is really a circle is to be drawn with 2 compasses or with one? Perhaps a picture, or more precise details, or a reference would resolve the lack of understanding. — Preceding unsigned comment added by 31.44.140.246 ( talk) 19:29, 19 December 2012 (UTC)
I'm reverting the formula added today for the circumference, because the only citation given is http://ellipse-circumference3.blogspot.com/ , which is not a WP:reliable source according to Wikipedia standards. And that blog itself gives no other reference. Wikipedia requires that the formula be sourced to a refereed publication. Duoduoduo ( talk) 20:53, 2 October 2012 (UTC) Actually MrOllie beat me to it! Duoduoduo ( talk) 20:55, 2 October 2012 (UTC)
A formula of such a great breakthrough does not require a reference other than the formula as standing on its own. Show me one simplistic, one step formula capable of producing 8 significant digits, without the use of the Hoelder mean, and without using pi. Why reference a site that is not used as a blog as a blog? Was there any blogging on it? No! So why was the word "blog" mentioned"? If a person had a site named "mountainspot" would it really trip a person up if it wasn't about mountains? Get past the needless block and inspect the formula to see the tremendous contribution. If a person suddenly came up with an explicit formula, I suppose there will be haters of the contribution for wanting some encyclopedia to first publish it. Amazing. Please ask a mathematician about this - one who is versed in it, such as Michon or Cantrell. This amazing formula needs to be out in the open to be shared throughout the world to simulate investigation of the form used. Note that the error function listed on the site verifies the quality. Numbertruth ( talk) 03:45, 3 October 2012 (UTC)
Well, so it is, but some things just won't make it into journals when the one coming up with the material is not part of the status quo.
Look at these three new forms I derived using the artithmetic-geometric mean:
http://mathforum.org/kb/thread.jspa?threadID=2422080&messageID=7942499#7942499 C = 4*(pi*(e-1)*e*(e+1)*(d/dx)[agm(1,(1-x=e)/(1+x=e))]-pi*(e-1)*agm(1,(1-e)/(1+e)))/(2*(e+1)*agm(1,(1-e)/(1+e))^2)
http://mathforum.org/kb/thread.jspa?threadID=2420827&messageID=7937451#7937451 C=8e^2(1-e^2)*d/dk(K(k=e^2))+2(1-e^2)*pi/(AGM(1,sqrt(1-e^2)))
http://mathforum.org/kb/thread.jspa?threadID=2428401&messageID=8048879#8048879 C = 4*E((1-x)^2/(1+x)^2)= (2pi)*((((d/dx)[AGM(x,x^2)]|x=Q)*(1-x)*x^2)/agm(x,x^2)^2 + (2x^2-x)/agm(x,x^2))
Where Q = (1-e)/(1+e) = (1-sqrt(1-b^2))/(1-sqrt(1-b^2)); e= eccentricity, and b = length of minor axis.
These formulas make it easy for anyone to compute the ellipse circumference since the agm is simplistic and converges rapidly, thus so little effort would be required. The derivatives of the agm function require a good numeric processor, but other than that, these add to the theory on ellipses. As can be noticed, since I am not of the group that wants to reward those with credentials, progress will be slow. I saw this even when I was inventing in the sciences - it simply produced jealousies and then I get shut out. I hope this changes as humans need to look beyond certification to assess quality.
I will wait, even if I am long gone as this is what may happen.
I have noticed errors and some parts needing clarification regarding elliptical integrals on wiki but I'll leave it to the experts to spot. :) You may delete the earlier discussion regarding the pade variant or if you give me permission, I will do it. I don't want to do anything here I am not suppose to do. 184.100.17.31 ( talk) 07:44, 22 December 2012 (UTC) Numbertruth Numbertruth ( talk) 00:27, 11 January 2013 (UTC) A formula I just came up with today is simplistic, having a maximum relative error an entire magnitude lower than Ramanujan's second formula: C~4a[1+(pi/2-1)/(((1-sqrt(2)/2)+(sqrt(2)/2)*(b/a)^(-0.454))^(2pi-2))] Maximum absolute relative error: ~3.8E-5, note: b<>0.
This certainly is noteworthy to be inserted on the main page. Waiting around for this being shown in a popular publication shouldn't be the major point in preventing the world from seeing this. Numbertruth ( talk) 05:46, 18 January 2013 (UTC) Numbertruth ( talk) 19:35, 19 January 2013 (UTC)
I just edited the circumference section adding Bessel's series (which converges much more rapidly than the series in e). I also removed an elaborate series which is unlikely to be of much practical use. For people who need to compute the circumference accurately for very eccentric ellipses, I included the necessary AGM method (following the paper by Carlson). cffk ( talk) 18:25, 10 April 2013 (UTC)
The AGM method is good to have on this site. I went about it in a different fashion in three different forms, referenced above. I still think the direct and quite simplistic formula that has a maximum relative error an entire magnitude lower than Ramanujan's second formula should still get some mention here without first being touted by third-party "experts" since most anyone at the middle school level would be able to verify this and make the appropriate comparison: C~4a[1+(pi/2-1)/(((1-sqrt(2)/2)+(sqrt(2)/2)*(b/a)^(-0.454))^(2pi-2))] Maximum absolute relative error: ~3.8E-5, note: b<>0. Find a formula better than this that has just one number that isn't reduced into a simplified form (even the "0.454" could be altered to "5/11"), without converting a and b into some other number such as the Hoelder mean, and having this low of a maximum absolute relative error. Fellow Mathematicians, I am calling upon you! You may see the formula in normal format at http://ellipsesummary.blogspot.com/ I further contend that Zafary's formula deserves mention here as it's such a neat formula. Numbertruth ( talk) 04:15, 4 June 2013 (UTC)
In comparison with Ramanujan's formula that is shown on the main page versus the formula shown immediately above, the relative errors are (when a=1, b varies, with no loss of generality): b Ramanujan II Blankenhorn 0 4.023E-04 --- 0.00000001 4.023E-04 2.220E-16 0.000001 4.023E-04 1.952E-12 0.00001 4.021E-04 1.477E-10 0.0001 3.998E-04 9.635E-09 0.001 3.784E-04 4.750E-07 0.01 2.390E-04 1.206E-05 0.1 1.156E-05 8.321E-06 1 0 0
And so, one must wonder why there needs to be some sort of POPULAR recognition before allowing a very good formula to make it onto wiki? Please explain. This is so simple that just about any low level mathematician could verify. Numbertruth ( talk) 17:48, 14 July 2013 (UTC) Since the format is not easily seen for the relative erros shown above, a table form of the results are provided at: http://ellipsesummary.blogspot.com/ And so, the question still remains, why must there be a popularity contest to be able to provide such an advancement for all interested parties to be able to easily view? Again, this is something a low level mathematician could verify, in fact having just low level skills with a spreadsheet program and possessing a middle school math ability, one could easily verify what I am saying without requiring a vote of confidence from the math community. Numbertruth ( talk) 17:59, 14 July 2013 (UTC)
Unfortunately, nowadays there is a confusion in notation. Traditionally the linear eccentricity should be denoted with the latin e, and the numerical eccentricity should be denoted with the greek ε, where e=εa. Even on this page there is an obvious confusion in notation. Theodore Yoda ( talk) 15:29, 25 March 2013 (UTC)
I agree with the archived comment requesting labels on the diagramme http://en.wikipedia.org/wiki/File:Parametric_ellipse.gif. —DIV ( 138.194.10.62 ( talk) 06:54, 12 May 2013 (UTC))
Mathworld seems to have a different formula as their #58 compared to Ellipse#Parametric_form_in_canonical_position. I think WP is correct, but am puzzled at the discrepancy. —DIV ( 138.194.10.62 ( talk) 06:55, 12 May 2013 (UTC))
Quoth The name ἔλλειψις was given by Apollonius of Perga in his Conics, emphasizing the connection of the curve with "application of areas".
I don't get this sentence. Is an ellipse called so because it's a defective circle?
Thanks, Maikel ( talk) 09:48, 27 May 2013 (UTC)
Currently the lede says
This implies that parabolas are open but not unbounded, while hyperbolas are unbounded but not open. But it seems to me that both are open and both are unbounded -- if so, I think this needs to be reworded. Duoduoduo ( talk) 21:09, 19 July 2013 (UTC)
Duoduoduo: My proof that you just reverted was probably much more useful that the one that you replaced it by, since it used a "known" result (the area of a circle) and a simple geometric argument that anyone who's mathematically inclined could follow. Jacobians and integration seem like complete overkill. However, I don't care enough to press the case. cffk ( talk) 21:11, 21 July 2013 (UTC)
The "unproven" and "intuitive" result is proved as follows: the area is given by the integral
The second integral is just the area of a circle of radius , i.e., ; thus we have . In my book, this proof is so straight-forward that it doesn't need to be spelled out (and certainly the ancient Greeks found these results without resorting to calculus). cffk ( talk) 00:11, 22 July 2013 (UTC)
I have marked an uncited approximation formula as dubious. The article claimed it was 'better' but some simple comparisons with (1) Gnuplot's numerical elliptic integral routines, and (2) the Ramanujan approximation formula given in the article showed that the Ramanujan formula was always a better choice. I tested with a and b close to 1, close to 100, and various relative magnitudes of |a/b|.
It is possible that my little tests were somehow biased. I am not in a position to become more familiar with the various approximation schemes and cannot suggest a better formula. Frankly, the Ramanujan formula was a very good approximation. — Preceding unsigned comment added by 69.172.168.8 ( talk) 04:04, 31 July 2014 (UTC)
This is going nowhere and this is not the correct place for this. The correct approach has been explained repeatedly, far more times already than is necessary. |
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The following discussion has been closed. Please do not modify it. |
anyone who knows how to calculate area knows how to change the shape of a two dimensional object while keeping the same area I have proposed several times that these laws of geometry as I call them apply to calculating the circumference of an ellipse that by adding the x and y radii you will end up with the diameter of a circle with equal area as the ellipse as well as an equal perimeter who ever is calculating the circumference with these awful equations needs to realize there are 360 degree's in an ellipse no mater how you tweak it as long as the right and left half are mirror images this equation works so please check your math and make sure it works on a circle as well as an ellipse because your equation should cover a broad area and not focus on something we all learned in 3rd grade, that there is 360 degrees in a circle regardless of the dimensions. 68.185.82.178 ( talk) 02:24, 6 November 2014 (UTC) I calculated the area of a Ellipse (x-axis/2)(y-axis/2) 3.14= here is my equation (6/2)(3/2)3.14=14.14cm then I calculated the size of a circle with equal area using this equation √(6/2*3/2)^2 the problem I found is that the circumference of the circle with a diameter of 4.5cm is equal to the area of the ellipse but the area of the circle is not equal to the area of the ellipse. This is inconclusive because the areas should have been equal but instead the circumference was equal to the area. I need a further explanation because I'm sure my math was right so I think one of these equations are wrong (3.14 x r^2 = area for a circle) (x-axis/2)(y-axis/2) 3.14 = area of ellipse/oval 68.185.82.178 ( talk) 02:33, 24 November 2014 (UTC)
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I removed the image of Saturn as it is only an interesting tid bit, if you can even call it that. It is completely irrelevant to the actual article. I also removed the image of the conic section. I do not know what type of shape a conic section is classified as but i know it is not an ellipse. An ellipse is obtained by taking a cross section of a cylinder. (Angled cross section if you want a non-circular ellipse) — Preceding unsigned comment added by 173.166.173.129 ( talk) 16:39, 3 March 2015 (UTC)
Once again removed the image that attempts to show am ellipse as a conic section. — Preceding unsigned comment added by Dubsed ( talk • contribs) 05:18, 6 March 2015 (UTC)
My apologies. I stand corrected. — Preceding unsigned comment added by 173.166.173.129 ( talk) 12:54, 9 March 2015 (UTC)
Although I am a moderately good mathematician (MSEE) I may be missing something, but short of that it seems to me that the mixed case for the variables in this article is unconventional and misleading. Unless I hear otherwise, at some point I am going to clean it up. For one thing is looks sloppy and makes the article harder to read, but more than that novices to the subject should not have to wonder if there is some special implied meaning for versus . LaurentianShield ( talk) 15:32, 30 August 2015 (UTC)
In the "equations" part appears: "Then plotting x and y values for all angles of θ between 0 and 2π results in an ellipse (e.g. at θ = 0, x = a, y = 0 and at θ = π/2, y = b, x = 0)." It should be proven, it also speaks about a certain angle which should be defined — Preceding unsigned comment added by 88.13.77.209 ( talk) 14:37, 20 July 2015 (UTC)
The following subsection was put in on 4 May 2009 by an editor with no other math edits:
While this seems to be right at theta = 0 and theta = pi/2, I don't think it is correct in general. I believe the correct formula is
Am I just misinterpreting something here? I'm going to change this unless someone sets me straight. Loraof ( talk) 00:44, 20 November 2015 (UTC)
Baker, Chris (27 Nov 2015). "Elliptical Pool". Wired. Retrieved 29 Nov 2015. --User:Ceyockey ( talk to me) 16:40, 29 November 2015 (UTC)
Surely (except perhaps in the case of Jupiter) it is spurious accuracy to refer to the barycentre of the planet and the Sun (rather than just the Sun) while ignoring the masses of the other planets?---- Ehrenkater ( talk) 14:53, 11 December 2015 (UTC)
The article says: 'For an ellipse the eccentricity is between 0 and 1 (0 < e < 1)'. Shouldn't this be '(0 ≤ e <1)'? A circle is still an ellipse right? Bgst ( talk) 11:25, 11 June 2016 (UTC)
I agree with all previous editors that that mention of Adlaj and a link in-line is not appropriate. The formula is interesting and probably belongs in the article (although I have not checked the citation). However, the wording as of the last edit I reverted to is better. "Surprisingly" is a definite "says who" flag for one thing. Let's try to reach consensus here and not repeatedly edit-revert-edit. LaurentianShield ( talk) 19:02, 16 July 2016 (UTC)
I agree with LaurentianShield and Anita5192 and MrOllie that the result can be mentioned and reference, but that puffed-up style of the IP 2A00:'s edit is in appropriate. I think the consensus is to go back to before his edits, or a version without them, not put more about this until we have a consensus way to do so. We should ask for the protection to be lifted if he agrees to talk and hold off edits, or ask for him to be blocked and then lift the protection if he does not. Dicklyon ( talk) 22:10, 16 July 2016 (UTC)
What about changing the words "A surprisingly simple and exact formula" to "A much more efficient formula"?
Jrheller1 (
talk)
01:51, 17 July 2016 (UTC)
I think it's safe to assume that NAMS would have verified that the formula is actually correct, although there may be a very small chance they did not. As far as the efficiency goes, it is not very difficult to check this. For a=2, b=1, computing in double precision, the Ivory-Bessel series requires 23 terms to reach machine epsilon of approximately 1e-16 (the 22nd term is above machine epsilon, the 23rd term is below machine epsilon). Both MAGM and AGM only require 4 iterations to get below machine epsilon difference between the upper and lower bound, for a total of 8 square root operations. I have compared the speed of trig operations and square roots on my computer: a cosine operation takes approximately 6 times longer than a square root. Computing a cosine to machine precision requires summing only 10 terms of the Taylor series (up to the term with coefficient 1/20!, which is less than machine epsilon). So this indicates that evaluating the Ivory-Bessel series would take approximately the same time as computing 14 square roots.
So in double precision, the iterative algorithm would be slightly more efficient than Ivory-Bessel (8 square roots + some multiplications and additions versus 14 square roots). But for quad precision (which some applications need), the iterative algorithm would be much more efficient. It would only take one more iteration to get to a precision of 1e-32, whereas the Ivory-Bessel series would take 54 terms to get to this same precision.
So maybe "A formula that is more efficient for high-precision calculations" would be better wording. Jrheller1 ( talk) 06:17, 17 July 2016 (UTC)
I have protected this article (in the wrong version) for three days, because of the recent edit warring. Please take advantage of the enforced hiatus on editing to discuss the issue and formulate a concensus rather than merely going back and forth between the two disputed versions. If a consensus does form, please notify me so I can remove the protection earlier. — David Eppstein ( talk) 20:24, 16 July 2016 (UTC)
The article states: "In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve." By that definition every ellipse can be defined by the perimeter of an intersection between a cylinder and a plane, with the semi-minor axis of the ellipse being equal to the cylinder radius and the semi-major axis of the ellipse being equal to the cosine of the angle between the intersecting plane and the circular cross-section (perpendicular to the tube's length).
The article also states: "Ellipses are the closed type of conic section: a plane curve resulting from the intersection of a cone by a plane (see figure to the right)." But the curvature in any plane near the top of a cone is greater than the curvature at the bottom of a cone. Wouldn't the intersection would be egg shaped?
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I've been saying this all along the ellipse is just a elongated circle and everyone knows how to find the area and circumference of a circle you have the area equation right now you just need to fix the circumference equation the circumference of an ellipse is a+b/2*3.14 were a and b are the semi-major and semi-minor axes adding them and dividing by 2 gives the average diameter 2001:558:6012:1B:5CF8:2FED:C347:768E ( talk) 17:56, 26 February 2018 (UTC)
Let a = 1, and b = cos(β), where β is the angle of tilt, of the minor axis from the z-axis, φ is an angle of rotation of the major axis around the z-axis, angle α =(nθ-φ) is rotation around the z-axis, in a counter clockwise direction.
r(θ) = 2cos(β)/√(3+cos(2β)(1+cos(α))-cos(α)).
The view of a unit circle becomes more elliptical as it is tilted away from the z-axis, or line of sight. Cuberoottheo ( talk) 19:23, 29 April 2018 (UTC)
Its the articles polar equation, using angles α and β instead of lengths a and b, the number n = 2 for an ellipse, if n = 1 the ellipse is egg shaped, if n ≥ 2 the ellipse has 2 or more major axis . Cuberoottheo ( talk) 10:30, 30 April 2018 (UTC)
The unit circle's circumference is 2π radian, tilted its 3√(3+cos(n*β)) radian, viewed area is also reduced to 0.5π(a^2)√(3+cos(n*β))*cos(β). Cuberoottheo ( talk) 21:20, 3 May 2018 (UTC)
@ Cuberoottheo: This discussion isn't pertinent to the ellipse but, apparently, to some generalization of an ellipse. In addition, your thoughts are evidently still evolving (you've made a dozen or so edits over the past month, with next to no engagement from other editors). I recommend that you move your work to your user sandbox until you're able to articulate some concrete proposals for the article. cffk ( talk) 13:39, 10 June 2018 (UTC)
This section is either not clear (to me), or incorrect.
The proof included in this section says, "Line w is the bisector of the angle between the lines ..."
The very next sentence says, "In order to prove that w is the tangent line ..."
So is w the bisector or the tangent?
The accompanying graphic on the right indicates w is the tangent, in which case the text should be changed.
The accompanying graphic on the right also includes the statement, "... the tangent bisects the angle between the lines to the foci."
I think it means to say, "... the normal to the tangent at P ...".
David Binner ( talk) 19:48, 10 July 2017 (UTC)
In engineering drawing class I was taught to draw an ellipse by drawing circles centred on the centre of the ellipse with radii same as the semi axes. One then draws sum diameters and where the diameter intersects the smaller circle draw a line through this parallel to the major axis and similarly the larger one and the minor axis. These meet at a point on the ellipse. This is in textbooks on geometrical and mechanical drawing and I wondered why it does not feature in this article. Billlion ( talk) 12:34, 19 March 2019 (UTC)
I was looking for some guidance in finding the equations for an axis-aligned bounding box of a general (rotated) ellipse. This clearly involves tangents to extreme points in the plane. That is, the uppermost, lowermost, rightmost and leftmost points. Is there a simple set of relations involving the coefficients of the general ellipse equation? 2600:1700:9C91:2620:B0E9:DDEE:CBD0:F60D ( talk) 22:56, 29 May 2019 (UTC)
The formula for the canonical form coefficients a,b is not correct. I tested it with A=4, B=0, C=1, D=0, E=0, and F=-4, which corresponds to x2/12 + y2/22 = 1, meaning a=1, b=2, but the formula as written gives a=1, b=1/2. Engineer editor ( talk) 18:03, 29 May 2019 (UTC)
Kleuske That page is entirely clear. It contradicts nothing that I have done but you do not seem to understand your own recommendation. And now that your fealth is exposed and after looking up your profile I can tell that you have nothing to do with the editing of the ellipse page. You are in violation of wiki rules trying to conceal someone else, nicknamed Wcherowi who fears to be have his edit warring further exposed. You two seem to be members of some mafia gang parasiting here who will be fleeing elsewhere as soon as they are exposed. What a shame! 83.149.239.125 ( talk) 17:04, 25 September 2018 (UTC)
def carlson(a, b):
"""
Compute the circumference of an ellipse with semi-axes a and b.
This implements Eqs. (2.36) to (2.39) of Carlson (1995). Require
a >= 0 and b >= 0. Relative accuracy is about 0.5^53.
"""
import math
x, y = max(a, b), min(a, b)
digits = 53; tol = math.sqrt(math.pow(0.5, digits))
if digits * y < tol * x: return 4 * x
s = 0; m = 1
while x - y > tol * y:
x, y = 0.5 * (x + y), math.sqrt(x * y)
m *= 2; s += m * math.pow(x - y, 2)
return math.pi * (math.pow(a + b, 2) - s) / (x + y)
def adlaj(a, b):
"""
Compute the circumference of an ellipse with semi-axes a and b.
This implements Eq. (2) of Adlaj (2012). Require a >= 0 and b >= 0.
"""
import math
x, y = max(a, b), min(a, b)
digits = 53; tol = math.sqrt(math.pow(0.5, digits))
if digits * y < tol * x: return 4 * x
X, Y, Z = math.pow(x, 2), math.pow(y, 2), 0
while x - y > tol * y:
x, y = 0.5 * (x + y), math.sqrt(x * y)
t = math.sqrt( (X - Z) * (Y - Z) )
X, Y, Z = 0.5 * (X + Y), Z + t, Z - t
return 2 * math.pi * (X + Y) / (x + y)
cffk ( talk) questions indicate that the single link provided was ignored, so I repost it for its crucial importance https://math.stackexchange.com/questions/391382/modified-arithmetic-geometric-mean. Both questions were anticipated and answered. Of course, there is only one best method for calculating the circumfernce of an ellipse. It is Gauss's method which you should not call Carlson's. There is no convincing "philosophical" justification for "short-changing" Gauss's method, as you are not obliged to call this original method, based on AGM, after anyone other than Gauss. We have NO Carlson's method here to be discussed. There is, however, Adlaj's formula for the circumference of an ellipse as ratio of two "means". Being a ratio, it must be calculated via Gauss's method as told in Adlaj's paper and further explained in the link which you must see. Calculating the ratio does not restrict you to calculating the numerator and the denominator separately. That calculation completely coincides your first implementation which you keep on wrongfully calling Carlson's method. The additional concept of the modified arithmetic-geometric mean is however important for its own sake as it can be used in other physical applications such as calculating the length of a tether in a linear parallel field of repelling forces as told in another freely available Adlaj's paper http://www.ccas.ru/depart/mechanics/TUMUS/Adlaj/ImaginaryTension16.pdf. Your second implementation and the definition of MAGM would then become relevant since the AGM no longer appears in the denominator, as you can tell, if you'd be as keen as to look up that second paper. The calculation then may resemble your second implementation. So, sorry again, Adlaj's contribution is more important than you care to admit whereas Carlson need not necessarily be mentioned here. That answers your second question. Yet, I have no strong feelings concerning this issue as I was not as lazy to dismiss your link to Calson's 1971 paper which I find quite nice but irrelevant to our discussion. So please do look up what I am sending you before you reask and find all your questions already answered. I know you would not be able to read the story on the origin of MAGM since it's not available in English. But just in case I'm wrong I'll give you the link to a Russian book https://www.morebooks.de/store/gb/book/%D0%A0%D0%B0%D0%B2%D0%BD%D0%BE%D0%B2%D0%B5%D1%81%D0%B8%D0%B5-%D0%BD%D0%B8%D1%82%D0%B8-%D0%B2-%D0%BB%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%BE%D0%BC-%D0%BF%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D0%BC-%D0%BF%D0%BE%D0%BB%D0%B5-%D1%81%D0%B8%D0%BB/isbn/978-3-659-53542-0. 83.149.239.125 ( talk) 12:40, 28 September 2018 (UTC)
The circumference of the ellipse may be evaluated in term of the complete elliptic integral of the second kind using Gauss's arithmetic-geometric mean (cite NIST Handbook + Python code).
If this is a repeat submission, I apologize. For some reason it didn't get added yesterday.
The radius of curvature (as opposed to curvature k) is generally defined as the radius of a circle that fits tangent to a graph at a vertex.
In the Ellipse article (and in the hyperbola article), the semi lattice rectum is defined as a length that is the radius of curvature. (see Section 2:2:4) This has to be a mistake of some kind ... for the images and hyperlinks in the ellipse article show a correct definition of radius of curvature and osculating circles by hyperlink; However, in the common image showing the semi-lattice rectum for conic sections having a common vertex and semi-lattice rectum (called 'p' or cursive 'l'); clearly the semi-lattice rectum of the ellipse is LARGER than the radius of curvature for the circle in the same image. Clearly, the focus of the ellipse can NOT coincide with the origin of the circle ... so the semi-lattice rectum and the radius of curvature for an ellipse can not be the same size.
But that's what section 2.2.4 implies by it's wording about "radius" of curvature.
How do we go about deciding which to correct, the picture or the text? Andrew3Robinson ( talk) 21:00, 23 April 2020 (UTC)
{{
Citation needed}}
template. If no one cites this statement in a reasonable amount of time, the statement can be removed. I also cited Protter & Morrey for the statement thatThe disputed statement is correct. Please see any text book on this subject. (I have only access to German books). I deleted "osculating circle". "radius of curvature" is enough.-- Ag2gaeh ( talk) 08:25, 24 April 2020 (UTC)
Let’s say we want to represent an ellipse in the three-dimensional space. If it’s centered at the origin and in the (x, y) plane, then its equation is obviously (x²/a²)+(y²/b²)+z=1, where z would be zero if it’s on the (x, y) plane and any real number if it’s parallel to the (x, y) plane.
Now, let’s rotate and move our ellipse on the (x, y) plane or parallel to it, by (X, Y) and angle θ. (This is equivalent to a rotation around the z axis.) We thus have:
{[(x-x₀)cosθ+(y-y₀)sinθ]/a²}+{[(x-x₀)sinθ+(y-y₀)cosθ]/b²}+z=1
which is basically what we have on the main page.
But what if our ellipse is also rotated around the x axis or around the y axis? Or even around both of these axes. Let’s call these angles φ (rotation around the x axis) and ξ (rotation around the y axis).
What would we get? My math skills are not advanced enough to get there, so I would appreciate any help possible.
Thanks in advance!
CielProfond ( talk) 21:23, 16 August 2020 (UTC)
The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.
check the math it does not add up so I changed it the original was Area = I changed it to Area = because when stretched by a factor of did not match the scale of the area by the same factor relevant to with a starting radius (r) of 3 — Preceding unsigned comment added by 2601:203:101:BD0:78F1:5839:7DCC:FC03 ( talk) 18:16, 24 September 2020 (UTC)
You are Half right I already made the mistake so I won't try changing it again however check the math use the radius value 3 the equation is wrong but this one is right a radius of 3 with the stretch factor a=2 b=4 keep in mind there are two radii now that is the original of 3 so it was my mistake I multiplied instead of dividing if Pi does not = 1.2275 if you changed the value of Pi then it works but Pi is a constant it is the relation to how many time the radius of a Circle will wrap around the perimeter I know algebra and I know how people that use algebra to solve simple equations make mistakes just like I did multiplying instead of dividing but this boils down to the value of Pi. 19dreiundachtzig ( talk) 20:07, 24 September 2020 (UTC) — Preceding unsigned comment added by 2601:203:101:BD0:78F1:5839:7DCC:FC03 ( talk) 19:40, 24 September 2020 (UTC)
what is the value of Pi maybe you know algebra but do you know how to do basic math 19dreiundachtzig ( talk) 20:10, 24 September 2020 (UTC)
What year was your textbook written I think stretch factor is someone else's technique and not being used properly this article has bean changed so many times there's always a different equation and a new textbook the underline principal of stretch factor is that an ellipse is an elongated circle nothing more 2601:203:101:BD0:78F1:5839:7DCC:FC03 ( talk) 20:38, 24 September 2020 (UTC)
I'm going to make this simple the source is not cited cite the source or I delete it you have one week to cite the source or I will deleted it on 10/01/20 — Preceding unsigned comment added by 2601:203:101:BD0:78F1:5839:7DCC:FC03 ( talk) 20:50, 24 September 2020 (UTC)
No not just a citation for but the stretch factor in relation to the equation show the value of and if you can't prove that stretch factor has actual values as you would in algebra just like any algebra equation if you cant prove the stretch factor or the values are true then I will just delete that and leave the rest 2601:203:101:BD0:78F1:5839:7DCC:FC03 ( talk) 21:10, 24 September 2020 (UTC)
Thank you for adding a citation I can't access the material from the link I will check my University Library later to see if they have it you said pg.115 I hope this is not just the area equation because what I am challenging is weather or not the area equation coincides with the stretch factor equations without changing the value of Pi as determined by non other than Archimedes the person you cited 2601:203:101:BD0:78F1:5839:7DCC:FC03 ( talk) 21:46, 24 September 2020 (UTC)
okay this is just area it does not mention the stretch factor maybe you think I will just see something that is not there if that is what you were trying to do you seceded what do you think about deleting the stretch factor part and just adding some type of definition about area and Pi maybe it could held set a example or outline to me it seems the author uses Pi without understanding it I could change the value of Pi and this explanation of stretch would work and tie up everything that is being held together so loose. 2601:203:101:BD0:78F1:5839:7DCC:FC03 ( talk) 23:28, 24 September 2020 (UTC)
I'm trying to submit changes through the sandbox here is a preview Where and are the lengths of the semi-major and semi-minor axis, respectively. The area formula is intuitive start with a circle of area and scale it by a factor of to make an ellipse, this scales the area by the same factor of π=π 19dreiundachtzig ( talk) 00:18, 25 September 2020 (UTC)
The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.
This article has allot of gatekeeping. I would love to discuss improvements with out being attacked by gatekeepers 10 years ago when there was no mention in this article about a Ellipse being an Elongated Circle I was attacked by a Gatekeeper when I said an Oval is just an elongated Circle because I said Oval and not Ellipse now the article is littered with the Idea of Elongation. The improvement I would like to see is used for radius Under the Area section you see the equation π in the main article for Circle (r) is used Second improvement I would like to see is the use of brackets in the Area equation π I think it should look like this π thank you for taking the time to consider these improvements please keep an open mind. 19dreiundachtzig ( talk) 02:06, 27 September 2020 (UTC)
"Crank" doesn't help us move forward though. He might be back, complaining more about how he can't say his bit. Challenging him to explain himself seems worth doing, but requires waiting on his response, to see if the confusion behind it can be revealed. Dicklyon ( talk) 04:29, 29 September 2020 (UTC)
The area formula inserted by User Bernard Gottschalk can be seen as a consequence of a theorem of Apollonius on conjugated half diameters of an ellipse (see Bronshtein and Semendyayev, or https://de.wikipedia.org/wiki/Satz_von_Apollonios) or can be calculated straight forward (see section Theorem of Apollonios on conjugate diameters).-- Ag2gaeh ( talk) 15:53, 2 February 2021 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 |
The section General polar form gives without citation a general formula. See this math reference desk discussion of 1 Oct. 2012 about whether the formula could be correct. In particular, it seems to me that the radical should have a plus minus sign before it. Otherwise, when the origin is outside the ellipse and the angle theta is such that the ray cuts through the ellipse twice (entering and exiting), the formula fails to give two solutions. Duoduoduo ( talk) 15:16, 1 October 2012 (UTC)
Several pages on Wikipedia including this page and the page Conic Sections claim that ellipses are one type of conic section. My idea of such a conic section would result in a curve like an ellipse but with one end with a smaller radius, or 'pointier' than the other. This is backed up by an edit in this talk pages archives which suggests that an ellipse being a conic section is fallacious and the true conic section would be an egg-shape. If this is true, could this be clarified? Otherwise, am I (and the commenter in the archives) just wrong? 124.177.190.63 ( talk) 12:57, 19 November 2012 (UTC)
Being Ben BernankGrinch of the Federal Reserve stole yet another Christmas from me, I had time to improve upon the Blankenhorn-Ramanujan ellipse circumference formula. This formula provides exact end-points and with a maximum relative error of several magnitudes better than the Ramanujan formula. Some day, the Wiki editors will recognize the improvement, but I am not holding my breath for it.
C~pi(a+b)*(1+3h/(10+sqrt(4-3h)) +(1.5*h^6-0.5*h^12)/((11pi/(44-14pi))+24100(1-h)))
Where a and b are the half length axes and h = (a-b)^2/(a+b)^2
Referencing http://i39.photobucket.com/albums/e191/toomers/ell1.jpg, replace the h^5.20114 with 1.5*h^6-0.5*h^12, and the 19176 with 24100.
See: http://ellipse-circumference.blogspot.com
BEING THIS IS FACTUAL, as anyone can check out and I can provide a spreadsheet to anyone interested this should be able to be showing on wiki's site for people to have easy access to. When there is no doubt about the veracity, there should not be power mongers preventing publication. Numbertruth ( talk) 00:41, 26 December 2012 (UTC)
It would be nice for the general public to see an equivalent form of the Ramanujan II formula without the Hoelder mean:
C~pi(a+b)[1+((x-1)^2(10(x+1)-sqrt(x^2+14x+1)))/((x+1)(33x^2+62x+33))], where x=b/a
Anyone here should be able to verify this. If the power structure here still doesn't want this shown to the public, well, I made my appeal. Numbertruth ( talk) 08:41, 9 January 2013 (UTC)
I simply didn't understand what exactly was meant here. Is the approximate ellipse which is really a circle is to be drawn with 2 compasses or with one? Perhaps a picture, or more precise details, or a reference would resolve the lack of understanding. — Preceding unsigned comment added by 31.44.140.246 ( talk) 19:29, 19 December 2012 (UTC)
I'm reverting the formula added today for the circumference, because the only citation given is http://ellipse-circumference3.blogspot.com/ , which is not a WP:reliable source according to Wikipedia standards. And that blog itself gives no other reference. Wikipedia requires that the formula be sourced to a refereed publication. Duoduoduo ( talk) 20:53, 2 October 2012 (UTC) Actually MrOllie beat me to it! Duoduoduo ( talk) 20:55, 2 October 2012 (UTC)
A formula of such a great breakthrough does not require a reference other than the formula as standing on its own. Show me one simplistic, one step formula capable of producing 8 significant digits, without the use of the Hoelder mean, and without using pi. Why reference a site that is not used as a blog as a blog? Was there any blogging on it? No! So why was the word "blog" mentioned"? If a person had a site named "mountainspot" would it really trip a person up if it wasn't about mountains? Get past the needless block and inspect the formula to see the tremendous contribution. If a person suddenly came up with an explicit formula, I suppose there will be haters of the contribution for wanting some encyclopedia to first publish it. Amazing. Please ask a mathematician about this - one who is versed in it, such as Michon or Cantrell. This amazing formula needs to be out in the open to be shared throughout the world to simulate investigation of the form used. Note that the error function listed on the site verifies the quality. Numbertruth ( talk) 03:45, 3 October 2012 (UTC)
Well, so it is, but some things just won't make it into journals when the one coming up with the material is not part of the status quo.
Look at these three new forms I derived using the artithmetic-geometric mean:
http://mathforum.org/kb/thread.jspa?threadID=2422080&messageID=7942499#7942499 C = 4*(pi*(e-1)*e*(e+1)*(d/dx)[agm(1,(1-x=e)/(1+x=e))]-pi*(e-1)*agm(1,(1-e)/(1+e)))/(2*(e+1)*agm(1,(1-e)/(1+e))^2)
http://mathforum.org/kb/thread.jspa?threadID=2420827&messageID=7937451#7937451 C=8e^2(1-e^2)*d/dk(K(k=e^2))+2(1-e^2)*pi/(AGM(1,sqrt(1-e^2)))
http://mathforum.org/kb/thread.jspa?threadID=2428401&messageID=8048879#8048879 C = 4*E((1-x)^2/(1+x)^2)= (2pi)*((((d/dx)[AGM(x,x^2)]|x=Q)*(1-x)*x^2)/agm(x,x^2)^2 + (2x^2-x)/agm(x,x^2))
Where Q = (1-e)/(1+e) = (1-sqrt(1-b^2))/(1-sqrt(1-b^2)); e= eccentricity, and b = length of minor axis.
These formulas make it easy for anyone to compute the ellipse circumference since the agm is simplistic and converges rapidly, thus so little effort would be required. The derivatives of the agm function require a good numeric processor, but other than that, these add to the theory on ellipses. As can be noticed, since I am not of the group that wants to reward those with credentials, progress will be slow. I saw this even when I was inventing in the sciences - it simply produced jealousies and then I get shut out. I hope this changes as humans need to look beyond certification to assess quality.
I will wait, even if I am long gone as this is what may happen.
I have noticed errors and some parts needing clarification regarding elliptical integrals on wiki but I'll leave it to the experts to spot. :) You may delete the earlier discussion regarding the pade variant or if you give me permission, I will do it. I don't want to do anything here I am not suppose to do. 184.100.17.31 ( talk) 07:44, 22 December 2012 (UTC) Numbertruth Numbertruth ( talk) 00:27, 11 January 2013 (UTC) A formula I just came up with today is simplistic, having a maximum relative error an entire magnitude lower than Ramanujan's second formula: C~4a[1+(pi/2-1)/(((1-sqrt(2)/2)+(sqrt(2)/2)*(b/a)^(-0.454))^(2pi-2))] Maximum absolute relative error: ~3.8E-5, note: b<>0.
This certainly is noteworthy to be inserted on the main page. Waiting around for this being shown in a popular publication shouldn't be the major point in preventing the world from seeing this. Numbertruth ( talk) 05:46, 18 January 2013 (UTC) Numbertruth ( talk) 19:35, 19 January 2013 (UTC)
I just edited the circumference section adding Bessel's series (which converges much more rapidly than the series in e). I also removed an elaborate series which is unlikely to be of much practical use. For people who need to compute the circumference accurately for very eccentric ellipses, I included the necessary AGM method (following the paper by Carlson). cffk ( talk) 18:25, 10 April 2013 (UTC)
The AGM method is good to have on this site. I went about it in a different fashion in three different forms, referenced above. I still think the direct and quite simplistic formula that has a maximum relative error an entire magnitude lower than Ramanujan's second formula should still get some mention here without first being touted by third-party "experts" since most anyone at the middle school level would be able to verify this and make the appropriate comparison: C~4a[1+(pi/2-1)/(((1-sqrt(2)/2)+(sqrt(2)/2)*(b/a)^(-0.454))^(2pi-2))] Maximum absolute relative error: ~3.8E-5, note: b<>0. Find a formula better than this that has just one number that isn't reduced into a simplified form (even the "0.454" could be altered to "5/11"), without converting a and b into some other number such as the Hoelder mean, and having this low of a maximum absolute relative error. Fellow Mathematicians, I am calling upon you! You may see the formula in normal format at http://ellipsesummary.blogspot.com/ I further contend that Zafary's formula deserves mention here as it's such a neat formula. Numbertruth ( talk) 04:15, 4 June 2013 (UTC)
In comparison with Ramanujan's formula that is shown on the main page versus the formula shown immediately above, the relative errors are (when a=1, b varies, with no loss of generality): b Ramanujan II Blankenhorn 0 4.023E-04 --- 0.00000001 4.023E-04 2.220E-16 0.000001 4.023E-04 1.952E-12 0.00001 4.021E-04 1.477E-10 0.0001 3.998E-04 9.635E-09 0.001 3.784E-04 4.750E-07 0.01 2.390E-04 1.206E-05 0.1 1.156E-05 8.321E-06 1 0 0
And so, one must wonder why there needs to be some sort of POPULAR recognition before allowing a very good formula to make it onto wiki? Please explain. This is so simple that just about any low level mathematician could verify. Numbertruth ( talk) 17:48, 14 July 2013 (UTC) Since the format is not easily seen for the relative erros shown above, a table form of the results are provided at: http://ellipsesummary.blogspot.com/ And so, the question still remains, why must there be a popularity contest to be able to provide such an advancement for all interested parties to be able to easily view? Again, this is something a low level mathematician could verify, in fact having just low level skills with a spreadsheet program and possessing a middle school math ability, one could easily verify what I am saying without requiring a vote of confidence from the math community. Numbertruth ( talk) 17:59, 14 July 2013 (UTC)
Unfortunately, nowadays there is a confusion in notation. Traditionally the linear eccentricity should be denoted with the latin e, and the numerical eccentricity should be denoted with the greek ε, where e=εa. Even on this page there is an obvious confusion in notation. Theodore Yoda ( talk) 15:29, 25 March 2013 (UTC)
I agree with the archived comment requesting labels on the diagramme http://en.wikipedia.org/wiki/File:Parametric_ellipse.gif. —DIV ( 138.194.10.62 ( talk) 06:54, 12 May 2013 (UTC))
Mathworld seems to have a different formula as their #58 compared to Ellipse#Parametric_form_in_canonical_position. I think WP is correct, but am puzzled at the discrepancy. —DIV ( 138.194.10.62 ( talk) 06:55, 12 May 2013 (UTC))
Quoth The name ἔλλειψις was given by Apollonius of Perga in his Conics, emphasizing the connection of the curve with "application of areas".
I don't get this sentence. Is an ellipse called so because it's a defective circle?
Thanks, Maikel ( talk) 09:48, 27 May 2013 (UTC)
Currently the lede says
This implies that parabolas are open but not unbounded, while hyperbolas are unbounded but not open. But it seems to me that both are open and both are unbounded -- if so, I think this needs to be reworded. Duoduoduo ( talk) 21:09, 19 July 2013 (UTC)
Duoduoduo: My proof that you just reverted was probably much more useful that the one that you replaced it by, since it used a "known" result (the area of a circle) and a simple geometric argument that anyone who's mathematically inclined could follow. Jacobians and integration seem like complete overkill. However, I don't care enough to press the case. cffk ( talk) 21:11, 21 July 2013 (UTC)
The "unproven" and "intuitive" result is proved as follows: the area is given by the integral
The second integral is just the area of a circle of radius , i.e., ; thus we have . In my book, this proof is so straight-forward that it doesn't need to be spelled out (and certainly the ancient Greeks found these results without resorting to calculus). cffk ( talk) 00:11, 22 July 2013 (UTC)
I have marked an uncited approximation formula as dubious. The article claimed it was 'better' but some simple comparisons with (1) Gnuplot's numerical elliptic integral routines, and (2) the Ramanujan approximation formula given in the article showed that the Ramanujan formula was always a better choice. I tested with a and b close to 1, close to 100, and various relative magnitudes of |a/b|.
It is possible that my little tests were somehow biased. I am not in a position to become more familiar with the various approximation schemes and cannot suggest a better formula. Frankly, the Ramanujan formula was a very good approximation. — Preceding unsigned comment added by 69.172.168.8 ( talk) 04:04, 31 July 2014 (UTC)
This is going nowhere and this is not the correct place for this. The correct approach has been explained repeatedly, far more times already than is necessary. |
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anyone who knows how to calculate area knows how to change the shape of a two dimensional object while keeping the same area I have proposed several times that these laws of geometry as I call them apply to calculating the circumference of an ellipse that by adding the x and y radii you will end up with the diameter of a circle with equal area as the ellipse as well as an equal perimeter who ever is calculating the circumference with these awful equations needs to realize there are 360 degree's in an ellipse no mater how you tweak it as long as the right and left half are mirror images this equation works so please check your math and make sure it works on a circle as well as an ellipse because your equation should cover a broad area and not focus on something we all learned in 3rd grade, that there is 360 degrees in a circle regardless of the dimensions. 68.185.82.178 ( talk) 02:24, 6 November 2014 (UTC) I calculated the area of a Ellipse (x-axis/2)(y-axis/2) 3.14= here is my equation (6/2)(3/2)3.14=14.14cm then I calculated the size of a circle with equal area using this equation √(6/2*3/2)^2 the problem I found is that the circumference of the circle with a diameter of 4.5cm is equal to the area of the ellipse but the area of the circle is not equal to the area of the ellipse. This is inconclusive because the areas should have been equal but instead the circumference was equal to the area. I need a further explanation because I'm sure my math was right so I think one of these equations are wrong (3.14 x r^2 = area for a circle) (x-axis/2)(y-axis/2) 3.14 = area of ellipse/oval 68.185.82.178 ( talk) 02:33, 24 November 2014 (UTC)
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I removed the image of Saturn as it is only an interesting tid bit, if you can even call it that. It is completely irrelevant to the actual article. I also removed the image of the conic section. I do not know what type of shape a conic section is classified as but i know it is not an ellipse. An ellipse is obtained by taking a cross section of a cylinder. (Angled cross section if you want a non-circular ellipse) — Preceding unsigned comment added by 173.166.173.129 ( talk) 16:39, 3 March 2015 (UTC)
Once again removed the image that attempts to show am ellipse as a conic section. — Preceding unsigned comment added by Dubsed ( talk • contribs) 05:18, 6 March 2015 (UTC)
My apologies. I stand corrected. — Preceding unsigned comment added by 173.166.173.129 ( talk) 12:54, 9 March 2015 (UTC)
Although I am a moderately good mathematician (MSEE) I may be missing something, but short of that it seems to me that the mixed case for the variables in this article is unconventional and misleading. Unless I hear otherwise, at some point I am going to clean it up. For one thing is looks sloppy and makes the article harder to read, but more than that novices to the subject should not have to wonder if there is some special implied meaning for versus . LaurentianShield ( talk) 15:32, 30 August 2015 (UTC)
In the "equations" part appears: "Then plotting x and y values for all angles of θ between 0 and 2π results in an ellipse (e.g. at θ = 0, x = a, y = 0 and at θ = π/2, y = b, x = 0)." It should be proven, it also speaks about a certain angle which should be defined — Preceding unsigned comment added by 88.13.77.209 ( talk) 14:37, 20 July 2015 (UTC)
The following subsection was put in on 4 May 2009 by an editor with no other math edits:
While this seems to be right at theta = 0 and theta = pi/2, I don't think it is correct in general. I believe the correct formula is
Am I just misinterpreting something here? I'm going to change this unless someone sets me straight. Loraof ( talk) 00:44, 20 November 2015 (UTC)
Baker, Chris (27 Nov 2015). "Elliptical Pool". Wired. Retrieved 29 Nov 2015. --User:Ceyockey ( talk to me) 16:40, 29 November 2015 (UTC)
Surely (except perhaps in the case of Jupiter) it is spurious accuracy to refer to the barycentre of the planet and the Sun (rather than just the Sun) while ignoring the masses of the other planets?---- Ehrenkater ( talk) 14:53, 11 December 2015 (UTC)
The article says: 'For an ellipse the eccentricity is between 0 and 1 (0 < e < 1)'. Shouldn't this be '(0 ≤ e <1)'? A circle is still an ellipse right? Bgst ( talk) 11:25, 11 June 2016 (UTC)
I agree with all previous editors that that mention of Adlaj and a link in-line is not appropriate. The formula is interesting and probably belongs in the article (although I have not checked the citation). However, the wording as of the last edit I reverted to is better. "Surprisingly" is a definite "says who" flag for one thing. Let's try to reach consensus here and not repeatedly edit-revert-edit. LaurentianShield ( talk) 19:02, 16 July 2016 (UTC)
I agree with LaurentianShield and Anita5192 and MrOllie that the result can be mentioned and reference, but that puffed-up style of the IP 2A00:'s edit is in appropriate. I think the consensus is to go back to before his edits, or a version without them, not put more about this until we have a consensus way to do so. We should ask for the protection to be lifted if he agrees to talk and hold off edits, or ask for him to be blocked and then lift the protection if he does not. Dicklyon ( talk) 22:10, 16 July 2016 (UTC)
What about changing the words "A surprisingly simple and exact formula" to "A much more efficient formula"?
Jrheller1 (
talk)
01:51, 17 July 2016 (UTC)
I think it's safe to assume that NAMS would have verified that the formula is actually correct, although there may be a very small chance they did not. As far as the efficiency goes, it is not very difficult to check this. For a=2, b=1, computing in double precision, the Ivory-Bessel series requires 23 terms to reach machine epsilon of approximately 1e-16 (the 22nd term is above machine epsilon, the 23rd term is below machine epsilon). Both MAGM and AGM only require 4 iterations to get below machine epsilon difference between the upper and lower bound, for a total of 8 square root operations. I have compared the speed of trig operations and square roots on my computer: a cosine operation takes approximately 6 times longer than a square root. Computing a cosine to machine precision requires summing only 10 terms of the Taylor series (up to the term with coefficient 1/20!, which is less than machine epsilon). So this indicates that evaluating the Ivory-Bessel series would take approximately the same time as computing 14 square roots.
So in double precision, the iterative algorithm would be slightly more efficient than Ivory-Bessel (8 square roots + some multiplications and additions versus 14 square roots). But for quad precision (which some applications need), the iterative algorithm would be much more efficient. It would only take one more iteration to get to a precision of 1e-32, whereas the Ivory-Bessel series would take 54 terms to get to this same precision.
So maybe "A formula that is more efficient for high-precision calculations" would be better wording. Jrheller1 ( talk) 06:17, 17 July 2016 (UTC)
I have protected this article (in the wrong version) for three days, because of the recent edit warring. Please take advantage of the enforced hiatus on editing to discuss the issue and formulate a concensus rather than merely going back and forth between the two disputed versions. If a consensus does form, please notify me so I can remove the protection earlier. — David Eppstein ( talk) 20:24, 16 July 2016 (UTC)
The article states: "In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve." By that definition every ellipse can be defined by the perimeter of an intersection between a cylinder and a plane, with the semi-minor axis of the ellipse being equal to the cylinder radius and the semi-major axis of the ellipse being equal to the cosine of the angle between the intersecting plane and the circular cross-section (perpendicular to the tube's length).
The article also states: "Ellipses are the closed type of conic section: a plane curve resulting from the intersection of a cone by a plane (see figure to the right)." But the curvature in any plane near the top of a cone is greater than the curvature at the bottom of a cone. Wouldn't the intersection would be egg shaped?
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I've been saying this all along the ellipse is just a elongated circle and everyone knows how to find the area and circumference of a circle you have the area equation right now you just need to fix the circumference equation the circumference of an ellipse is a+b/2*3.14 were a and b are the semi-major and semi-minor axes adding them and dividing by 2 gives the average diameter 2001:558:6012:1B:5CF8:2FED:C347:768E ( talk) 17:56, 26 February 2018 (UTC)
Let a = 1, and b = cos(β), where β is the angle of tilt, of the minor axis from the z-axis, φ is an angle of rotation of the major axis around the z-axis, angle α =(nθ-φ) is rotation around the z-axis, in a counter clockwise direction.
r(θ) = 2cos(β)/√(3+cos(2β)(1+cos(α))-cos(α)).
The view of a unit circle becomes more elliptical as it is tilted away from the z-axis, or line of sight. Cuberoottheo ( talk) 19:23, 29 April 2018 (UTC)
Its the articles polar equation, using angles α and β instead of lengths a and b, the number n = 2 for an ellipse, if n = 1 the ellipse is egg shaped, if n ≥ 2 the ellipse has 2 or more major axis . Cuberoottheo ( talk) 10:30, 30 April 2018 (UTC)
The unit circle's circumference is 2π radian, tilted its 3√(3+cos(n*β)) radian, viewed area is also reduced to 0.5π(a^2)√(3+cos(n*β))*cos(β). Cuberoottheo ( talk) 21:20, 3 May 2018 (UTC)
@ Cuberoottheo: This discussion isn't pertinent to the ellipse but, apparently, to some generalization of an ellipse. In addition, your thoughts are evidently still evolving (you've made a dozen or so edits over the past month, with next to no engagement from other editors). I recommend that you move your work to your user sandbox until you're able to articulate some concrete proposals for the article. cffk ( talk) 13:39, 10 June 2018 (UTC)
This section is either not clear (to me), or incorrect.
The proof included in this section says, "Line w is the bisector of the angle between the lines ..."
The very next sentence says, "In order to prove that w is the tangent line ..."
So is w the bisector or the tangent?
The accompanying graphic on the right indicates w is the tangent, in which case the text should be changed.
The accompanying graphic on the right also includes the statement, "... the tangent bisects the angle between the lines to the foci."
I think it means to say, "... the normal to the tangent at P ...".
David Binner ( talk) 19:48, 10 July 2017 (UTC)
In engineering drawing class I was taught to draw an ellipse by drawing circles centred on the centre of the ellipse with radii same as the semi axes. One then draws sum diameters and where the diameter intersects the smaller circle draw a line through this parallel to the major axis and similarly the larger one and the minor axis. These meet at a point on the ellipse. This is in textbooks on geometrical and mechanical drawing and I wondered why it does not feature in this article. Billlion ( talk) 12:34, 19 March 2019 (UTC)
I was looking for some guidance in finding the equations for an axis-aligned bounding box of a general (rotated) ellipse. This clearly involves tangents to extreme points in the plane. That is, the uppermost, lowermost, rightmost and leftmost points. Is there a simple set of relations involving the coefficients of the general ellipse equation? 2600:1700:9C91:2620:B0E9:DDEE:CBD0:F60D ( talk) 22:56, 29 May 2019 (UTC)
The formula for the canonical form coefficients a,b is not correct. I tested it with A=4, B=0, C=1, D=0, E=0, and F=-4, which corresponds to x2/12 + y2/22 = 1, meaning a=1, b=2, but the formula as written gives a=1, b=1/2. Engineer editor ( talk) 18:03, 29 May 2019 (UTC)
Kleuske That page is entirely clear. It contradicts nothing that I have done but you do not seem to understand your own recommendation. And now that your fealth is exposed and after looking up your profile I can tell that you have nothing to do with the editing of the ellipse page. You are in violation of wiki rules trying to conceal someone else, nicknamed Wcherowi who fears to be have his edit warring further exposed. You two seem to be members of some mafia gang parasiting here who will be fleeing elsewhere as soon as they are exposed. What a shame! 83.149.239.125 ( talk) 17:04, 25 September 2018 (UTC)
def carlson(a, b):
"""
Compute the circumference of an ellipse with semi-axes a and b.
This implements Eqs. (2.36) to (2.39) of Carlson (1995). Require
a >= 0 and b >= 0. Relative accuracy is about 0.5^53.
"""
import math
x, y = max(a, b), min(a, b)
digits = 53; tol = math.sqrt(math.pow(0.5, digits))
if digits * y < tol * x: return 4 * x
s = 0; m = 1
while x - y > tol * y:
x, y = 0.5 * (x + y), math.sqrt(x * y)
m *= 2; s += m * math.pow(x - y, 2)
return math.pi * (math.pow(a + b, 2) - s) / (x + y)
def adlaj(a, b):
"""
Compute the circumference of an ellipse with semi-axes a and b.
This implements Eq. (2) of Adlaj (2012). Require a >= 0 and b >= 0.
"""
import math
x, y = max(a, b), min(a, b)
digits = 53; tol = math.sqrt(math.pow(0.5, digits))
if digits * y < tol * x: return 4 * x
X, Y, Z = math.pow(x, 2), math.pow(y, 2), 0
while x - y > tol * y:
x, y = 0.5 * (x + y), math.sqrt(x * y)
t = math.sqrt( (X - Z) * (Y - Z) )
X, Y, Z = 0.5 * (X + Y), Z + t, Z - t
return 2 * math.pi * (X + Y) / (x + y)
cffk ( talk) questions indicate that the single link provided was ignored, so I repost it for its crucial importance https://math.stackexchange.com/questions/391382/modified-arithmetic-geometric-mean. Both questions were anticipated and answered. Of course, there is only one best method for calculating the circumfernce of an ellipse. It is Gauss's method which you should not call Carlson's. There is no convincing "philosophical" justification for "short-changing" Gauss's method, as you are not obliged to call this original method, based on AGM, after anyone other than Gauss. We have NO Carlson's method here to be discussed. There is, however, Adlaj's formula for the circumference of an ellipse as ratio of two "means". Being a ratio, it must be calculated via Gauss's method as told in Adlaj's paper and further explained in the link which you must see. Calculating the ratio does not restrict you to calculating the numerator and the denominator separately. That calculation completely coincides your first implementation which you keep on wrongfully calling Carlson's method. The additional concept of the modified arithmetic-geometric mean is however important for its own sake as it can be used in other physical applications such as calculating the length of a tether in a linear parallel field of repelling forces as told in another freely available Adlaj's paper http://www.ccas.ru/depart/mechanics/TUMUS/Adlaj/ImaginaryTension16.pdf. Your second implementation and the definition of MAGM would then become relevant since the AGM no longer appears in the denominator, as you can tell, if you'd be as keen as to look up that second paper. The calculation then may resemble your second implementation. So, sorry again, Adlaj's contribution is more important than you care to admit whereas Carlson need not necessarily be mentioned here. That answers your second question. Yet, I have no strong feelings concerning this issue as I was not as lazy to dismiss your link to Calson's 1971 paper which I find quite nice but irrelevant to our discussion. So please do look up what I am sending you before you reask and find all your questions already answered. I know you would not be able to read the story on the origin of MAGM since it's not available in English. But just in case I'm wrong I'll give you the link to a Russian book https://www.morebooks.de/store/gb/book/%D0%A0%D0%B0%D0%B2%D0%BD%D0%BE%D0%B2%D0%B5%D1%81%D0%B8%D0%B5-%D0%BD%D0%B8%D1%82%D0%B8-%D0%B2-%D0%BB%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%BE%D0%BC-%D0%BF%D0%B0%D1%80%D0%B0%D0%BB%D0%BB%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D0%BC-%D0%BF%D0%BE%D0%BB%D0%B5-%D1%81%D0%B8%D0%BB/isbn/978-3-659-53542-0. 83.149.239.125 ( talk) 12:40, 28 September 2018 (UTC)
The circumference of the ellipse may be evaluated in term of the complete elliptic integral of the second kind using Gauss's arithmetic-geometric mean (cite NIST Handbook + Python code).
If this is a repeat submission, I apologize. For some reason it didn't get added yesterday.
The radius of curvature (as opposed to curvature k) is generally defined as the radius of a circle that fits tangent to a graph at a vertex.
In the Ellipse article (and in the hyperbola article), the semi lattice rectum is defined as a length that is the radius of curvature. (see Section 2:2:4) This has to be a mistake of some kind ... for the images and hyperlinks in the ellipse article show a correct definition of radius of curvature and osculating circles by hyperlink; However, in the common image showing the semi-lattice rectum for conic sections having a common vertex and semi-lattice rectum (called 'p' or cursive 'l'); clearly the semi-lattice rectum of the ellipse is LARGER than the radius of curvature for the circle in the same image. Clearly, the focus of the ellipse can NOT coincide with the origin of the circle ... so the semi-lattice rectum and the radius of curvature for an ellipse can not be the same size.
But that's what section 2.2.4 implies by it's wording about "radius" of curvature.
How do we go about deciding which to correct, the picture or the text? Andrew3Robinson ( talk) 21:00, 23 April 2020 (UTC)
{{
Citation needed}}
template. If no one cites this statement in a reasonable amount of time, the statement can be removed. I also cited Protter & Morrey for the statement thatThe disputed statement is correct. Please see any text book on this subject. (I have only access to German books). I deleted "osculating circle". "radius of curvature" is enough.-- Ag2gaeh ( talk) 08:25, 24 April 2020 (UTC)
Let’s say we want to represent an ellipse in the three-dimensional space. If it’s centered at the origin and in the (x, y) plane, then its equation is obviously (x²/a²)+(y²/b²)+z=1, where z would be zero if it’s on the (x, y) plane and any real number if it’s parallel to the (x, y) plane.
Now, let’s rotate and move our ellipse on the (x, y) plane or parallel to it, by (X, Y) and angle θ. (This is equivalent to a rotation around the z axis.) We thus have:
{[(x-x₀)cosθ+(y-y₀)sinθ]/a²}+{[(x-x₀)sinθ+(y-y₀)cosθ]/b²}+z=1
which is basically what we have on the main page.
But what if our ellipse is also rotated around the x axis or around the y axis? Or even around both of these axes. Let’s call these angles φ (rotation around the x axis) and ξ (rotation around the y axis).
What would we get? My math skills are not advanced enough to get there, so I would appreciate any help possible.
Thanks in advance!
CielProfond ( talk) 21:23, 16 August 2020 (UTC)
The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.
check the math it does not add up so I changed it the original was Area = I changed it to Area = because when stretched by a factor of did not match the scale of the area by the same factor relevant to with a starting radius (r) of 3 — Preceding unsigned comment added by 2601:203:101:BD0:78F1:5839:7DCC:FC03 ( talk) 18:16, 24 September 2020 (UTC)
You are Half right I already made the mistake so I won't try changing it again however check the math use the radius value 3 the equation is wrong but this one is right a radius of 3 with the stretch factor a=2 b=4 keep in mind there are two radii now that is the original of 3 so it was my mistake I multiplied instead of dividing if Pi does not = 1.2275 if you changed the value of Pi then it works but Pi is a constant it is the relation to how many time the radius of a Circle will wrap around the perimeter I know algebra and I know how people that use algebra to solve simple equations make mistakes just like I did multiplying instead of dividing but this boils down to the value of Pi. 19dreiundachtzig ( talk) 20:07, 24 September 2020 (UTC) — Preceding unsigned comment added by 2601:203:101:BD0:78F1:5839:7DCC:FC03 ( talk) 19:40, 24 September 2020 (UTC)
what is the value of Pi maybe you know algebra but do you know how to do basic math 19dreiundachtzig ( talk) 20:10, 24 September 2020 (UTC)
What year was your textbook written I think stretch factor is someone else's technique and not being used properly this article has bean changed so many times there's always a different equation and a new textbook the underline principal of stretch factor is that an ellipse is an elongated circle nothing more 2601:203:101:BD0:78F1:5839:7DCC:FC03 ( talk) 20:38, 24 September 2020 (UTC)
I'm going to make this simple the source is not cited cite the source or I delete it you have one week to cite the source or I will deleted it on 10/01/20 — Preceding unsigned comment added by 2601:203:101:BD0:78F1:5839:7DCC:FC03 ( talk) 20:50, 24 September 2020 (UTC)
No not just a citation for but the stretch factor in relation to the equation show the value of and if you can't prove that stretch factor has actual values as you would in algebra just like any algebra equation if you cant prove the stretch factor or the values are true then I will just delete that and leave the rest 2601:203:101:BD0:78F1:5839:7DCC:FC03 ( talk) 21:10, 24 September 2020 (UTC)
Thank you for adding a citation I can't access the material from the link I will check my University Library later to see if they have it you said pg.115 I hope this is not just the area equation because what I am challenging is weather or not the area equation coincides with the stretch factor equations without changing the value of Pi as determined by non other than Archimedes the person you cited 2601:203:101:BD0:78F1:5839:7DCC:FC03 ( talk) 21:46, 24 September 2020 (UTC)
okay this is just area it does not mention the stretch factor maybe you think I will just see something that is not there if that is what you were trying to do you seceded what do you think about deleting the stretch factor part and just adding some type of definition about area and Pi maybe it could held set a example or outline to me it seems the author uses Pi without understanding it I could change the value of Pi and this explanation of stretch would work and tie up everything that is being held together so loose. 2601:203:101:BD0:78F1:5839:7DCC:FC03 ( talk) 23:28, 24 September 2020 (UTC)
I'm trying to submit changes through the sandbox here is a preview Where and are the lengths of the semi-major and semi-minor axis, respectively. The area formula is intuitive start with a circle of area and scale it by a factor of to make an ellipse, this scales the area by the same factor of π=π 19dreiundachtzig ( talk) 00:18, 25 September 2020 (UTC)
The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.
This article has allot of gatekeeping. I would love to discuss improvements with out being attacked by gatekeepers 10 years ago when there was no mention in this article about a Ellipse being an Elongated Circle I was attacked by a Gatekeeper when I said an Oval is just an elongated Circle because I said Oval and not Ellipse now the article is littered with the Idea of Elongation. The improvement I would like to see is used for radius Under the Area section you see the equation π in the main article for Circle (r) is used Second improvement I would like to see is the use of brackets in the Area equation π I think it should look like this π thank you for taking the time to consider these improvements please keep an open mind. 19dreiundachtzig ( talk) 02:06, 27 September 2020 (UTC)
"Crank" doesn't help us move forward though. He might be back, complaining more about how he can't say his bit. Challenging him to explain himself seems worth doing, but requires waiting on his response, to see if the confusion behind it can be revealed. Dicklyon ( talk) 04:29, 29 September 2020 (UTC)
The area formula inserted by User Bernard Gottschalk can be seen as a consequence of a theorem of Apollonius on conjugated half diameters of an ellipse (see Bronshtein and Semendyayev, or https://de.wikipedia.org/wiki/Satz_von_Apollonios) or can be calculated straight forward (see section Theorem of Apollonios on conjugate diameters).-- Ag2gaeh ( talk) 15:53, 2 February 2021 (UTC)