Talk:Polar coordinate system

Conversions

The two Cartesian coordinates x and y can be converted to the two polar coordinates r and θ by using the trigonometric functions sine and cosine:

$x=r\cos \theta \,$

$y=r\sin \theta \,$

while the two polar coordinates r and θ can be converted to the Cartesian coordinates x and y by

$r={\sqrt {y^{2}+x^{2}}}\quad$ (as in the Pythagorean theorem), and

$\theta ={\begin{cases}\arctan({\frac {y}{x}})&{\mbox{if }}x>0\\\arctan({\frac {y}{x}})+\pi &{\mbox{if }}x<0{\mbox{ and }}y\geq 0\\\arctan({\frac {y}{x}})-\pi &{\mbox{if }}x<0{\mbox{ and }}y<0\\{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y>0\\-{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y<0\\0&{\mbox{if }}x=0{\mbox{ and }}y=0\end{cases}}$

Isn't this the wrong? Shouldn't it be:

The two Cartesian coordinates x and y can be converted to the two polar coordinates r and θ by using the Pythagorean theorem and the trigonometric function cosine:

$r={\sqrt {y^{2}+x^{2}}}$

$\theta =\arccos({\frac {2x^{2}}{2x{\sqrt {y^{2}+x^{2}}}}})$

while the two polar coordinates r and θ can be converted to the Cartesian coordinates x and y by

$x=r\cos \theta \,$

$y=r\sin \theta \,$

CJ ^{Drop me a line! • Contribs} 15:16, 12 May 2011 (UTC)

The arccos doesn't work, you can see that easily because there are two different angles which give the same value of cos separated by π. What gave you the idea that would work, it's not in some book is it? Dmcq (talk) 15:28, 12 May 2011 (UTC)

Looking over it, you're right; it should be:

The two Cartesian coordinates x and y can be converted to the two polar coordinates r and θ by using the Pythagorean theorem and the trigonometric function cosine:

$r={\sqrt {y^{2}+x^{2}}}$

$\theta ={\begin{cases}\arctan({\frac {y}{x}})&{\mbox{if }}x>0\\\arctan({\frac {y}{x}})+\pi &{\mbox{if }}x<0{\mbox{ and }}y\geq 0\\\arctan({\frac {y}{x}})-\pi &{\mbox{if }}x<0{\mbox{ and }}y<0\\{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y>0\\-{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y<0\\0&{\mbox{if }}x=0{\mbox{ and }}y=0\end{cases}}$

while the two polar coordinates r and θ can be converted to the Cartesian coordinates x and y by

$x=r\cos \theta \,$

$y=r\sin \theta \,$

basically the inverse of what the article says now. CJ ^{Drop me a line! • Contribs} 20:46, 12 May 2011 (UTC)

You're quite right, the descriptions before the two transformations look like they've been swapped round, I'll go and fix them. Dmcq (talk) 22:52, 12 May 2011 (UTC)

Thanks for that, I find problems like that quite hard to see, I read what I expect to read ;-) Dmcq (talk) 23:00, 12 May 2011 (UTC)

This is wrong. Talk to any mathematician, and (s)he will tell you that y=rsin(theta), x=rcos(theta), and that x^2+y^2=r^2 In addition, r=(y/sin(theta));r=(x/cos(theta));r=(sqrt(x^2+y^2)) and theta=arctan(y/x); theta=arctan(y/x+pi);theta=arctan(y/x-pi) (In addition, theta=(pi/2) if and only if y=0, and theta=0 if and only if x=0) It's very basic Calculus user:998walrus 1 March 2012 22:15

I'm not certain what you mean here. What does 'this' refer to. Plus the pi should not be within the bracket in your equations and theta is not the same as theta+pi plus you've hgot the x and y round he wrong way in the last bit. Dmcq (talk) 09:16, 2 March 2012 (UTC)

Misuse of sources

This article has been edited by a user who is known to have misused sources to unduly promote certain views (see WP:Jagged 85 cleanup). Examination of the sources used by this editor often reveals that the sources have been selectively interpreted or blatantly misrepresented, going beyond any reasonable interpretation of the authors' intent.

Please help by viewing the entry for this article shown at the page, and check the edits to ensure that any claims are valid, and that any references do in fact verify what is claimed.

I searched the page history, and found 3 edits by Jagged 85 (for example, see this edits). Tobby72 (talk) 21:00, 16 January 2012 (UTC)

I went ahead and removed the paragraph on Biruni which is the material in question. The first statement, citing MacTutor, uses a WP:WEASEL phrase ("are seen as"), and MacTutor doesn't give any additional information on this. I didn't verify the second statement on azimuthal equidistant projection since the source isn't freely available online, but it seems off-topic for this article anyway so the issue is moot. Perhaps someone with a Questia.com account can research this further but I think the statement, if verifiable, would belong the article on the projection rather than here.--RDBury (talk) 16:10, 17 January 2012 (UTC)

from theta to phi

Would someone kind enough to edit all the theta to phe which supposed to be the ISO standard for Azimuth Angle? Even though a lot of vector calculus & general physics textbooks use the former one. (Theta) It would make most students life less miserable. — Preceding unsigned comment added by 218.238.173.141 (talk) 14:26, 25 March 2012 (UTC)

Azimuth is normally measured from up in degrees. Angles here are measured from the x direction in radians normally. How have you come across azimuth and polar at the same time? Dmcq (talk) 14:46, 25 March 2012 (UTC)

Well I didn't know that the Azimuth Angle wasn't allowed in xy-plane 2D only space. Azimuth angle was used there on celestial coordinate system in Azimuth wiki article which was linked from Spherical coord wiki article; that's where I read it. It's just that some commonly used cylindrical system(phe for angle projected on xy-plane between xz ref plane through origin and another vertical plane through origin) & spherical system(phe for angle on xy-plane as well) are analogous to the polar coordinate system. Not very conform at all to use different variables for the same rotation. My old Calculus book by the way used theta for Azimuth angle & phe for angle from z axis with origin as pivot. One more thing I don't get is that on Vector Calculus part of the article it used cross product with one of the vector k normal to the projection plane! That involves z axis. Shouldn't that justify phe instead of theta? Also the term azimuth was used in the last sentense of the Introduction.— Preceding unsigned comment added by 218.238.173.141 (talk) 15:46, 25 March 2012 (UTC)

I don't have access to the citation in this article showing azimuth used for an angle in polar coordinates to see exactly what it is saying. As well as the azimuth I referred to, sorry I should have said north not up, which is measured from north clockwise in degrees, you do also get azimuth in spherical coordinates in mathematics measured in radians as in polar coordinates anticlockwise from the x line. I don't know where on earth you got the idea of some ISO standard applying in mathematics, mathematics don't go in for no stinking ISO standards ;-) but I can see navigators wanting standards for celestial naviggation and if that is what you mean that will the angle in degrees clockwise from north. I think I better edit the Azimuth article to make it clear that the 'polar coordinate' version is quite a different beast altogether to what is meant in the rest of the article. I really hope people reading that article know they need to use an appropriate version of sin or a conversion factor in the formulae there as sin more often uses radians in software. Dmcq (talk) 16:58, 25 March 2012 (UTC)

Actually No it(azimuth angle article) didn't say related to Polar coordinate it did relate itself to the spherical system though. I had no idea as well about ISO standard to some math coordinate system. But after looking at the spherical system wiki article under subtitle "Conventions" there is another article link of ISO 33-11 which is pretty convincing that there is made standard. Maybe these are too commonly used. And yes, haven't seen any programming language using degree in trig functions. Maybe maple if you can call that a programming language. By the way, I can keep the variables as before, no big deal. I was just checking r double primes(t) calculation in each coordinate system especially spherical coord and wiki's spherical coordinate & cylindrical coordinate had switched theta & phe. And reason being this way was you know why(Standardization of dummy variables. I don't get it.

No I don't think that you should try sticking in anything like that here. In fact I think that the spherical coordinate system article is a mess having had a look at it. People have stuck two different systems together. It should not have been done. There is a celestial and geographical coordinate systems and there is the maths spherical coordinate system. Just because they are spherical coordinate systems does not mean they should be covered in that article, there are geographical coordinate system and celestial coordinate system and a sectiopn deveoted to the differences from those other systems. Dmcq (talk) 20:03, 25 March 2012 (UTC)

Yes, I agree with a lot of what yall said. Using phi would certainly be more intuitive. After all, polar coordinates are a special case of cylindrical coordinates (for when z=0) and suddenly switching from "phi" to "theta" for the azimuthal angle is confusing. Also, it is true that "phi" is the ISO 31-11 standard, so shouldn't we be following it in this article? Let me know if you guys agree, and I'd be happy to change it! Monsterman222 (talk) 21:23, 12 October 2012 (UTC)

So, I haven't heard back from anyone in half a year... Should I go ahead and change "theta" to "phi" to conform with ISO 31-11? (which, by the way, the article on the spherical coordinate system also uses) Monsterman222 (talk) 06:51, 25 March 2013 (UTC)

I've converted the article to phi. Could anyone help by updated the images? Monsterman222 (talk) 09:22, 23 April 2013 (UTC)

I know I'm coming a bit late to the party, but I think this change should be reverted. The convention in mathematics is to use theta—compare the Azimuth article on this very topic:

In mathematics the azimuth angle of a point in cylindrical coordinates or spherical coordinates is the anticlockwise angle between the positive x-axis and the projection of the vector onto the xy-plane. The angle is the same as an angle in polar coordinates of the component of the vector in the xy-plane and is normally measured in radians rather than degrees. As well as measuring the angle differently, in mathematical applications theta, $\theta$, is very often used to represent the azimuth rather than the symbol phi $\phi$.

As far as I can tell (i.e. according to the Wikipedia article!), ISO 31-11 doesn't actually refer to 2D coordinate systems at all—and it uses both $\theta$ and $\phi$ in its spherical coordinates. And Spherical coordinates says that both symbols are used interchangeably (depending on whether you're a physicist or a mathematician) for azimuth angle and polar angle. In summary: it's the customary symbol in mathematics; people who prefer phi will need to understand when theta is used anyway; and ISO seems completely neutral on the subject... can we go back to theta? -- Perey (talk) 04:02, 28 November 2014 (UTC)

@Perey, about ISO 31-11, the reason I said that phi is the standard is that the Wikipedia page on it said "ρ and φ are the polar coordinates", although looking at it again, this is in the context of cylindrical coordinates, and it says: "lf z= 0, then ρ and φ are the polar coordinates". So, in fact, it's a bit ambiguous. Unfortunately, the original document ISO 31-11 (as well as the newer version which is ISO 80000-2:2009) is hidden behind a nasty paywall, so unless someone is willing to fork over a ridiculous $155, I guess we won't know... What are other people's thoughts? Monsterman222 (talk) 19:47, 10 January 2015 (UTC)

But one thing you can be certain of no matter what ISO says, is that most users of polar coordinates uses theta, and always have which is why it used to be what has used here.Carewolf (talk) 19:41, 13 May 2015 (UTC)

Well, right now, this bit is confusingly flipping back and forth between symbols ϕ & $\varphi$:

"The polar coordinates r and ϕ can be converted to the Cartesian coordinates x and y by using the trigonometric functions sine and cosine:

$x=r\cos \varphi \,$

$y=r\sin \varphi \,$"

There is no ϕ in the graphic at the right. One assumes the symbol should be the same throughout, rather than using one symbol in the description and another in the equations and/or graphic?? If so, can we get the symbol usage consistent, please? Currently I'm trying to figure something out and this is making it 'Greek to me...' 50.53.112.40 (talk) 06:25, 15 July 2015 (UTC)

Polar equations vs polar coordinates

An interesting technicality is whether there is a "standard" polar coordinate system and, if so, does it specify that the radius r must be non-negative? Clearly one can make sense of an expression like r cos(theta) if r is negative and convert such an (r,theta) to a cartesian coordinate (x,y). However, converting the point (x,y) back to a polar coordinate by the usual method would produce a point where r was positive.

I see nothing that prevents a "polar equation" from having a negative r value. So I wonder if polar equations are required to produce "standard" polar coordinates or whether they merely produce numbers than can be converted to standard polar coordinates.

Perhaps the people that know about the "chart and atlas" treatment of coordinate systems for manifolds can comment on whether there is a standard polar coordinate system.

Tashiro (talk) 19:38, 2 April 2012 (UTC)

It is standard to have r non-negative and θ in the range (−π, π]. You can of course have angles or radius outside of those ranges but that would be a non-standard representation. To produce an atlas the easiest thing is to use a different range of angles, however you have problems with that though for defining a manifold as the origin can have any angle, you' need to do that with some other coordinate system entirely and just leave that out otherwise. Dmcq (talk) 22:55, 2 April 2012 (UTC)

I'm curious whether there is any formal mathematical definition for "standard" polar coordinates. I agree that what one usually does when computing the polar coordinates of a point is compute r as positive and theta as (-pi, pi). But is this the "standard" way only in the sense of a social behavior or is there a formal mathematical definition of a system of polar coordinates that defines this standard?

If we allow the same point to be represented by several different pairs of numbers of the form (r, theta) then technically we should avoid statements that imply there is only one representation. For example, we shouldn't say that we are presenting "the" formulas for finding "the" polar coordinates of a point whose cartesian coordinates are (x,y). Instead we should say we are presenting "some" formulas for finding "a" polar coordinate of a point whose cartesian coordinates are (x,y). ( Of course, I suspect any attempt to reform this habitual language will fail!) — Preceding unsigned comment added by Tashiro (talk • contribs) 07:08, 3 April 2012 (UTC)

I think I mustn't be understanding what you mean by standard. It is not something that is dictated by some higher mathematical rule, it is simply what people do. What is in the article is as standard as one gets without getting an agreed ISO standard and that isn't done in maths. Some people have used an angle from 0 to 2π instead but that's a distinct minority and what's here is what you'll practically always get. Dmcq (talk) 07:40, 3 April 2012 (UTC)

I don't mean something like an ISO standard. I mean some established and precise definition within the field of mathematics, such as the definition of "Abelian group" or "vector space", which are not (as far as I know) ISO standards.

An example of my quibble about language is the passage in the current article that says "while the Cartesian coordinates x and y can be converted to the polar coordinates r and θ by:

   r = \sqrt{y^2 + x^2} ..."

It would be clearer not to say "the polar coordinates". It could say "a polar coordinate" or "one pair of polar coordinates for the point (x,y)" given that the article takes the viewpoint that polar coordinates are not unique.

Tashiro (talk) 21:32, 3 April 2012 (UTC)

The article gives the normal standard polar coordinates as r non-negative and θ in the range (−π, π]. What more do you need than that? That is as standard as one gets. The only other range that is sometimes used but is nowhere near so common is θ in the range [0,2π). There is no more to it. I really don't see what problem you see with that. It is possible to have an equation like r=θ which defines a spiral where θ goes out of that range but there's no difference between that and saying that the square root of 4 is 2 but that −2 is a square root of 4. That's referred to as the principal square root and the value in (−π, π] is the principal value here. Dmcq (talk) 22:59, 3 April 2012 (UTC)

I'm not disputing that that part of the article gives the usual formulae ( my example refers to the formulae, not the definition). My original question is whether there is a mathematical definition that defines the unique pair of numbers produced by those formulas as the "standard" or "normal" polar coordinates. As an analogy, an inverse trig like y = arcsin(x) isn't defined merely by the formula sin(y) = x since this doesn't define a unique value of y. To define arcsin(x) we can specify a definite range for y, but there is (or used to be, when I was in school) an alternative technical terminology about "principal" angles that we can use instead of specifying the range. So I am wondering if there is a similar terminology for the polar coordinates produced by those formulae. Is "standard" or "normal" a technical term in the same sense that "principal angle" is a technical term? (I don't know the answer to this legalistic question. I'm just asking!)

My point about the text surrounding the formulae is merely that the phrase "can be converted to the polar coordinates" could be changed to "can be converted to a pair of polar coordinates" in order to emphasize that the pair of coordinates produced by the formulae is not the only pair of polar coordinate for (x,y).

Tashiro (talk) 16:11, 8 April 2012 (UTC)

I think what you are saying is you'd like the text to say the conversion is to the standard range as specified in the section about a unique conversion. I'll try rephrasing it and see how that goes. Dmcq (talk) 16:39, 8 April 2012 (UTC)

Polar coordinates in $n$ dimensions

I could write something about this topic if someone thinks it could be useful / is not covered anywhere. Note that when i was studying this sort I thing i was unable to find any reference online (except for some research articles). --Agi 90 (talk) 11:48, 1 June 2012 (UTC)

Yes please, the anti-de Sitter space currently contains a red-link hyperpolar coordinates, as do some of the articles on geodesics. This is a standard topic for Riemannian geometry, it should have an article. Probably does, just under some name I can't think of ... Hmm. It would be great if you could make it clear that its for symmetric spaces with O(n) orthogonal group symmetry, and then explore the general case. Hmm. User:Linas (talk) 02:29, 3 November 2013 (UTC)

What is the extension / generalization to a spiral plane?

I am not sure if it is a space or a variation on the polar coordinates. For instance, using the notation (r, theta) for a point, the point (r0, theta) != (ro, theta+2*pi). Hence the plane/space/coordinate system is sort of a spiral. What is this called? — Preceding unsigned comment added by 68.228.41.185 (talk) 08:25, 1 November 2012 (UTC)

Perhaps you are thinking of a covering space or a universal cover ? User:Linas (talk) 02:31, 3 November 2013 (UTC)

FA?

This article is full of short paragraphs and sections and is, for the most part, devoid of inline citations. In addition, the intro is very short and the prose contains some awkwardly constructed sentences. Though I won't nominate it for an FAR right now, I don't believe it meets the 2014-era FA criteria. Tezero (talk) 02:47, 27 July 2014 (UTC)

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"arctan" vs. "Arctan"

Regardless to whom WP:BRD applies, I enjoy repeating the facts, given already in my edit summary.

1. "arctan" is a multivalued map —$\arctan :\;\mathbb {R} \to \mathbb {R}$— and is NOT a function.
2. The usual selection of a principal branch restricts the codomain to principal values from the interval $(-\pi /2,\pi /2).$ Except for the value $\pi /2$ this exhausts the full domain $(-\pi /2,\pi /2],$ admissible for the polar angle in this concept, suggesting the mentioned association.
3. This restriction generates a function, that maps $\mathbb {R} \to (-\pi /2,\pi /2).$ This function has the generally agreed upon denotation of "$\operatorname {Arctan}$", which justifies the use of this name at the place under discussion.

No rational reasoning for using "$\arctan$" instead was given yet. Purgy (talk) 07:41, 17 September 2018 (UTC)

---

Assertion does not a fact make. The distinction you are making isn't apparent in the article, of itself, nor in the arctan article. Nor do I see a more general convention of differentiating maps and functions in mathematics by the first letter of their name. If you want to have this convention on Wikipedia, you're going to have to go ahead and put the work in so that everyone knows what it means. 178.16.3.163 (talk) 18:29, 17 September 2018 (UTC)

I would suggest that the arctan article would be the only place to start. 178.16.3.163 (talk) 18:33, 17 September 2018 (UTC)

---

None has to be given. It is entirely up to you to justify your change on the talk page. That's how BRD works. 178.16.3.163 (talk) 18:24, 17 September 2018 (UTC)

Well this is a little confusing. As I read the history page the original change from "Arctan" to "arctan" was made by you and not Purgy, putting the onus of justification upon you. Be that as it may, I am familiar with the convention of capitalizing the first letter of these inverse trig multivalued expressions to indicate that a function is meant, but unfortunately I am on vacation at the moment and am unable to document that fact. --Bill Cherowitzo (talk) 18:54, 17 September 2018 (UTC)

I am happy to revert his entire change directly, if that makes it easier for the usual narrow application of WP:BRD to take place. 178.16.3.163 (talk) 18:59, 17 September 2018 (UTC)

I tried to (re-)state my opinion on this ridiculous quarrel about captioning one letter in my above OP, and expected some, at least semi-, professional argumentation against it. Instead, the IP rebuffs my statement with mathematical ignorance and bureaucratic, and imho incoherent, application of WP-rules, seriously suggesting to remove all my(!) improvements as a solution.

I am not interested in lowering my bar to this level. Purgy (talk) 07:08, 18 September 2018 (UTC)

Positive abscissa, progressing counterclockwise

It is conventional within mathematics to denominate the angle in radians, and to diagram polar coordinates by placing the zero angle along the positive abscissa, with increasing angles progressing counterclockwise.

Half a math degree thirty years ago, and this is now the best I can do. :-( But at least I know what needs to be said, if not necessarily how to best say it.

Likely this would benefit from some fine tuning. (Note also, I wasn't able to lift optimal text from the radians article on scant inspection, so there's a potential edit there, too.) — MaxEnt 22:31, 13 February 2019 (UTC)

I reverted your addition because there is a full section dealing with the various conventions, and because the "pos. abscissa" comes into existence only when referencing Cartesian coordinate systems, and this conversion is also covered in the article. I think one could discuss if the lead should contain hints to (1)different conventions re radians or degrees, or other units, (2)different orientational conventions, and (3)connections to various other coordinate systems, e.g. also the higher dimensional extensions. All these topics are addressed in the article. Purgy (talk) 08:06, 14 February 2019 (UTC)

FA concerns

Hi editors, as part of WP:URFA/2020 I reviewed this article and I do not believe it meets the featured article criteria anymore. I am concerned that there are multiple uncited sentences and paragraphs throughout the article, and the "General references" section should be used as footnotes throughout the article. Is anyone interested in bringing this back to FA quality? Z1720 (talk) 22:47, 10 April 2021 (UTC)

@Z1720: I would love to help, though I am quite certain my skills are lacking. Having seen the "review" of this article, I foresee a tough climb ahead. Please do call on me if necessary or appropriate. Horsesizedduck (talk) 23:54, 29 June 2021 (UTC)

@Horsesizedduck: If you are interested in working on the article, please comment on the article's FAR. I am not knowledgeable enough to edit this, but there are other editors with expertise who would be willing to help. Z1720 (talk) 23:59, 29 June 2021 (UTC)

Christoffel Symbols

Should Christoffel symbols be added?

$\Gamma _{ij}^{1}={\begin{pmatrix}0&0\\0&-r\end{pmatrix}},\Gamma _{ij}^{2}={\begin{pmatrix}0&{\frac {1}{r}}\\{\frac {1}{r}}&0\end{pmatrix}}$ Angulon (talk) 01:25, 25 January 2023 (UTC)

What about proofs

Proof of x= r×cos×phi Etc. Yuthfghds (talk) 16:07, 27 May 2023 (UTC)

suggested additional curve: folium of Descartes

The folium of Descartes is another nice application of polar coordinates and should perhaps be included in this article. Kontribuanto (talk) 21:18, 22 April 2024 (UTC)

Distance between points

Worth mentioning? By law of cosines, the distance between (r1, θ1) and (r2,θ2) is

$d^{2}=r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos(\theta _{1}-\theta _{2})$

Compare this to Cartesian coordinates

$d^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}$

https://math.stackexchange.com/questions/1506706/how-to-calculate-the-distance-between-two-points-with-polar-coordinates Wqwt (talk) 04:34, 15 June 2024 (UTC)