Talk:Polar coordinate system/Archive 1

Merge polar graph into polar coordinate system

I think that these two articles are essentially the same thing, so merging them is the best thing to do. Polar coordinate system is a more complete article, so polar graph should be merged into it. --Mets501^talk 03:19, 19 April 2006 (UTC)

I performed the merge. It seems like no one should have any objections. --Mets501^talk 18:47, 19 April 2006 (UTC)

Good articles

History, history, its always history! A lack of any discussion on the history of polar coordinates. Who first introduced them, how did the theory develop? Other than that a nice article. --Salix alba (talk) 08:05, 7 September 2006 (UTC)

I added a short history section, though I'm sure it could be expanded. Newyorkbrad 13:52, 8 September 2006 (UTC)

Elliptical coordinates

You can sort of think of Polar coordinates as being a limiting case of Elliptic coordinates. From the definition of

$x=a\ \cosh \mu \ \cos \nu ,y=a\ \sinh \mu \ \sin \nu$

if we let $\mu =\ln(2r/a)\,$ then $a\cosh(\mu )={a \over 2}(e^{\ln(2r/a)}+e^{-\ln(2r/a)})={a \over 2}(2r/a+a/(2r))=r+a^{2}/(4r)$. In the limit as a tends to zero, this becomes simply r. The same holds for $a\sinh(\mu )\to r\,$, and hence the the equations become those of polar coordinates. --Salix alba (talk) 15:44, 9 September 2006 (UTC)

Another GA Reviews

• The lead section is not enough to summarize the article. A context about this article should be given in the lead section, including some advantages and disadvantages using Polar coordinate system. The lead section should explain what the article is about for common readers, that does not have specific knowledge about the article. Guideline is given here: WP:LS.
• In Polar Equations section, the statement A polar line/curve is symmetric about the 0°/180° line if replacing θ by −θ in its equation produces in an equivalent equation, symmetric about the 90°/270° line if replacing θ by π−θ produces an equivalent equation, and symmetric about the pole if replacing r by −r produces an equivalent equation. is unclear for a common reader to understand it.
• I think the article needs to be expanded further and deeper, rather than just listing Polar Equations for some popular geometrical shapes.

— Indon (reply) — 08:36, 10 September 2006 (UTC)

I've dealt with the first two issues (and Newyorkbrad helped copyedit), but I'm not quite sure what to do about the third. —Mets501 (talk) 14:20, 10 September 2006 (UTC)

Either get to the library and do some research, or ask Salix alba to help out. :) Newyorkbrad 14:26, 10 September 2006 (UTC)

A real library, what's that? :-) Salix alba looks like he knows what he's talking about, though, he should be able to help. —Mets501 (talk) 14:32, 10 September 2006 (UTC)

What me know what I'm talking about, just commiting a bit of original research and shouting history!

Anyway I did a bit of digging in google which turned up

1. Gregory St. Vincent (Grégoire de Saint-Vincent), with -

   The Greeks described a spiral using an angle and a radius vector, but it was St. Vincent and Cavalieri who simultaneously and independently introduced them as a separate coordinate system. In an article The Origin of polar coordinates, J. J. Coolidge refers to the priority dispute between Cavalieri and St. Vincent over their discovery. St. Vincent wrote about this new coordinate system in a letter to Grienberger in 1625 and published the process in 1647. On the other hand Cavalieri's publication appeared in 1635 and the corrected version in 1653.

2. and Milestones in the History of Thematic Cartography

   Development of the use of polar coordinates for the representation of functions. Newton's Method of Fluxions was written about 1671, but not published until 1736. Jacob Bernoulli published a derivation of the idea in 1691. [238,p. 324] attributes the development of polar coordinates to Fontana, with no date.- Isaac Newton (1643-1727), England, and Gregorio Fontana (1735-1803) and Jacob Bernoulli (1654-1705) [238,p. 324]. 1671 is probably too early; 1736--1755 would probably be more appropriate. There are earlier references to Hipparchus (190-120BC) regarding the use of polar coordinates in establishing stellar positions, and Abu Arrayhan Muhammad ibn Ahmad al-Biruni (1021) regarding the use of three rectangular coordinates to establish a point in space.

   1843 Use of polar coordinates in a graph(frequency of wind directions)- Léon Lalanne (1811-1892), France [148].

3. La Habra High School Math History Timeline

   Jacob Bernoulli invents polar coordinates, a method of describing the location of points in space using angles and distances.

4. Sherlock Holmes in Babylon and Other Tales of Mathematical History has a section

   Newton as an originator of polar coordinates

5. Jacob Amsler http://www-history.mcs.st-and.ac.uk/~history/Printonly/Amsler.html

   [Amsler] invented the polar planimeter, a device for measuring areas enclosed by plane curves. It was based on polar coordinates whereas earlier instruments were based on cartesian coordinates. In 1856 Amsler published a paper Über das Planimeter in which he gave details of his idea. As Mahoney writes in [1], Amsler's planimeter:- ... adapted easily to the determination of static and inertial moments and to the coefficients of Fourier series: it proved especially useful to shipbuilders and railway engineers.

So it seems there is considerable different accounts of their introduction, and quite a few notable applications. --Salix alba (talk) 16:10, 10 September 2006 (UTC)

Nice research! Do you want to add that to the article (the fact that there are differences in opinion)? —Mets501 (talk) 19:25, 10 September 2006 (UTC)

I think he's leaving it for you to do. :) Although I'm not sure whether it's a matter of differences of opinion, or just that different mathematicians used the concept at different times to do different things. Also regarding expanding the article, you might want to explain in more detail what types of real-world problems are solved more easily using polar coordinates than Cartesian ones. Newyorkbrad 19:33, 10 September 2006 (UTC)

Good Article

After reading the changes, this article qualifies as Good Article status.

Per WP:WIAGA,

1. Well written

I enjoy reading this article, as there's no complicated mathematical statements. Non specialist readers can benefit for this kind of article to learn about the subject. Structure is quite logical and the lead section has been improved to state the context of this article. I like how editors put mathematical equations. Technical jargons are simply and briefly explained.

2. Factually accurate and verifiability

References are enough and reliable. I cannot really check whether mathematical equations given are accurate, but I hope there is no mathematical typos in the equations. Inline citations are given in the History section.

3. Broad in its coverage.

Broad enough and not too much unnecessary details.

4. Neutral point of view policy.

Passes.

5. Stable

Passes.

6. Images

Very good illustrations. Geometrical shapes are displayed properly to give a good illustration of the polar equations.

Congratulation to the editors' hard work for this article. Further improvements can lead this article to FA status. — Indon (reply) — 09:46, 14 September 2006 (UTC)

Thanks for evaluating it, Indon! —Mets501 (talk) 11:05, 14 September 2006 (UTC)

Polar coordinates is indeed going to feature on the main page someday, although Mr. Mets may not yet appreciate how much work he gets to do on it between now and then. I'll help too, of course. Newyorkbrad 21:05, 14 September 2006 (UTC)

Don't worry, I realise this article isn't near featured status yet :-) —Mets501 (talk) 21:16, 14 September 2006 (UTC)

References

Format of authors' names in references should be consistent -- either last name first or first name first. Not sure which one you want but we need to pick one or the other. Newyorkbrad 20:01, 16 September 2006 (UTC)

Last, first seems to be the standard (as per the cite templates) so I've changed it all to that. —Mets501 (talk) 20:54, 16 September 2006 (UTC)

Spherical coordinates

Adding the section on extension to three dimensions is good. Note that cylindrical coordinates have their own article, but spherical coordinates are discussed under coordinate system. I'm not much of a math editor but there should probably be more consistent cross-referencing between all these articles. In a perfect world, either spherical would be a separate article or cylindrical would not, for parallelism, but making either change would be a little bit too BOLD for me. Newyorkbrad 16:58, 17 September 2006 (UTC)

Yes the various coordinate articles are messy, for starters we have Coordinate system and coordinates (mathematics) which have a large overlap, perhaphs they should be merged. I certainly think that spherical coordinates deserves its own article. Theres also less well know systems like Oblate spheroidal coordinates which don't get a mention, Orthogonal coordinates seems to have a more comprehensive list. A bit of boldness and some cutting, pasting, moving and merging would not go amis. --Salix alba (talk) 17:23, 17 September 2006 (UTC)

I'm being bold and creating a spherical coordinate article. The geographic coordinate system article should probably get a mention as well. I didn't know about it when I originally wrote the spherical coordinate section. --Carl ^{(talk|contribs)} 17:42, 17 September 2006 (UTC)

Partial (trapezoidal) section of sphere/ellipsoid surface

What is a trapezoidal shaped section of a sphere or ellipsoid's surface called? I think it is something like "quadratic angle" or "quadrilateral section"—?  ~Kaimbridge~ 19:37, 12 December 2006 (UTC)

A better place for this would be the Mathematics reference desk. —Mets501 (talk) 21:26, 12 December 2006 (UTC)

Yes - answer is provided there.83.100.254.21 17:58, 13 December 2006 (UTC)

Comments

This is in response to the request for comment User:Mets501 posted at WPM.

Some comments after a brief scan:

• There is no discussion of the problems with the polar coordinate system such as the (coordinate) singularity at the origin or the difficulties associated with the multivaluedness of the θ coordinate.
• There should be some mention of the Euclidean metric in these coordinates as well as a discussion of integrating functions in polar coordinates, i.e.

  $\int _{r_{0}}^{r_{1}}\int _{\theta _{0}}^{\theta _{1}}f(r,\theta )\,rd\theta \,dr$

• I view as suspect any article using MathWorld as a reference. Some of the references could be improved.
• I find the cis(θ) notation to be bizarre and nonstandard. Mention Euler's formula and then just use e^iθ like everyone else.
• There is some inconsistency with writing sin θ vs. sin(θ). Not a big deal, just looks weird.

-- Fropuff 00:50, 21 December 2006 (UTC)

Thanks for the comments; I'll place them in a todo section above. It's funny how you say that the cis(theta) notation is "bizarre and nonstandard", it's the only way that I wrote it in school. I will change it however, because if it's awkward to you it most likely is for others as well. —Mets501 (talk) 03:07, 21 December 2006 (UTC)

Note in particular that the "cis" notation receives a serious thrashing in Trigonometric_identities#Pedagogy_and_.22cis.22. I guess it's something of a political issue that we shouldn't get bogged down in, but people might want to look at that article for guidance/ideas on how to handle this. (I saw it in high school, and hated it immediately.) William Ackerman 15:12, 21 December 2006 (UTC)

I don't interpret the text in List of trigonometric identities as "thrashing". I agree with avoiding the use of this cis function here, though.  --Lambiam^Talk 17:11, 21 December 2006 (UTC)

Interesting. Maybe it is standard in some places. Honestly, I've never seen it outside Wikipedia. I do think the article looks much better now with the exponential form. -- Fropuff 03:42, 21 December 2006 (UTC)

We used "cis" notation in school too (late 1970's) so I agree it's been the standard (at least in New York State). Historically, I believe "cis" notation was recommended for publication because it can all sit on one line, as opposed to the alternative. This was important when typesetting was done manually and setting subscripts or exponents could require handwork by the printer with associated costs in time, expense, and risk of error. See any pre-computer-age edition of the Chicago Manual of Style etc. Those considerations, of course, no longer apply today. Newyorkbrad 17:25, 21 December 2006 (UTC)

Never heard of "cis' before. linas 22:43, 21 December 2006 (UTC)

See cis (which is a disambiguation page, look for "in mathematics" near the bottom). Newyorkbrad 23:07, 21 December 2006 (UTC)

Some more GA-article comments: Applications: the applications section should mention the use of polar plots for antenna (radio) radiation patterns (see also side lobe), and for microphone polar paterns (sensitivity plots_. These applications are far more "common" than any of the others mentioned: in particular, they actually have a number of actual polar plots in the articles, whereas the articles on the other applications (robots, compressors, etc) do not. linas 22:43, 21 December 2006 (UTC)

Alternative formula for the angular coordinate

Currently this article only has a formula to calculate the angular coordinate in the interval [0, 2π). However, in many cases, e.g. complex numbers, the angular coordinate usually is calculated in the interval (-π, π]. Therefore I suggest to add a formula for this case, so other articles like "complex number" can reference to this article. As a compensation the section on complex numbers could be merged into the article on complex numbers. --IP 01:35, 2 January 2007 (UTC)

Section on complex numbers

I suggest to remove the section on complex numbers, because all this is treated in the article complex number. Perhaps a note could be made that polar coordinates are used e.g. in complex numbers. Instead, it would be useful to extend the section on calculating the angular coordinate with a formula that provides the angle in the interval (-п, п] as it is used in complex numbers. --IP 12:55, 2 January 2007 (UTC)

I think the section is good. It happens to be one of the major uses of the polar coordinate system in mathematics, so it should belong in the polar coordinate system article. It can also have the same information as complex number with no problems. —Mets501 (talk) 21:49, 2 January 2007 (UTC)

Yea, it would be a glaring ommision to talk about the polar coordinate system without talking about Complex Numbers. It's be like the article on Newton having no mention of the theory of gravity. Brentt 01:35, 17 February 2007 (UTC)

Applications

Following discussion on Wikipedia:Featured article candidates/Polar coordinate system I'm working on a draft of the applications section here. Its very sketchy at the moment, feel free to kick it into shape. --Salix alba (talk) 20:02, 17 January 2007 (UTC)

I'll edit it a bit now. I'm a little confused by:

Measurment

When the distance and angle are measured

Communication

Polar coordinates can be used for communicating positions

Storage

Is that part of a draft or something? —Mets501 (talk) 22:21, 17 January 2007 (UTC)

Yes I'm not quite happy with it yet. Their seem to be a few main uses: measurement, storage, communication, modeling, calculation, probably others I haven't identified yet. There seems to be quite an overlap between the first three and I'm not sure of how best to word it. Apologies for bad spelling and grammar, I was on a computer which didn't have a spellchecker installed. --Salix alba (talk) 22:36, 17 January 2007 (UTC)

No problem; I've cleaned it up quite a bit, and think it's basically ready to go in the article. —Mets501 (talk) 23:09, 17 January 2007 (UTC)

No time to really edit this now. But it should be noted that the robot thing is a bit misleading. The robot will have to do non-trivial trigonometry no matter what: once it starts moving, the new coordinates of an object that does not lie on the ray that the bot moves on will have to be recalculated. Pascal.Tesson 23:17, 17 January 2007 (UTC)

Yes I'm concerned about the robots, referenece is one paper of some guys personal way of doing things, not representatitive of robotics in general. Possibly quite POV and borderline OR.

As for calculation, the example I had in mind was finding the integral of $e^{-x^{2}}$, which was a well remembered A-level exercise. This is not intergratable my normal means, but you can consider the 2D version $e^{-x^{2}-y^{2}}=e^{-x^{2}}e^{-y^{2}}=e^{-r^{2}}$ switching back and forward to polar coords makes the problem tractable, see Gaussian integral for the proof. It seems to me that there are probably quite a few similar formula where using polar coords really help. --Salix alba (talk) 00:09, 18 January 2007 (UTC)

I've redone the robot section and merged this into the article. —Mets501 (talk) 02:48, 19 January 2007 (UTC)

Area swept out

I removed the following fragment from the article:

"Let $\mathbf {A}$ be the area swept out by a line joining the focus to a point on the curve. In the limit $d\mathbf {A}$ is half the area of the parallelogram formed by $\mathbf {r}$ and $d\mathbf {r}$,

$dA={\begin{matrix}{\frac {1}{2}}\end{matrix}}|\mathbf {r} \times d\mathbf {r} |,$

and the total area will be the integral of $d\mathbf {A}$ with respect to time."

As far as I can remember, this is only used when deriving Kepler's second law. While undoubtedly an important application, I don't think it is important enough to be mentioned here, especially as a slightly different form is used in the article on Kepler's law. Additionally, I think the fragment as it is, is hard to understand because it lacks precision. -- Jitse Niesen (talk) 12:42, 30 January 2007 (UTC)

No problem. —Mets501 (talk) 16:55, 31 January 2007 (UTC)

Other Shapes

The page shows a spiral and a rose, but I think pictures of Lemniscates, Limaçons, & Cardioids should also be added. Would do it myself, but I'm not too swell at Wiki-ing.YeoungBraxx 17:23, 16 February 2007

This article is probably long enough, but you can find information on those other shapes at the articles about them—see lemniscate, limacon, cardioid. Newyorkbrad 16:19, 17 February 2007 (UTC)

NPOV

0 degrees should be on the left hand side of the y axis. I know us leftists are in the minority, but minority views should be mentioned. The article is clearly positive-right-centric. Brentt 06:08, 17 February 2007 (UTC)

No. 0 degrees (or 0 radians, whatever, they're the same angle at zero...) goes to the right hand side because when you draw a polar function you pipe r and θ into the parametric x=r cos θ, y=r sin θ, and that kind of just wants to put zero on the right hand side (and you can't argue with it, it's the nature of the Universe, not a man-made convention!) Perhaps when god designed the triangle (and therefore sin and cos), it was discriminating against lefties... :P Not likely. Not a NPOV issue (as far as I know, anyway. In my opinion, etc). Stuart Morrow 21:05, 20 February 2007 (UTC)

While I agree with Stuart that we should follow the standard convention, the choice of which axis goes first and indeed which direction is considered to be positive is a convention. There are eight equally valid arangments for the axis. I don't know who chose the particular system we use which is a good question. cartesian coordinates does not seem to have the answer for this, nor does the Number line. --Salix alba (talk) 22:06, 20 February 2007 (UTC)

Guys, it was a joke. But Stuart, your wrong about it not being a convention. The universe has no preference for left or right in physical laws or math. All left right or left hand representations are by convention (yes, even the right-hand rule. Math and physics would look exactly the same if it was the "left-hand rule"). See Feynman's Symmetry in Physical Law lecture, it talks about this. Parametric plots would work just as well if the 0 was on the left. As a matter of fact, if you look at a parametric plot in the mirror its not like it won't make any sense. All the same information would be there and would mean exactly the same thing. Brentt 17:47, 26 February 2007 (UTC)

Polar coordinates!

They are awesome! Yeah! 132.161.164.75 06:46, 17 February 2007 (UTC)

Congrats to everyone that has contributed to this article! I have one suggestion: it would be helpful if the unit circle graph included radians as well as degrees. User:swimguy112

2D ?

Are there no 3D polar coordinate systems (article says its 2D only)? Seems easy enough to add another parameter (I'll call it the 'up-down angle' in my own unscientific way) and you have a 3D system. Surely someone invented that too? MadMaxDog 13:33, 17 February 2007 (UTC)

Did you find the section called "Three dimensions"? It seems to be in there. -- Jitse Niesen (talk) 13:38, 17 February 2007 (UTC)

Madmaxdog, see Spherical coordinate system for the 3d analog of the polar coordinate system. Raul654 14:21, 17 February 2007 (UTC)

See also Cylindrical coordinate system which is a direct translation of polar coordinates into 3d. Raul654 14:21, 17 February 2007 (UTC)

What about polar coordinates in higher dimensions than 3? Jeltz talk 16:35, 17 February 2007 (UTC)

Applications

"Polar coordinates are two-dimensional and thus they can be used..."

My fault for not seeing that one (the "Three dimensions") - I was only skimming and landed in the above subsection, which contradicts it, then... MadMaxDog 23:41, 17 February 2007 (UTC)

Factual error on the page

While translating the article into Esperanto for Esperanto-Wiki we found a crude factual error in the English version. The statement

If k is irrational, the curve forms a disc since every point in the coordinate plane with r < a will satisfy the equation for some value of θ.

seems to be wrong. Here's the spontaneous disprove of this statement: Let's take the equation r=a*cos(k*t) (a - radius of the disk) let's take z<a and greater than 0 let's explore the circle with radius z. We look for points of this curve on this circle, they should be derived from the equation z=a*cos(k*t) The set of possible solutions of this equations is a denumerable set. The set of all points of a circle with radius s is continuous and thus greater than the denumerable set. Therefore there must be some points on the circle (and these points constitute the majority) which are not crossed by the curve. The solution of the equation z/a=cos(k*t) is the following: t=(2*pi*integer+acos(z/a))/k t=(2*pi*integer-acos(z/a))/k The whole set of solutions is denumerable. Thus the set of points on this curve is not more than denumerable. The set of a circle is continuous hence bigger than any denumerable set. Alaudo 17:57, 30 March 2007 (UTC)

I removed the claim. Whenever you find something like that, you might as well delete it from the article at the same time you bring it up on the talk page. Although that claim is not right (otherwise it would be a very simple example of a space filling curve!), the graph ought to be dense in the disc, which will look like it is the entire disc when you try to graph it on a calculator. This could be the source of the mistake. CMummert · talk 18:40, 30 March 2007 (UTC)

The graph is really "dense everywhere" (as we write in in Esperanto-version), still there are some points which it does not cross. I did not dare to delete this very dubious statement just because the article was selected as "an excellent article" and was tagged as "reviewed". Alaudo 19:47, 30 March 2007 (UTC)

Featured article review checks many things but does not include a line-by-line fact check. In this case, the error was hidden in the phrasing - if it had said "the graph includes the entire unit disc" then someone would certainly have noticed it. CMummert · talk 00:36, 31 March 2007 (UTC)

As a matter of fact, I did pause at the sentence when it was up for review but I considered it to be an acceptable formulation to convey "dense everywhere", which to me seems too technical for this article. In hindsight, it is better to remove the claim, as CMummert did, because the point it makes is not germane to the topic. -- Jitse Niesen (talk) 05:03, 31 March 2007 (UTC)

For me, as a non-native speaker, the wording used in the article was in no way meaning "dense everywhere", but rather as "completing the whole space". For those, who do not know the difference it does not matter, for those, who does, it can be crucial! Please, notice, that this equation strictly speaking does not result the subspace which is "dense everywhere" (we already corrected it in our Esperanto-version), but rather simply "dense". There are further problems in this article which I report in several minutes (I need some time to formulate that in English), if not, than tomorrow afternoon. Alaudo 22:54, 1 April 2007 (UTC)

Further problems in the article

(I am writing on behalf of several contributors and use "me" only for convenience).

Problem 1. The article correctly states

To get a unique representation of a point, it is usual to limit r to non-negative numbers r ≥ 0 and θ to the interval [0, 360°) or (−180°, 180°] (or, in radian measure, [0, 2π) or (−π, π]).[9]

but if we look at the polar rose equation r(t)=a*cos(k*t+t0) we can notice, that in some cases the sign of the radius can turn negative. For the polar rose it means, that some "petals" disappear, because by some t there are NO "petals" in this direction. Basically, if we look at the illustration in the article, we expect only one petal in the quote between 12 and 3AM, namely that pointing approximately to 2AM. That's true for other quoters as well. So, it is evident for me, that the author either used the modulus for the angle between pi/4 and pi/2 (that's not stated!), or they took the negative value, thus placing the "petal" which had to be put on 1 AM to 7AM (and visa versa, hence getting the full rose). Both would be quite voluntary and must be duly described. Alaudo 23:05, 1 April 2007 (UTC)

This is not a conflict. We clearly state: to get a unique representation of a point; this has nothing to do with curves. It is understood when the domain or range is not stated that it is over the entire domain of theta or entire domain of the radius. We often don't want a unique representation of points, either, for curves, such as lemniscates. The same thing can be observed in the Cartesian coordinate system with the equation for a circle: $x^{2}+y^{2}=r^{2}$. —METS501 (talk) 00:03, 2 April 2007 (UTC)

I am not talking about a conflict. I am just trying to emphasize that maybe one should describe these points more explicitly. I do understand the idea and the assumptions which are taken for granted, but some other readers could stumble. Alaudo 07:36, 2 April 2007 (UTC)

Problem 2. The article states:

Note that these equations never define a rose with 2, 6, 10, 14, etc. petals.

That's in my opinion not true. Carefully dealing with the definition domain one can define these roses. Alaudo 23:09, 1 April 2007 (UTC)

Again, we assume that these equations are over the entire domain. I'll edit the article to reflect that. —METS501 (talk) 00:03, 2 April 2007 (UTC)

That should be explicitly mentioned. Alaudo 07:36, 2 April 2007 (UTC)

You might like to cast your eye over Rose (mathematics) which may have similar problems, feel free to fix as appropriate. --Salix alba (talk) 07:54, 2 April 2007 (UTC)

If the assumption that the entire domain be used is explicitly articulated, then I suppose the issue is solved. Nevertheless, it it worth pointing out here (probably not in the article, though) that the 2, 6, 10, etc. petalled roses obtained by restricting the domain do not exhibit nice symmetry, and therefore might be ruled out. VectorPosse 23:28, 2 April 2007 (UTC)

This article within the scope of the WikiProjects Systems

Hi, I have put this article under the scope of the WikiProject Systems because of the formal relation, but more because of the inspiring and motivating example this article can give our project and it's participants (to come). We are still a small and beginning group, and working to get our own toko going. In due time I hope we can also deliver a valuable contributions here from our point of view. In the mean time I wish all of you all te best. Best regards - Mdd 21:36, 4 May 2007 (UTC)

fix Polar_Angle

http://en.wikipedia.org/wiki/Polar_Angle is a dead end page and a stub. So maybe somebody do the apropriate merge/redirect.

I just made it into a plain old redirect. It had nothing really new anyway. —METS501 (talk) 06:40, 17 June 2007 (UTC)

Use of atan2?

Would it be advisable to use atan2 in addition to, or in replace of atan? The advantages would include shorter definitions as the function takes into account the quadrant of the angle. --Skytopia (talk) 19:49, 20 November 2007 (UTC)

The problem with atan2 is that is well know to people who know C and its derivatives. For non computer scientists it justs adds another thing people have not heard about. That said there is a short sentence on the subject which could be expanded. --Salix alba (talk) 00:40, 21 November 2007 (UTC)

But the space devoted here to explaining how to determine $\theta \,$ in terms of $\arctan()\,$ is just a repeat of the same information available at atan2. If we link to that article, yes, some readers will have to see a new "name" for a concept that they might already understand. But if we don't link to it, then everybody will have to be subjected to the gory detail of a 4-quadrant arctangent, which most of them probably don't need to see.

--Bob K (talk) 04:57, 21 November 2007 (UTC)

I guess it's okay to mention atan2 if this can be done easily; that would resolve Bob K's concern that we make things unnecessarily hard for those that do know atan2. However, I don't like replacing the arctan formula with atan2. Firstly, most calculus books that I know (perhaps even all) use arctan and not atan2. Secondly, I think that logically, polar coordinates come first. So atan2 should be explained in terms of polar coordinates and not the other way around. -- Jitse Niesen (talk) 13:55, 21 November 2007 (UTC)

Polar transformation in multivariable functions

Dear colleagues, would you please answere the following questions.

1) Is this true? Let f be R × R $\to$ R, G be the map G(r,φ)=(r cos(φ),r sin(φ)), and A ∈ R. Then

$\exists \lim \limits _{(0,0)}f=A\quad \Longleftrightarrow \quad \forall (r_{n},\varphi _{n})\in (\mathrm {Dom} (f\circ G)\setminus \{0\}\times [0,2\pi ))^{\mathbf {Z} ^{+}}\quad (\;\exists \lim(r_{n})=0\quad \Rightarrow \quad \exists \lim(f(G(r_{n},\varphi _{n})))=A\;)$

2) And this?

$\exists \lim \limits _{(0,0)}f=A\quad \Longleftrightarrow \quad (\forall \varphi [0,2\pi ))(\exists \lim _{(0,\varphi )}(f\circ G)=A\;)$

I can prove the first one, but I don't belive in the second. Mozó (talk) 10:59, 29 June 2008 (UTC)

Your notation is a bit non-standard. If I understand your question correctly, I would write $\lim _{(0,0)}f=A$ or possibly $\exists A\in \mathbb {R} \colon \lim _{(0,0)}f=A$ instead of $\exists \lim _{(0,0)}f=A$, and similarly for the other limits.

I think 2) is the statement: the limit of f at the origin is A if and only if the limit of f along every ray is A. Multivariable calculus says that a counterexample is given by the function

$f(x,y)={\frac {x^{2}y}{x^{4}+y^{2}}}$

(see the start of that article). -- Jitse Niesen (talk) 12:48, 29 June 2008 (UTC)

On ' ∃lim₀ f = A ' I mean 'there exists a limit and it equals to A '. And of course it is a kind of home notation. But, I think 1) is an important lemma in the topic 'solve limits using polar coordinates' and it can be used for example to show that the function above is a good counterexample for 2). Since, using substitution we have

$f(x(r,\varphi ),y(r,\varphi ))={\frac {r\cos ^{2}(\varphi )\sin(\varphi )}{r^{2}\cos ^{4}(\varphi )+\sin ^{2}(\varphi )}}$

and if

$r(\phi )={\frac {\sin(\varphi )}{\cos ^{2}(\varphi )}}$

i.e. we give sequences of the form (sin(φ)/cos²(φ) , φ) for example (r_n,φ_n)=(sin(π+1/n)/cos²(π+1/n) , π+1/n) then r_n $\to$ 0, however

$f(x(r,\varphi ),y(r,\varphi ))={\frac {\sin ^{2}(\varphi )}{2\sin ^{2}(\varphi )}}={\frac {1}{2}}$

(and not 0 that occures when φ_n ≡ 0 and r_n say 1/n).

It is interesting that sequence (φ_n) doesn't have to be a zero sequence (in the lemma it is arbitrary choosen). So, my question is: 'Is 1) true or not?' (I think: yes). Mozó (talk) 14:52, 29 June 2008 (UTC)

Yes, it's true. The following three statements are equivalent:

a) $\lim _{(x,y)\to (0,0)}f(x,y)=A$

b) if sequences $(x_{n})$ and $(y_{n})$ with $x_{n},y_{n}\in \mathbb {R}$ and $(x_{n},y_{n})\neq (0,0)$ for all $n\in \mathbb {Z} ^{+}$ satisfy $\lim _{n\to \infty }x_{n}=\lim _{n\to \infty }y_{n}=0$, then $\lim _{n\to \infty }f(x_{n},y_{n})=A$

c) if sequences $(r_{n})$ and $(\varphi _{n})$ with $r_{n}\in \mathbb {R} \setminus \{0\}$ and $(\varphi _{n})\in [0,2\pi )$ for all $n\in \mathbb {Z} ^{+}$ satisfy $\lim _{n\to \infty }r_{n}=0$, then $\lim _{n\to \infty }f(G(r_{n},\varphi _{n}))=A$

The limit in b) is called a sequential limit (see Limit of a function#Sequential limits). The equivalence between a) and b) is a fairly standard result in topology; see for instance the section on the Cauchy and Heine definition of continuity in continuous function for the same result in R, or the section on sequential continuity in continuous function (topology) for a more general discussion (that latter article says that in some spaces, normal limits and sequential limits are not the same, but R² is a metric space and in such spaces, normal limits and sequential limits are the same). Statements b) and c) are equivalent because the coordinate transformation G is a bijection between R² \ {(0,0)} and (R \ {0}) × [0, 2π). And statement c) is the conclusion of your 1). -- Jitse Niesen (talk) 17:11, 29 June 2008 (UTC)

Mechanics

Section added 15.07.2008 by Stamcose —Preceding unsigned comment added by Stamcose (talk • contribs) 14:19, 15 July 2008 (UTC)

Here it is: I removed it for cleanup. Re-add when ready. —METS501 (talk) 18:57, 23 July 2008 (UTC)

Mets501

Who is Mets501? Editor? What do you mean with cleanup? Please explain! This text should be put back precisely as it was! To avoid "ping pong" I have not pressed the undo button yet. First should be agreed that you do not remove it again!! Stamcose (talk) 19:29, 23 July 2008 (UTC)

Actually there is one thing that is not correct!

"Brews ohare" changed the original text from

Note that

$m\cdot r\cdot {\dot {\theta }}^{2}=m\cdot {\frac {h^{2}}{r^{3}}}={\frac {H^{2}}{m\cdot r^{3}}}$

is the "centrifugal force"

to:

Note that

$m\cdot r\cdot {\dot {\theta }}^{2}=m\cdot {\frac {h^{2}}{r^{3}}}={\frac {H^{2}}{m\cdot r^{3}}}$

resembles the expression for "centrifugal force" for paths with constant radius r.

It does not "resemble" the centrifugal force, it is the centrifugal force! And it has nothing to do with constant radius! Why not read the article "Centrifugal force", you might learn something!

Stamcose (talk) 20:03, 23 July 2008 (UTC)

Hi Stamcose. Please try to keep civil, alright? Anyway, I was the editor who brought this article up to featured status and wrote the majority of the article, and I am just trying to make sure that it maintains its high quality. By cleanup (WP:CLEANUP) I was just referring to the fact that it needs to be written better, not that it should be removed and never added back, or that it was wrong or anything. I'm going to clean it up now in the article instead of reverting again; please let me know if you have any problems with that. Thank you. :-) —METS501 (talk) 16:41, 25 July 2008 (UTC)

Dear Mets501

The formulas:

${\frac {d\mathbf {r} }{dt}}={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\theta }}{\hat {\boldsymbol {\theta }}},$

${\frac {d^{2}\mathbf {r} }{dt^{2}}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}.$

of

"Vector calculus"

and

${\dot {\bar {r}}}={\dot {r}}\cdot {\hat {r}}+r\cdot {\dot {\theta }}\cdot {\hat {t}}$

${\ddot {\bar {r}}}=({\ddot {r}}-r\cdot \ {\dot {\theta }}^{2})\cdot {\hat {r}}+(r\cdot {\ddot {\theta }}+2\cdot {\dot {r}}\cdot {\dot {\theta }})\cdot {\hat {t}}$

of

"Mechanics"

are obviously the same.

"Vector calculus"and "Mechanics" could be combined! Anyway, the formulas above is what is most important!

"maintains its high quality"! I think it would be better if written more compactly, i.e. shorter. After all the subject is rather trivial! And a lot of the stuff is too "talkative" Stamcose (talk) 19:42, 25 July 2008 (UTC)

Dear Mets501

The problem with "Vector calculus" is essentially that the definition of ${\hat {\boldsymbol {\theta }}}$ as a "unit vector at right angles to $\mathbf {r}$" is a bit sloppy and un-precise.

There are infinitely many unit vectors at right angles to $\mathbf {r}$

Even if it is understood that ${\hat {\boldsymbol {\theta }}}$ should be in the plane in which polar coordinates are used there are two alternative directions for ${\hat {\boldsymbol {\theta }}}$. It must be said that that ${\hat {\boldsymbol {\theta }}}$ should correspond to the direction of an increasing $\theta$

To bring it up to the mathematical standard of "Mechanics" you should write:

Vector calculus

Vector calculus can also be applied to polar coordinates. Let $\mathbf {r}$ be the position vector $(r\cos(\theta ),r\sin(\theta ))\,$, with r and $\theta$ depending on time t, ${\hat {\mathbf {r} }}=(\cos(\theta ),\sin(\theta ))$ (a unit vector in the direction $\mathbf {r}$) and ${\hat {\boldsymbol {\theta }}}=(-\sin(\theta ),\cos(\theta ))$ (a unit vector at right angles to $\mathbf {r}$). The first and second derivatives of position are

${\frac {d\mathbf {r} }{dt}}={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\theta }}{\hat {\boldsymbol {\theta }}},$

${\frac {d^{2}\mathbf {r} }{dt^{2}}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}.$

I peronally would prefer if you replaced the symbol

${\hat {\boldsymbol {\theta }}}$

with

${\hat {\boldsymbol {t}}}$

that normally is used for this!

If this is done "Mechanics" can be skipped, this was written mainly because I did not consider "Vector calculus" to be of the standard that I could refer to it!

Ps: I do not know if for example "Complex numbers" should belong to "Polar coordinates"!

Just because they are in the "Gaussian plane"?

Stamcose (talk) 00:53, 26 July 2008 (UTC)

Mechanics

Let ${\bar {r}}$ be radius vector for a mass point with mass $m$ moving in a plane and let ${\hat {x}}\ ,\ {\hat {y}}$ be a cartesian coordinate system in this plane. With polar coordinates $r\ ,\ \theta$ one has that

${\bar {r}}=r\cdot (\cos \theta \cdot {\hat {x}}+\sin \theta \cdot {\hat {y}})$

Defining the orthogonal unit vectors ${\hat {r}}\ ,\ {\hat {t}}$ as

${\hat {r}}=\cos \theta \cdot {\hat {x}}+\sin \theta \cdot {\hat {y}}$

${\hat {t}}=-\sin \theta \cdot {\hat {x}}+\cos \theta \cdot {\hat {y}}$

and taking the derivatives with respect to time one has that

${\bar {r}}=r\cdot {\hat {r}}$

${\dot {\bar {r}}}={\dot {r}}\cdot {\hat {r}}+r\cdot {\dot {\theta }}\cdot {\hat {t}}$

${\ddot {\bar {r}}}=({\ddot {r}}-r\cdot \ {\dot {\theta }}^{2})\cdot {\hat {r}}+(r\cdot {\ddot {\theta }}+2\cdot {\dot {r}}\cdot {\dot {\theta }})\cdot {\hat {t}}$

The radial and tangential velocity components $V_{r},\ V_{t}$ of the velocity vector

${\bar {v}}={\dot {\bar {r}}}=V_{r}\cdot {\hat {r}}+V_{t}\cdot {\hat {t}}$

have the values

$V_{r}={\dot {r}}$

$V_{t}=r\cdot {\dot {\theta }}$

If a force

${\bar {F}}=F_{r}\cdot {\hat {r}}+F_{t}\cdot {\hat {t}}$

is acting on the mass point the differential equation system defining its motion is

$m\cdot ({\ddot {r}}-r\cdot {\dot {\theta }}^{2})=F_{r}$

$m\cdot (r\cdot {\ddot {\theta }}+2\cdot {\dot {r}}\cdot {\dot {\theta }})=F_{t}$

As the angular momentum $H$ is defined as

$H=m\cdot r^{2}\cdot {\dot {\theta }}=m\cdot r\cdot V_{t}$

one has that

${\dot {\bar {H}}}={\dot {\overbrace {m\cdot r^{2}\cdot {\dot {\theta }}} }}=m\cdot r\cdot (r\cdot {\ddot {\theta }}+2\cdot {\dot {r}}\cdot {\dot {\theta }})=r\cdot F_{t}$

Note that $r\cdot F_{t}$ is the torque caused by the force ${\bar {F}}=F_{r}\cdot {\hat {r}}+F_{t}\cdot {\hat {t}}$

Defining $h$ as the angular momentum per unit mass

$h={\frac {H}{m}}=r^{2}\cdot {\dot {\theta }}$

and $f_{r},\ f_{t}$ as force per unit mass

$f_{r}={\frac {F_{r}}{m}}$

$f_{t}={\frac {F_{t}}{m}}$

the first order differential equation system that must be integrated to find the path ${\bar {r}}(t)$ of the mass particle as a function of time $t$ takes the form

${\dot {r}}=V_{r}$

${\dot {\theta }}={\frac {h}{r^{2}}}$

${\dot {V_{r}}}=f_{r}+{\frac {h^{2}}{r^{3}}}$

${\dot {h}}=r\cdot f_{t}$

where the dependent variables are

$r,\ \theta ,\ V_{r},\ h$

Note that

$m\cdot r\cdot {\dot {\theta }}^{2}=m\cdot {\frac {h^{2}}{r^{3}}}={\frac {H^{2}}{m\cdot r^{3}}}$

resembles the expression for "centrifugal force" for paths with constant radius r.

Centrifugal and Coriolis terms

"Brews ohare" introduced this paragraph. It is similar to my "Mechanics" of "20:32, 23 July 2008" that I removed "20:46, 27 July 2008" as I agree with "Mets501" that this does not fit here. The interpretation of these terms as "virtual forces" should be by setting up the physical "equation of motion" in a rotating 3-dimensional coordinate system. And if these physical concepts should be discussed here it would be more important to point out that $r^{2}{\dot {\theta }}$ is the angular momentum (per unit mass)and that the time derivative of the angular momentum is the torque.

Stamcose (talk) 19:49, 10 August 2008 (UTC)

With the recent edits, I think that this section has become too long. This is not the right place to discuss fictitious forces and inertial frames. -- Jitse Niesen (talk) 22:19, 26 August 2008 (UTC)

Okay, this has to stop. This article is not about physics. I thus reverted the recently added bits. If you think that extra discussion is necessary beyond the articles we already have, then start a new article called "polar coordinates in dynamics" or something like that and link to it from here. -- Jitse Niesen (talk) 12:09, 13 November 2008 (UTC)

Font

I noticed that there are some minor font inconsistencies on this page. In some places the coordinate r is written as $r$ and in other places as r. I personally prefer the second for inline text, but thought I should ask here first. In addition I have seen $(r,\theta )$ and ($r$, θ) and (r, θ). Changing it would be no problem for me as I have scripts to automate the process. Does anyone else have a particular preference? Thanks! Plastikspork (talk) 03:35, 17 February 2009 (UTC)

Chinese contribution

Any body found any Chinese contribution to polar coordinate? I am thinking of Lai Zhide in particular through his development of the Taijitsu and the I Ching.Jehovajah (talk) 12:54, 12 September 2010 (UTC) repositioned Jehovajah (talk) 07:35, 15 September 2010 (UTC)

Incorrect Conversion Formula

In Converting between polar and Cartesian coordinates, the formula is partly given as this:

θ = arctan(y/x), x >= 0

But this can't apply to x = 0 without dividing by zero!

I believe the complete formula should be this:

	⎧	arctan(y/x)	if x > 0
	⎪	-arctan(y/x) + π	if x < 0
θ =	⎨	π/2	if x=0 and y>0
	⎪	-π/2	if x=0 and y<0
	⎩	0	if x=y=0

I would correct it myself, but the formula is an image, not a block of text.

(The order of these items doesn't match the form of the original, but it matches the order of the second and third formulations in the original.) —MiguelMunoz (talk) 10:25, 19 December 2010 (UTC)

Position and Navigation

In the section titled Position and Navigation, 'heading' is described as if it was necessarily related to magnetic north. This is incorrect in the case of 'true heading'. It is only correct when discussing 'magnetic heading'. Thus, the statement in this section that Heading 360 corresponds to magnetic north, without qualifying it specifically as 'magnetic heading', is not accurate from a navigation standpoint. A 'true heading' of 360 corresponds to the geographic north pole, not to magnetic north. For the same reason, most of the subsequent discussion is not technically correct.

The example concerning aircraft also ignores the fact that the 'heading' of an aircraft must also take into consideration the wind. In fact, it is precisely the influence of a cross-wind force vector upon an aircraft that defines the difference between the terms 'heading' and 'course' in navigation. 'Heading' designates the direction in which the vehicle is oriented, relative to the polar coordinate system. 'Course' is the correct term for the actual direction of travel of the vehicle, as tracked against the surface of the globe and its coordinate system. Course and heading are only the same when there is no crosswind vector force in play.

Therefore, it would be simpler, and much clearer, to limit the discussion to true course rather than magnetic heading. This would avoid bringing in factors such as wind and magnetic deviation, which have no direct bearing (pardon the pun) on the polar coordinate system as applied to navigation of the globe. These extraneous factors are only likely to confuse the uninitiated reader. MisterSquirrel 19:02, 22 December 2010 (UTC) — Preceding unsigned comment added by MisterSquirrel (talk • contribs)