Talk:Coordinates (mathematics)

Moving this from the article page to here:

FURTHER WORK REQUIRED
 * theta=arctan(y/x) need a fix to make values in the adequate range.
 * Some conversion matrices of derivatives need fact-checks.
 * Simple figures would be a great help.
 * Should we mention that Cartesian really means the oblique coordinates?

RickK 04:22, 18 Jan 2004 (UTC)


 * Linked an existing figure for cartesian coordinates.
 * Made a new one for circular coordinates. Tell me if I should change something.
 * Switched two lines in the last matrix in the cartesian-spherical section. Could someone doublecheck?
 * DrZ 22:05, 20 Feb 2004 (UTC)

Previously unheadered
Let theta= the degree of angle '' arctan= the inverse tangent ''   y  =  the value on the y-axis ''   x  =  the value on the x-axis

Therefore, Theta = arctan (y/x)

(y/x) is tangent function, so the angle is figured out by inversing it.

(literally r is the hyptoneus of the triangle.) 03:19, 8 Apr 2005 (UTC)

I have changed all radian angles to degrees. While I prefer radians, too, I guess people not yet familiar with coordinate systems might have difficulties with them. Also, I'd have to change my pic otherwise :-) DrZ 22:05, 20 Feb 2004 (UTC)

(My suggestion is my property; if you wanted to copy it, note that's it's me, 11 years old Harris Chan)

I have some suggestion to add on the conversion between the the polar coordinate and the circluar (may be someone thought of this already?): r= y / sin Ø = x / cos Ø.

And (thetha)= tan ^-1(y / x)= sin^-1; (y / r)= cos^-1 (x / r)

Changes
Aren't the "polar coordinates" and "circular coordinates" listed here the same thing? It appears to be a holdover from when separate pages were merged. Also, what would people think of rearranging so that all the 2-D is in one major section, and the 3-D in another major section? --zandperl 23:51, 15 Apr 2004 (UTC)

http://mathworld.wolfram.com/SphericalCoordinates.html gives quite different values for the derivative conversion matrices. It would be good to straighten this out. --Mmm 06:36, Apr 20, 2004 (UTC)
 * Could you specify which matrix you mean? I could only find one common matrix on both sites (the jacobian for the spherical - cartesian conversion), which seems to be essentially the same. DrZ 14:35, 28 Jun 2004 (UTC)
 * The spherical-cartesian matrices were the same, I agree. When checking it, though, the intermixing of "engineering" and "math" (different phi-theta convention) coordinates was extremely unhelpful, so I deleted the set inconsistent with the above article.  Might want to bring back the vector relations in some form if it can be done without confusion, though.  11 Jul 2004

Mistake
In the chapter "Conversion between.." the vector of the drivates is calculated as the matrix-product of the Jacobian matrix and the vector of given derivates. Hence, you should use parenthesis instead of the lines denoting determinates (\pmatrix instead of \vmatrix). Refer also to "mathworld.wolfram.com", where brackets are used. This mistake was transfered to German wikipedia, where i am going to correct it. -- 81.210.158.250 13:36, 12 Feb 2005 (UTC)

This is now repaired. Problem - the links under "credit to original articles" in the "see also" section seem to be broken. Paul Reiser 18:21, 13 Feb 2005 (UTC)

Mistake in Cylindrical and spherical?
Don't understand why, but Cylindrical to spherical transformation is (part of):
 * $${h}=\rho \,\cos\phi$$

but this worked for me:
 * $${h}=\rho \,\cos\phi \, \cos\theta$$

-- Pok2 21:23, 14 March 2006 (UTC)

Math vs. Physics Convention
Hi, I posted an item to the Wikiproject Math talk page, but I thought I should put a comment here too.

It seems that the convention used here is that common in mathematics, while most other pages, including most importantly perhaps Nabla in cylindrical and spherical coordinates, use the convention common in physics, whereby $$r$$ is the length of the vector, and $$\rho$$ is the length of the projection onto the $$xy$$-plane. I think that this should probably be the convention adopted here as well (at least those that have discussed so far at the wikiproject page seem to think so, and I'll yield to their judgement). So I'm changing all of the material at this page to match what is at the Nabla in coords page, and putting the result at user:jacobolus/coordinates temporarily. It will take a while, and I'll post back when I'm done. Anyway, it really seems to me that we benefit from consistency, with prominent disclaimers at the top (right now there are tiny notes tucked in the bottom of both pages). --jacobolus (t)  20:17, 6 Mar 2005 (UTC)

This done yet? For I was very confused when wiki disagreed with my, non-American physicist, conventions. In any case you definitely need bigger disclaimers. 84.67.9.198 21:51, 16 September 2005 (UTC)

$$d$$ in "Conversion between coordinate systems"
Can someone explain the $$d$$ in the conversion equations, for example (taken from Cartesian to cylindrical):



\begin{bmatrix}dx\\dy\\dz\end{bmatrix}= \begin{bmatrix} \cos\theta&-r\sin\theta&0\\ \sin\theta&r\cos\theta&0\\ 0&0&1 \end{bmatrix} \begin{bmatrix}dr\\d\theta\\dh\end{bmatrix} $$



\begin{bmatrix}dr\\d\theta\\dh\end{bmatrix}= \begin{bmatrix} \frac{x}{\sqrt{x^2+y^2}}&\frac{y}{\sqrt{x^2+y^2}}&0\\ \frac{-y}{x^2+y^2}&\frac{x}{x^2+y^2}&0\\ 0&0&1 \end{bmatrix} \begin{bmatrix}dx\\dy\\dz\end{bmatrix} $$

Is the $$d$$ supposed to be the derivative (if so, with respect to what?) or does it represent something else? Pardon me, but I'm rather new to this topic and I feel that this ambiguity should be cleared up.

AstroNox 05:36, 13 September 2005 (UTC)

They aren't really derivatives but more like 'small' distances in the x,y or z direction. rex_the_first 18:54 Nov 05 (GMT)


 * The notation is meant to convey that the partial derivative matrix $$\frac{\partial{(x, y, z)}}{\partial{(\rho, \theta, \phi)}}$$ is the given matrix. I will replace the bad notation shortly --MarSch 13:22, 30 April 2006 (UTC)

Coordinate systems having their own articles
I noticed that Cylindrical coordinate system, Cartesian coordinate system and Parabolic coordinates are all developing independently of this one, with some overlap. Would it be best if they are merged into this one? Or maybe the contents of this article split into each of those pages? --Anguis 16:39, 21 February 2006 (UTC)


 * I don't think a merge is possible, but the articles should be synced and this article shouldn't duplicate much and so should probably contain a lot less formulas. --MarSch 13:26, 30 April 2006 (UTC)


 * perhaps it is a good idea to collect all conversion formulas and stick them on a new page list of canonical coordinate transformations or so. --MarSch 13:29, 30 April 2006 (UTC)


 * I have now done this. --MarSch 14:53, 30 April 2006 (UTC)


 * I don't think there should be extensive discussion of individual coordinate systems on this page if there are main pages for those systems. Also, I think having conversion to cartesian coordinates put on this page, and then a link to the list of canonical transformations. One reason for that is that, without having a conversion, its difficult for a reader to come across the list of transformations. If we put the conversion to cartesian, not only could someone derive all the other conversions from those, but it would be an apt place to put a "see also the list of more conversions". I just now put some conversions to cartesian coordinates because it would have been useful, but reverted myself after reading the discussion page, so that we can discuss it more before I/someone-else does that edit. Fresheneesz 08:15, 3 May 2006 (UTC)

Creators
Can we get some info on who first discovered/used each system? -- ʀ6ʍ ɑ  ʏ89  11:22, 14 March 2006 (UTC)

Link to proposal
The comment that the proposal did not mention a formula confuses me. It's a proposal for how to write the coordinates, not a formula. --Mmm 15:43, 30 April 2006 (UTC)


 * How else do you define coordinates than with a formula defining them. You can use words. But words just aren't good enough. Especially when it is so easy to give a formula which _says it all_. Such as $$z = \cos \theta$$ or $$z = \cos \phi$$ and its relatives.--MarSch 14:01, 1 May 2006 (UTC)


 * The proposal isn't about defining coordinates. It's about standardizing the symbols to use when referring to the angles in spherical coordinates. --Mmm 03:57, 31 May 2006 (UTC)

Wrong image
This image is wrong: the y-axis is in front of the green z-coordinate line. It should be the other way around. --Abdull 07:58, 29 May 2006 (UTC)
 * It is still righthanded, all angles still go in the right direction, shortly there is no problem with it whatsoever. It is just in a question of perspective, perhaps it is different from what you are accustomed to (x to front, y to the right?). Feel free to change it into a more familiar view. --Tauʻolunga 19:49, 29 May 2006 (UTC)
 * I guess you didn't see the mistake i am talking about. It is in the upper-right corner. It definitely is wrong. --Abdull 09:13, 30 May 2006 (UTC)
 * Yes, now I see what you mean, the Y axis seems to come towards you. You are right. Well let us do something about it then. Done, some extra improvements as well --Tauʻolunga 04:58, 31 May 2006 (UTC)
 * Great, thank you very much! I'm not that confident in PNG editing, thanks for your job! --Abdull 08:37, 4 June 2006 (UTC)

merge from List of canonical coordinate transformations
I think that the transformations should be merged here, to keep all the information easily findable without scouring the page for the link to List of canonical coordinate transformations. Fresheneesz 22:34, 14 June 2006 (UTC)
 * I personally agree, however the original List of canonical coordinate transformations page was actually cut-and-pasted from Coordinates (mathematics), so perhaps there was some consensus to separate them (although it may have just been one user). — M e ts 501 (talk) 22:52, 14 June 2006 (UTC)
 * I disagree. Adding two big articles together results in nothing but clutter. I would suggest that instead a small section be created in this article talking about canonical coordinates, and having a main template at the beginning of section referring to List of canonical coordinate transformations for more info. Oleg Alexandrov (talk) 03:09, 15 June 2006 (UTC)


 * Something like that would work. I think that perhaps a good idea might be to have a section for transformations for each type of coordinates, and list only the cartesian transformation (since cartesian is pretty standard) - and *then* link to the big page. Comments on that? Fresheneesz 06:05, 15 June 2006 (UTC)

Canonical dreibeins of the coordinate systems
I want to ask whether the canonical dreibeins of the coordinate systems shall be included?

$$ r=r e_r $$

$$ e_r=(\cos\phi \cos\theta,\sin\phi \cos\theta, \sin\theta) $$

$$ e_{\theta}=(-\cos\phi \sin\theta,-\sin\phi \sin\theta, \cos\theta) $$

$$ e_{\phi}=(-\sin\phi,\cos\phi,0) $$

What's so nice about this dreibein? The fun thing is that if your are doing computations in classical mechanis, and $$ \theta, \phi, r$$ change over time, you simply get a moving frame. Inside this frame everything looks simple, only when you take derivatives, you must differentiate the dreibein vectors also.

$$ \dot{e}_r=\dot{\theta}e_{\theta}+\sin\theta\dot{\phi}e_{\phi} $$

$$ \dot{e}_{\theta}=-\dot{\theta}e_r+\cos\theta\dot{\phi}e_{\phi} $$

$$ \dot{e}_{\phi}=-sin\theta\dot{\phi}e_r-\cos\theta\dot{\phi}e_{\theta} $$

I found them very convenient, but realized that many students in class were not familar with them.

Caution: I used the European conventions for spherical coordinates, so my formulaes do not fit to the article rightaway.

--Benjamin.friedrich 10:00, 14 September 2006 (UTC)

Minor edits
I've made some edits which I hope improve this article, but I think there is still quite a lot to be done, and more detail is needed. I hope someone takes it up. I plan to work a bit on coordinate system and may move some of that material here. Geometry guy 22:21, 23 February 2007 (UTC)