# Talk:Spherical coordinate system

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One of the 500 most frequently viewed mathematics articles.

Previous discussions pertaining to the old article have been archived to Talk:Spherical coordinate system/Archive. --Carl (talk|contribs) 01:25, 18 September 2006 (UTC)

## Printing

I printed out this page and the images came out as black squares. get new images, these ones are garbage! —Preceding unsigned comment added by 62.136.133.147 (talk) 08:26, 5 May 2008 (UTC)

The main article would be improved considerably by changing the images to pairs of stereoptic 3D images. 216.99.219.151 (talk) 22:14, 15 June 2009 (UTC)

## Elements

This needs to be fixed, its why is there no info for spherical coordinates like they have for cylindrical, such as dV and dS:

(From cylindrical article) Line and volume elements

In many problems involving cylindrical polar coordinates, it is useful to know the line and volume elements; these are used in integration to solve problems involving paths and volumes.

The line element is dl = dr\,\mathbf{\hat r} + r\,d\theta\,\boldsymbol{\hat\theta} + dz\,\mathbf{\hat z}.

The volume element is dV = r\,dr\,d\theta\,dz.

The gradient is \nabla = \mathbf{\hat r}\frac{\partial}{\partial r} + \boldsymbol{\hat \theta}\frac{1}{r}\frac{\partial}{\partial \theta} + \mathbf{\hat z}\frac{\partial}{\partial z}.

Even mathworld has dS......

I have added surface and volume. –Pomte 11:13, 10 April 2007 (UTC)

## American convention

This article itself states that the international recommended standard for spherical polars is r, ϑ, φ for distance, zenith, and azimuth. If this is so, why use the american convention for the rest of the article? Shouldn't we use the international convention on the basis of the international applicability of the article?

In the diagram, the "r" length isn't the "r" mentioned in the text right? It's confusing that it was labelled r.

Perhaps there should be different diagrams for the different coordinate systems. Unfortunately, someone has to draw them. Shinobu 10:15, 16 November 2006 (UTC)

The symbols in the text and in the figure don't match. for instance: the azimuth angle in the text is referred to as θ, but the same symbol is taken for the polar angle in the diagramm. The polar angle in the text is a small phi (φ) and a capital phi (Φ) is taken for the azimuth angle in the figure. That is very confusing. (RolandRo 12:37, 11 January 2007 (UTC))

I have about the same problem, but all phi:s in the text are small (lowercase). I think it's the weird TeX small phi, that looks very much like a capital phi. It's a matter of different fonts. Rursus 18:04, 4 February 2007 (UTC)
I changed all φ to Φ to avoid confusion. TeX is showing the uppercase phi. If someone feels φ should be used instead, please revert all instances of them, even in TeX. –Pomte 11:13, 10 April 2007 (UTC)
But TeX isn't showing a capital phi $\Phi\,$, it's showing a lowercase phi $\phi\,$. If you want to use the other TeX lowercase phi $\varphi\,$, that's OK, but please don't use uppercase - the convention is to use lowercase (as in the diagram). --Zundark 11:26, 10 April 2007 (UTC)
Ugh, sorry, I got confused. Should I revert despite visual discrepancy? If the \varphi symbol is indeed convention, I'll convert all \phi to \varphi. I've seen \phi more often than \varphi, but that's no good indicator. –Pomte 11:37, 10 April 2007 (UTC)
I've already reverted. (My revert also reverted your later edit - you can restore that, of course.) I think \phi is more usual than \varphi for this purpose. The visual discrepancy depends on what font you are using - there's no way to fix it for everyone (except by doing everything in TeX), so I think we have to live with it. --Zundark 11:43, 10 April 2007 (UTC)

I do not understand the statement that the so-called American notation makes things more compatible with polar or cylindrical coordinates. On the contrary: using the notation in the article (with ranges indicated for $\phi,\theta$), cylindrical coordinates would be of the form $(\rho,z,\theta)$. Whereas starting with the notation $(r,\theta)$ (or $(\rho,\theta)$) for polar coordinates, cylindrical and spherical coordinates are obtained by simply making the coordinate pair a coordinate triple (with the caveat that $\rho$ would be $\rho=\sqrt{x^2+y^2}$ rather than $\rho=\sqrt{x^2+y^2+z^2}$ with both cylindrical or polar, i.e., the projection onto the $xy$-plane).

It seems to me that the debate is more between how physicists and mathematicians use different notations, rather than between an American notation and a notation elsewhere. Looking at three calculus textbooks written by reputable American authors, all use the (length,azimuth,zenith) notation. Since I also read French, I took a look at the literature, and there the issue is the same: mathematicians tend to write (length,azimuth,zenith), physicists tend to write (length,zenith,azimuth). The idea is that, making a broad generalizing statement, in mathematics we do not really work with spherical coordinates, they are merely a tool, just another change of basis. Whereas in physics, they are commonly used as a coordinate system proper, in which case it is important that such things as the right hand rule applies, because it makes computations easier to manage. Jarino1 18:44, 9 July 2007 (UTC)

The difference isn't in the placement of the coordinates; it's which letters are used for which coordinates. In mathematics θ is used for the azimuthal angle and φ is used for the zenith angle. Whereas in physics θ is used for the zenith angle and φ is used for the azimuthal angle. This should be made clear on the article.
I myself have studied under the mathematical convention in the United States and it is always denoted (ρ, θ, φ); so I disagree with the placement of the coordinates on the article; but this is a separate issue.
The mathematical convention is more compatible with polar and cylindrical coordinates because on the xy-plane, the azimuthal angle (which is denoted by θ) is the same as the polar angle used in polar and cylindrical coordinates (which is also denoted by θ). --Spoon! 21:32, 12 September 2007 (UTC)
I really like the ISO 31-11 notation, but for it to be compatible with polar coordinates it would be a great idea to start using φ for the polar angle. That way all the systems would be compatible with each others, and there would be minimal confusion. 78.91.38.146 (talk) 11:38, 27 November 2008 (UTC)
I'm an unlucky physicist who started in math, so the ISO 31-11 notation always confuses me. That said, I've had a few textbooks and professors / colleges who use the math convention, or other stranger conventions, so I didn't even know that someone had tried to make a physics convention before reading this article. I'd be surprised if the ISO 31-11 notation was actually widely dominant, but it is certainly possible. As a result, I'd definitely prefer using the more consistent (and personally less confusing) math convention. naturalnumber (talk) 04:27, 16 January 2010 (UTC)
If it were a mater of 99 to 1, the most common notation should definitely be used. However, from these responses and a few check to sources, it seems to me that both variants are fairly common. Trying to decide which is more common would be both hopeless and pointless. Thus I would say that either choice is equally good (or equally bad, take your pick), as long as the existence of variant notations is clearly noted, and the same notation is consistently used for the whole article. (On the other hand, due to the way that Wikipedia works, consistency of notation between separate articles so unlikely to be achieved that it is not even worth trying.) Readers who are used to the other notation will be inconvenienced, but that will happen either way. In any case, both sides must be prepared to read books and articles written in the "wrong" notation; so the unlucky readers may regard this article as a good exercise on that art 8-). As for following the ISO standard, I would quote the saying "the nice thing about standards is that there are so many of them to choose from" 8-) All the best, --Jorge Stolfi (talk) 15:59, 16 January 2010 (UTC)
I think there is no real conflict between the choice of φ for the azimuth and the polar planar coordinates. When you go from 2D to 3D through a revolution (for instance, when moving from elliptic coordinates to prolate spheroidal coordinates; or from Bipolar coordinates to Bispherical coordinates) you simply rotate around an axis and call the new coordinate φ. In the same way, spherical coordinates are a rotation of polar planar coordinates around the Z axis. θ is the same in both systems and the azimuth φ is the new coordinate.--Gonfer (talk) 19:56, 12 February 2010 (UTC)
The 'conflict' is that the angular coordinate of polar coordinates (which ranges over a full circle) seems to be commonly denoted by θ. Cylindrical coordinates are essentially polar + height, and its only angular coordinate is the azimuth. In spherical coordinates there are two angular coordinates, but only one of them ranges over a full 360° circle, namely the azimuth. So consistency would dictate using the same symbol for these three coordinates, and a distinct symbol φ for elevation (which ranges from -90° to +90°) or inclination (which ranges from 0 to 180°). Alas, the world is not consistent, and it is not Wikipedia's role to make it so. All the best, --Jorge Stolfi (talk) 20:45, 14 February 2010 (UTC)

## Longitude

I don't think this statement is correct:

The longitude is the azimuth angle shifted 180° from θ to give a domain of -180° ≤ θ ≤ 180°.

Assuming the x axis goes from the center of the Earth through the equator at 0 deg longitude, then I believe the correct wording is:

The longitude is the azimuth angle but for 180° < θ < 360°, subtract 360° so that -180° < longitude ≤ 180°.

Actually, it would probably be more correct to bring up East and West when talking about longitude, but in any case, saying a shift by 180 deg is wrong (unless the x axis is supposed to point through the 180 deg longitude point which I doubt) DaraParsavand 19:01, 18 January 2007 (UTC)

You're right, it's only a shift on half the values. I think it's pointless to discern whether any shift starts at θ = 0° or θ = 90° or anywhere else, so I've left it as an east-west distinction. –Pomte 11:37, 10 April 2007 (UTC)

## Angles

Radians should be used, not degrees. Jarino1 18:55, 9 July 2007 (UTC)

• Both are used, depending on the field of application. As said in the article, mathematicians and physicists seem to prefer radians, while engineers and and geographers often use degrees. --Jorge Stolfi (talk) 20:12, 24 November 2009 (UTC)

## Range of azimuth

There seems to be a mistake in the page. atan2 is said to return a result between pi and -pi and yet phi is said to be in the range of 0 and 2pi. What's going on here? —Preceding unsigned comment added by 132.67.97.20 (talk) 11:30, 19 February 2009 (UTC)

• As explained in the beginning of the article, the range of the azimuth is arbitrary; some use 0 to 360, others use -180 to +180, possibly others use still other ranges. Actually the range matters only if it is important to have unique coordinates; then the chosen range should span 360 degrees, but may start with any angle. --Jorge Stolfi (talk) 20:12, 24 November 2009 (UTC)
Actually it matters so that one can truly use a given coordinate system to find referenced loci, not just for uniqueness. But of course that is exactly the level of detail that someone would look to this article for, and which it fails to provide because of its over-generality. Dlw20070716 (talk) 09:05, 19 July 2011 (UTC)

## Comparsion with Euler angles

Conventional spherical coordinates as described in this article are a combination of rotations along the z and y axes. However, Euler angles are most commonly combinations of z and y rotations. Maybe someone could mention a few words about this issue? I was a bit confused by this difference. I propose to discuss this at Talk:Euler angles. Han-Kwang (t) 10:50, 4 December 2007 (UTC)

## Angle symbols swapped?

Compared to the first diagram, has the theta and phi swapped when it gets to this point

${x}=r \, \sin\varphi \, \cos\theta \quad$
${y}=r \, \sin\varphi \, \sin\theta \quad$
${z}=r \, \cos\varphi. \quad$

137.205.76.233 (talk) 18:34, 9 March 2008 (UTC)

I also think that this is the case. I have been working in OpenGL with cartesian and spherical coordinates and the current symbols are definately swapped. Z must be invariant with $\theta$. At least my math classes nomiated them so. -67.171.122.139 (talk) 18:23, 13 May 2008 (UTC)

• The naming of the coordinates is a matter of convention. Different authors/systems may use different conventions. --Jorge Stolfi (talk) 18:53, 24 November 2009 (UTC)
• The triple integral article uses phi for elevation and theta for azimuthal. Is there a wikipedia standard for these we could apply to all articles? —Preceding unsigned comment added by Ban Bridges (talkcontribs) 19:54, 18 March 2010 (UTC)
Yeah, they look swapped to me.
http://tutorial.math.lamar.edu/Classes/CalcIII/SphericalCoords.aspx
Methinks the coordinate system on this page needs to be standardized. --Secruss (talk) 20:56, 18 May 2011 (UTC)
• As always, the wikipedia standard is to use published conventions from reliable publications and document same with references (which this article fails to do). Dlw20070716 (talk) 08:57, 19 July 2011 (UTC)

The formulae in this section use the less-used unsigned spherical coordinates. For formulae using the more common geographical system, see the Great Circle article, this website. The article distinguishes from "inclination" (represented by theta) which starts at zero at the south pole, and "elevation" which starts at zero at the equator. Similarly, the rotation angle given here goes from zero to 360° (2π) whereas longitude starts at the Prime Meridian and extends to plus and minus 180°. L e cox (talk) 23:06, 4 September 2011 (UTC)

## θ is referred to elevation

φ is referred to as the azimuth and θ is referred to elevation, right? —Preceding unsigned comment added by 131.180.34.78 (talk) 13:01, 22 May 2008 (UTC)

• This is explained at the beginning of the article. It depends, some authors may swap the letters and/or use inclination instead of elevation.

## Immediate Action

Is this article going to get fixed soon, by someone smart enough? It is full off errors. Anyone using this as a reference (and not knowing any better) will have a very hard time learning accurately. —Preceding unsigned comment added by 98.179.13.179 (talk) 22:58, 4 October 2009 (UTC)

• Please point out the errors. --Jorge Stolfi (talk) 18:49, 24 November 2009 (UTC)
• The errors are everywhere! I just fixed a handful of them. Somebody destroyed this page. —Preceding unsigned comment added by 129.65.149.8 (talk) 00:57, 8 December 2009 (UTC)
• Please check the conventions used in cylindrical coordinate system and in this article (in the introduction, and right below the formulas in question). In both articles the azimuth angle is denoted by φ, not θ; the latter is spherical inclination or elevation. Thus the conversion preserves φ, while θ must be computed from z, of vice versa. (There was one θ = θ which should have been φ = φ; it's fixed now.) Also, the spherical radial coordinate is denoted here by r, while the cylindrical radial coordinate is ρ, here and in the other article. Am I missing something?
Perhaps you are used to a different convention? I am sorry that you have wasted your time, but the conventions used in those two articles appear to be fairly common (if not predominant) and seem acceptable to most editors. All the best, --Jorge Stolfi (talk) 03:25, 8 December 2009 (UTC)
• This article is not consistent in is notation. The transformation formulas to go from Cartesian to spherical coordinates and vice-versa (that I wrote some time ago) use the angle θ as the polar angle, measured from the zenith (with is the ISO standard). If at the beginning of the article the elevation is called θ, the formulas are confusing. The same for the images on the right. Elevation (latitude) should be called λ and θ reserved for the polar angle. —Preceding unsigned comment added by Gonfer (talkcontribs) 15:27, 12 February 2010 (UTC)
• I agree that using theta for both elevation and inclination is confusing, but that unfortunately seems to be the situation in the real world. The use of lambda for elevation seems common in geography, but geographic coordinates (lat/lon/alt) are not quite spherical coordinates, due to the way latitude and altitude are defined (reference spheroid and all that). Outside geography, theta and phi seems to be used for [inclination or elevation] and [azimuth] — and not even "respectively"! The article tries to be consistent (if it is not, it should be fixed) and to give formulas for both conventions when possible; but even so it is necessary to say every time what theta is, for the benefit of readers who are used to the other convention.
• I think for purposes of this article, we can ignore that the earth is only a spheroid. A spherical coordinate system can be readily projected onto a spheroid. Latitude and longitude together with altitude are indeed a complete spherical coordinate system, and as they are the only one most people are familiar with it behooves any wikipedia author working on this article to treat it with respect and expertise. With zero altitude defined as a theoretical mean sea level most people never notice that they don't live on a sphere. Dlw20070716 (talk) 08:51, 19 July 2011 (UTC)

## Error in Cartesian formulas?

Anonymous user 146.87.52.54 added a note to the "Geographic coordinates" section saying "There is a mistake on the 3rd equation of the formula of the cartesian coordinates!". Would the author please clarify? (Perhaps he/she assumed that θ in the Cartesian formulas was latitude? It is actually inclination, that is, 90° minus the latitude.) --Jorge Stolfi (talk) 18:48, 24 November 2009 (UTC)

Since geographical coordinates use latitude and not inclination, there is an obvious error to be addressed. A wikipedia author is not free to redefine the geographical coordinate system to include inclination! If it is mentioned at all, you'd better have an ironclad reference for it. Dlw20070716 (talk) 08:22, 19 July 2011 (UTC)

Maybe he's referring to the fact that the equations for theta and phi appear to be swapped? See Mathworld. mitch_feaster (talk) 17:08, 10 August 2010 (UTC)

## Directrix?

Someone has introduced the name "directrix" for the reference direction on the equatorial plane (azimuth zero). However the directrix article does not mention such a sense, and I have never seen the word used for that. Is there a reference for it? Thanks and all the best, --Jorge Stolfi (talk) 05:53, 8 December 2009 (UTC)

Hi there, professor Jorge! It was me who edited this page without first signing in. I was looking for the standard for spherical coordinates which I was about to use in my own mathematical derivation project, when I noticed this article could use some improvement, to state in clear, unambiguous, and elegant form the most popular of the standards used to date. I'm honored to see that you adopted some of my suggestions (though I fail to see why you removed the limits in the definition of the angular coordinates). As you point out, I used the name "directrix" incorrectly to name a reference line, something from my distant, confused memory: it should be taken out and I'll do so right away.
I made several minor contributions to various articles, previously, without noticing that I had offended anyone, until now. This is my first attempt at having a discussion with another contributor and I'm getting a better picture of how it all works. I now also have my own, very brief, User Page. Regards.--Toolnut (talk) 08:24, 8 December 2009 (UTC)
However, note that the ranges of the coordinates need not be the ones you put in. The ranges are relevant only if it is important to have unique coordinates for each point; and, even then, there are several choices in common use. This issue is discussed in a later section.
More generally, we must avoid putting things in the definition (such as the letters z and x for the axes) that are not universal. One can do that in a textbook or a paper, but an encyclopedia article must limit the definition to the really essential concepts, and discuss conventions separately.
Indeed, I am a bit unhappy with the current "Definition" section, because it uses the letter r, θ, φ as if they were part of the definition. Actually there are several conventions in use, and Wikipedia should not take sides on that (just as we should not take sides on degrees vs. radians). We must eventually ] pick a notation, but it should be clear that it is only one of various common choices. All the best, --Jorge Stolfi (talk) 15:28, 8 December 2009 (UTC)
PS. Also, the term "clockwise" only makes sense for physical three-space, and is not defined in mathematics or for more abstract three-dimensional manifolds (say, Pressure/Temperature/Time). In general, one must explicitly define the positive sense for azimuth. Moreover, the the right-hand convention is not universal (if I am not mistaken, on planets other than the Earth longitudes are measured in the opposite sense). --Jorge Stolfi (talk) 16:32, 8 December 2009 (UTC)

## Linear algebra, analytic geometry, or geometry?

### Unit Vectors, Position Vectors, Frame of Reference, Orthonormal Basis

The above words should be incorporated in the definitions of any coordinate system. There is a reference zenith direction which is more accurately represented by a "unit vector," and it should be called exactly that, and may be impartially named $\hat z$ for succinctness in references to it. There is a reference azimuth direction, also, more precisely a unit vector, which may be called $\hat x$, again, for the purposes of succinctness in the descriptions of the various conventions used. We need to adopt a convention in order to describe other conventions in precise mathematical terms. These two constitute an orthonormal basis, a frame of reference, relative to which the coordinates of a point, represented by a position vector OP may be defined. "Spherical coordinates" is a mathematical phrase, more than geographical or astronomical, and it should therefore be treated in precise terms. Three-dimensional coordinate systems have their roots in the Cartesian system, which universally uses $\hat x$, $\hat y$, & $\hat z$ as its unit vectors, and which are just as universally mapped to the spherical coordinate system as I suggested.

• "Mathematical precision" does not mean "mathematical formalism". The latter may be more coincise, but I am not sure that it is more understandable or even more precise.
You and I are used to 'algebraic prose' like "let x be a direction on the plane P orthogonal to z"; and, for instance, we understand that this sentence is defining not only x but also P. Well, I do not think we can assume that much from readers who come to this article to know what "spherical coordinates" are. I suspect that a 'plain English' definition, like the current one in the article, will actually be easier for them to understand (and no less precise) than an 'algebraic prose' definition.
The term "direction", in particular, is mathematically precise (in this context, it has is no possible interpretation other than the one intended). It is no less precise than "plane" and "point". Ditto for the other terms. A "unit vector" is a vector with unit length; but that length is not needed anywhere in the definition of spherical coordinates, so using a unit vector to define a direction (outside of linear algebra) is a mathematical overkill. (My favorite way of defining "direction" is a point at infinity in oriented projective space; but I will not use that here, either 8-).
Moreover, the concept of "vector" is actually much harder to define than a spherical coordinate system.
If we were defining spherical coordinates *relative to a previously defined Cartesian coordinate system*, it would be natural to use the unit vectors x, y, and z; but it is not the case here. The reasonable foundation for the article is Euclidean geometry. We cannot assume (or pretend, or postulate) that Cartesian coordinates are more fundamental than spherical ones.
Furthermore, two orthogonal vectors do not define a Cartesian system in Euclidean 3-space, because the direction of the third vector cannot be defined mathematically; it must be chosen explicitly. (In Euclidean geometry there is no absolute handedness.) In spherical coordinate system, the third vector (y) does not appear anywhere except to define the positive sense of turning on the reference plane. So, again, if one is defining spherical coords directly, rather than though a prior Cartesian system, it is cleaner and simpler to choose the sense of positive rotation directly. All the best, --Jorge Stolfi (talk) 21:26, 8 December 2009 (UTC)
The term "direction" is no less imprecise than "vector", it's just the fact that you don't use the former so much when you're actually establishing the foundations of 3-D geometry. I have attempted to introduce the concept of a vector through illustration, for now, making do with the less-than-ideal graphics that are there.
My reason for the introduction and use of (a) unit vectors $\hat z$ & $\hat x$ to represent the reference directions and (b) the position vector OP, from the outset, has to do with the fact that Cartesian, spherical, or any other system's coordinates may later be more conveniently derived from, and defined by, just these two orthonormal (basis) vectors and the position vector:
$\hat y = \hat z \times \hat x$
$\hat r = \frac{OP}{|OP|}$
$\hat \phi = \frac {\hat z \times \hat r}{|\hat z \times \hat r|} \text{, unit vector in } \hat z \times \hat r \text{ direction}$
$\hat \theta = \hat \phi \times \hat r$
$x=OP \cdot \hat x,\,y = OP \cdot \hat y,\,z = OP \cdot \hat z$
$r = OP \cdot \hat r$
$\theta = \arccos ( \hat r \cdot \hat z )$
$\phi = \arg ( \hat x \cdot \hat r + j( \hat x \times \hat r \cdot \hat z) ),\, j=\sqrt{-1} \text{, or}$
$\sin \theta \cos \phi = \hat x \cdot \hat r ,\,\sin \theta \sin \phi = \hat x \times \hat r \cdot \hat z$
All it takes is the definition of dot product and cross product to put things in the above, unambiguous, symbolic form: it is immeasurably harder to solve problems in 3-D geometry without these well-established concepts. The use of these two basis vectors is essential regardless of what orthogonal coordinate system you're deriving. Our only connection to the Cartesian system is the reuse of two of its coordinate names. The reason why we cannot use $\hat r,\,\hat\theta,\,\hat\phi$ as basis vectors is that they are position-dependent and not fixed in space; they are only used to describe the orientation of a vector in a vector field.
Also, as I've shown above, two orthogonal vectors do define the Cartesian (or any other) system, as the third vector is derived from the first two. Finally, the concept of "position vector" is in every textbook in engineering and physics I've ever read, so I don't understand why you are taking pains not to use it here, in favor of "Euclidean distance" (is that more understandable?) or "line segment." I know, not everyone would like to get deeper into this subject, but for the benefit of those who might, it doesn't hurt to put this different perspective out there. Regards.Toolnut (talk) 08:56, 9 December 2009 (UTC)
• You are assuming that one can or should define spherical coordinates only after defining Cartesian coordinates, dot product, cross product etc. This approach is a big overkill, and not appropriate for an encyclopedia! It may be adequate for a textbook on calculus or a university math curriculum, where students are assumed to learn one before going on to the other. But that is not a reasonable assumption for Wikipedia readers.
In any case, Cartesian coodinates and algebra (much less linear agebra!) are absolutely NOT necessary to define spherical coordinates, and do NOT make them easier to understand. In geography, for instance, people have always used and defined spherical coordinates directly, without even mentioning Cartesian ones. Linear algebra (vectors, dot product, cross product, etc.) are definitely NOT "the foundation of 3D geometry", only of analytical geometry. People have been and are doing some pretty advanced 3D geometry, including speherical coordinates, without reference to any of those concepts.
Finally, the cross product formula can be used ONLY after you have chosen all THREE Cartesian basis vectors, so you cannot use it to define the third vector in terms of the other two. Either you choose the third vector explicitly, or you choose a handedness for 3-space (which is not "built in" in Eucliean geometry). You cannot escape that.
So please keep the original simple, direct, algebra-free, non-circular definition; there was nothing wrong with it. Please. All the best, --Jorge Stolfi (talk) 15:29, 9 December 2009 (UTC)
PS. I added a note on the "position vector". However, note that it too is a concept of linear algebra that is useful only if you are going to use linear algebra operations on it (which is surely the case of all the technical texts that you mention). The local "spherical" frame vectors $\hat r$, $\hat \varphi$, $\hat \theta$ would be fine in the "calculus" section of the article (together with line element, etc.), but are not necessary to define spherical coordinates per se. And one must note the degeneracy at the poles. All the best, --Jorge Stolfi (talk) 16:03, 9 December 2009 (UTC)

### Dot product, cross product, handedness; abstract and concrete

1. You insist that one "cannot use [the concept of cross product] to define the third vector in terms of the other two." Do we at least agree that any 3-D frame of reference can be represented (i.e. fixed in space) by exactly two basis vectors? Does the concept of "right angle" assume knowledge of the Cartesian coordinate system? The third basis vector in establishing such a system is simply defined as being simultaneously at right angles to the first two, with a particular handedness: you fix the system in space by only fixing two of these basis vectors; the last one follows in a predetermined way.
• Yes, *once you choose a "right" handedness for the space*, then two orthogonal directions define a third direction orhtogonal to both, unambigously, by that "right hand" rule.
But classical Euclidean 3D geometry usually does not distingish "right handed" from "left handed". I presume that this "hand neutrality" is a consequence of Euc.Geom. having been developed first for the plane, in contexts where the handedness is irrelevant. Consider for example the theorem or axiom "two triangles are equal if corresponding sides are equal": it implicitly assumes that you can flip one triangle over to match it with the other triangle. For *physical* 3D space (the one whe move in) we of course cannot "flip" a solid over, so one may assume that "right-handed" is well defined and use that to define the third axis. (But you should read Martin Gardner's discussion in The Ambidextrous Universe. And if the physical universe turns out to have the topology of a non-orientable 3D manifold, then we are in trouble again...) However, if we are choosing coordinates for some abstract 3D space (such as the state of a compressed gas), then there is no natural definition of "right-handed", and one must specify the three axes explicitly. I suppose that there are people who define and work with "oriented Euclidean 3D space" (classical E. space plus a distinguished handedness), but that does not seem to be common.
• PS. I forgot to say: yes, in Euclidean geometry the concept of "perpendicular" is defined (and almost primitive). In the plane, two lines are perpendicular if all four angles between them are congruent (and "congruent" is a primitive concept). "Parallel" is a more primitive notion: two lines are parallel if they lie on the same plane but never meet. The existence of parallel lines is an axiom (much disputed until Lobachevski/Riemman).
• Thanks!
I guess I was confusing the definition of "frame of reference" with the definition of a "coordinate system," which requires further elaboration, such as the definition of the third basis vector (in Cartesian coordinates) even though it is based on the first two; the function that defines the third basis is what adds that third dimension. In Spherical coordinates, there are three functions of the same basis vectors defining three alternate dimensions (not basis vectors, nevertheless having the ability to uniqely define a point in space, except at the degeneracies.Toolnut (talk) 21:48, 9 December 2009 (UTC)
2. Can you show me the derivation of the most general equations for 3-D rotation, a complex geometry problem, without the use of vector geometry algebra? Is it just as easy to accomplish?
Toolnut (talk) 18:30, 9 December 2009 (UTC)
• In geometry, a rotation is defined as a mapping of points to points that preserves all distances between point pairs and has only one line's worth of fixed points. So there are no equations to derive. (The last condition is only needed to exclude reflections and screw motions; the first condition alone defines the class of congruences or isometries, which are much more important in geometry.) Given enough data to determine the rotation (for example, the fixed axis line and two directions perpendicular to it) there is a simple 3D geometric construction that will yield the rotated image of any given point. Again without any formulas.
What you ask is a problem of analytic geometry. All the best, --Jorge Stolfi (talk) 19:36, 9 December 2009 (UTC)
• Right, I was talking about 3D rotation about an arbitrary axis and the derivation of the new coordinates of an arbitrary point (x, y, z), or (r, θ, φ), from knowledge of the direction numbers of the axis of rotation (represented by a unit vector). Is "analytic geometry" really what I'm after, not so much "linear algebra"? I'm not aware that linear algebra deals with cross products and their physical meaning, or the physical meaning of dot products, either, especially because it is more general than just 3D space. Cross products don't carry over to higher dimensions very well, even without a physical meaning. (My last attempt to do that implied that a cross product becomes a trinary operation in 4D, an operation on three 4D vectors.) Anyway, there is no easier way to do this task (derive the coords of a rotated point) than through the use of vectors and their dot and cross products, is there?Toolnut (talk) 21:48, 9 December 2009 (UTC)
• Cross products can be generalized to any dimension n and any number mn of vectors as follows: assemble an m by n matrix M where each row is one of the vectors, in the order given. Then take the m by m minors, that is, the determinants of all m by m submatrices (formed by choosing m out of the n columns) of M, each multiplied by (−1)p where p is the "parity" of the corresponding column subset. List these minors in some canonical order, to get a vector with choose(n,m) coordinates. These numbers identify the linear subspace spanned by the given vectors --- uniquely, except by a common scale factor. In projective geometry these are called the Grassman or Plücker coordinates of that subspace; I don't know what they are called in linear algebra.
The parity p is the number of column swaps that one must perform to bring the chosen columns to columns 1..m, without changing their relative order or that of the remaining columns.
In particular, if m = 2 and n = 3, and one lists the determinants in the order {2,3}, {1,3}, {1,2}, one gets the standard 3D cross product; the respective parities are 2, 1, 0.
If m=1 and n=2, one gets the formula for rotating a 2D vector by 90 degrees.
If m=n−1, as you noted, one gets another n-vector that is orthogonal to the given ones. If m=n one gets a single number, the determinant of M; which is the volume of the parallelotope whose sides are the given vectors, with a sign that depends on their n-dimensional handedness (in the given order).
All the best, --Jorge Stolfi (talk) 22:17, 9 December 2009 (UTC)
• As for your other questions: in linear algebra 'vectors' are either abstract objects defined indirectly by their properties (the usual mathematicians' view), or lists of real numbers (a more pdestrian, let's say "engineers'" view). Analytic geometry ties algebra and linear algebra to geometry, usually by choosing a Cartesian coordinate system, mapping points to Cartesian triplets, and then using algebra and engineers' linear algebra on those. However one can also define a vector in geometric terms (as an Euclidean "translation", for example) and use abstract linear algebra on them, without ever choosing a coordinate system. Of course, if you want coordinates out, you need coordinates in. In abstract liner algebra one can define an abstract dot product too, as being a binary operation with certain properties. Once you have a dot product you can define "distance" and "perpendicular", and then you can do with abstract vectors anything that you can do in Euclidean geometry, for any dimension n --- all abstractly. (But you don't get handedness yet; for that you need an n-ary "mixed product" operation, the n by n signed volume determinant, in the engineers' view.) You can define what a "rotation" is in abstract linear algebra, and "compute" its effect from given data (as in the Euclidean rotation example above), all without coordinates; but again, if you want coordinates out, you need coordinates in. Hope it helps. All the best, --Jorge Stolfi (talk) 22:54, 9 December 2009 (UTC)

## Inclination or elevation, which should come first

I have reworded the lead to define the 'inclination' variant first, then 'elevation'. The rest of the article uses mostly the inclination variant, and that seems to be the preference of some editors (however the 'elevation' fans may have been a silent majority). On the other hand, the 'inclination' variant is a bit more awkward to define, since one needs to introduce both the zenith and the reference plane; whereas to define the 'elevation' variant one needs only the reference plane.

I think the inclination variant is pretty unnatural for the reason you stated (it requires a zenith not otherwise needed). I have only encountered elevation variants (I think). Both the world coordinate system and the celestial coordinate system are elevational, and because of their importance and familiarity I would recommend that they should come first in an article that attempts to be generally applicable like this one does. By the way, the celestial coordinate system uses the term declination, not elevation, and not inclination. Dlw20070716 (talk) 07:58, 19 July 2011 (UTC)
I first encountered elevation, in undergraduate studies. Only later did I encounter instances of the inclination variant. I would summarise them as follows:
The inclination variant is convenient for emphasising symmetry about the pole. Conversely, the elevation is convenient for emphasising symmetry about the reference plane.
...Feel free to quote (and cite) me.  :-)
—DIV (138.194.11.244 (talk) 07:53, 29 August 2011 (UTC))

## Figure order

The fisrt three figures were recently reordered. I have partially restored their original order. The two line diagrams should come first since they illustrate the two variants of spherical coordinates (inclination vs. elevation) defined in the first paragraph. The raytraced figure is somewhat redundant, it does not show the coordinates proper (only their isosurfaces for one point), and requires a much longer caption to make sense.

On the other hand, I have retained the new order between the two line diagrams (with inclination before elevation), since I have also reordered the definitions of the two variants in the lead paragraph (see above). Note that swapping the two figures requires rewriting their captions.

All the best, --Jorge Stolfi (talk) 21:30, 14 February 2010 (UTC)

## Unique spherical coordinates for origin?

The section "If it is necessary to define a unique set of spherical coordinates for each point..." does not appear to deal with what happens at the origin. Could someone who knows comment on any approaches or conventions there might be for this? Thanks! Gwideman (talk) 04:33, 17 March 2010 (UTC)

## Describe to me why one of the angles has to be 0 to pie

The range of the angles bothers me.


0<r< 0<θ —Preceding unsigned comment added by 168.18.148.126 (talk) 23:16, 11 August 2010 (UTC)

Consider longitude and latitude, longitude goes from 180° east to 180° west, i.e. -π to π, or equivalently 0 to 2 *π. Latitude goes from 90° south to 90° north, i.e -π/2 to π/2. If you measure the angle from the north pole then this will be 0 to π. If they both went 0 to 2π the you would actually cover the sphere twice.--Salix (talk): 23:29, 11 August 2010 (UTC)

## Uniqueness of Representation

The section on unique representations is not quite correct. For example, $r = 1$ and $\theta = 0$, then we are identifying the north pole of the unit sphere for any value of $\phi$. Austinmohr (talk) 01:28, 2 September 2010 (UTC)

I fixed it.--Patrick (talk) 07:07, 2 September 2010 (UTC)

## Inconsistency with Dimensions

Using the stated formulas for converting cartesian to spherical coordinate returns results I believe are not consistent with dimensions or more precisely the spacial dimensions in "Dimensions". ("The spherical coordinates (r, θ, φ) of a point can be obtained from its Cartesian coordinates (x, y, z) by the formulas .... ")

e.g. converting (1,0,0) returns θ =π/2 and φ =π/2 , according to dimension it ought to be θ = 0. The inconsistency can well be in dimensions, θ and φ might adhere to a different convention with respect to their relation to x,y,y than this article.

Could somebody with knowledge please check and align if required?

Thanks, it is appreciated. Aporio (talk) 20:50, 13 October 2010 (UTC)

## dxdy

$dx=r\cos\theta\cos\phi d\theta-r\sin\theta\sin\phi d\phi$

$dy=r\cos\theta\sin\phi d\theta+r\sin\theta\cos\phi d\phi$

then $dxdy=r^{2}\cos\theta\sin\theta\left(\cos^{2}\phi-\sin^{2}\phi\right)d\theta d\phi\ne r^{2}\cos\theta\sin\theta d\theta d\phi$ anyone knows the reason why like this???

anyway, just replace dA in one system with dA in another system. one dV with another dV. no other adjustment is required. Jackzhp (talk) 13:45, 28 March 2011 (UTC)
It's $dx\wedge dy$. See differential form. Sławomir Biały (talk) 14:54, 28 March 2011 (UTC)

## Distinguishment between inclination and elevation variables

The distinguishment would be achieved by using $\, \theta_{inc}$ (or $\, \theta$) and $\ \theta_{el}$ instead of using the same $\ \theta$'s in this article. Adding the description of the order of the 3D variables would be, furthermore, useful: $\ (r,\ \phi,\ \theta_{el})$ in a right-handed coordinate system and $\ (r,\ \theta_{el},\ \phi,)$ in a left-handed coordinate system. Kkddkkdd (talk) 11:07, 21 May 2011 (UTC)

As evidenced by all the discussion above, this article has major problems, mostly because of conflicting usages by various authors and communities. That, and it has no references at all (and nobody so far seems to have noticed because of all the other problems)!

Because of all the conflicting conventions, I recommend turning it into a disambiguation-like page and redirecting to pages that better discuss specific spherical coordinate systems as they are actually used in specific systems, like geographical location, mathematics, computer graphics, etc. The page as it stands is too confusing and just plain erroneous to be useful. It would be useful to have an article that covers the conventions used in mathematics if done well, but that appears to be hopeless because of the confusion reigning therein. Dlw20070716 (talk) 07:44, 19 July 2011 (UTC)

## Incorrect Formula from Cartesian to Spherical

The spherical coordinates (r, θ, φ) of a point can be obtained from its Cartesian coordinates (x, y, z) by the formulae

$r=\sqrt{x^2+y^2+z^2}$
$\theta=\cos^{-1} \left( \frac{z}{r}\right)$
$\varphi = \tan^{-1} \left( \frac{y}{x} \right)$

Wolfram disagrees with this here: http://mathworld.wolfram.com/SphericalCoordinates.html

Images:

http://mathworld.wolfram.com/images/equations/SphericalCoordinates/Inline33.gif

http://mathworld.wolfram.com/images/equations/SphericalCoordinates/Inline36.gif

http://mathworld.wolfram.com/images/equations/SphericalCoordinates/Inline39.gif — Preceding unsigned comment added by 129.116.33.62 (talk) 21:34, 5 September 2011 (UTC)

-No wolfram and this article are agreeing. You just need to study up on your trigonometry to understand the relation. Cos(theta) = Adjacent Leg of triangle divided by hypotenuse of triangle. So cos(theta) = z/r and theta = cos-1(z/r) — Preceding unsigned comment added by 99.137.50.90 (talk) 12:35, 23 October 2012 (UTC)
Well, in our version of these formulae theta and phi are swapped compared to wolfram. But that is not incorrect, there are two common ways to define them, as explained in our article. We also explicitly state which of the definitions we use for the formulae just above them. I think that's all we can do. — HHHIPPO 17:22, 23 October 2012 (UTC)
My apologies for not knowing the proper way to point this out to you WikiPeople, but it seems the "theta = arccos(-z/r)" equation should read "theta = arccos(z/r)", no? — Preceding unsigned comment added by 71.237.117.249 (talk) 16:09, 28 December 2013 (UTC)

## Need to deal in the main with the main topic

The main topic here is the mathematical spherical coordinate system. It should not have the stuff from celestial coordinate system] and geographical coordinate system mixed into the main article. There are separate articles on those. I think it is a good idea to have a separate section on those pointing out the differences from what is done in maths but this article has become a mess by mixing them into the main article. Dmcq (talk) 20:08, 25 March 2012 (UTC)

Agreed. a13ean (talk) 20:13, 25 March 2012 (UTC)