Talk:Spherical coordinate system

848:

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 Knowledge 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, --

3396:), 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? 3555:

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

767:

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.

3298:. 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. 108: 98: 77: 44: 5108: 1661:

latitude and altitude are defined (reference spheroid and all that). Outside geography, theta and phi seems to be used for and — 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.

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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.

3294:"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 35: 899:

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 Knowledge's role to make it so. All the best, --

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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.

2070:. 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. 2454: 3571:

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

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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

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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

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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

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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

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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'

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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"

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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

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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

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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

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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,

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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.

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as if they were part of the definition. Actually there are several conventions in use, and Knowledge 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.

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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

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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

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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

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You insist that one "cannot use 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?

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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

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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

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However, there's a variant of the spherical coordinate system, which can actually be more practical in some applications (namely, when you don't want your vertical angle theta to be 0 on the z axis, but rather have theta be 0 on the XY plane). In this case, the definitions of the variables are as

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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,

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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

2100:) 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, -- 1660:

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

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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

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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

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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

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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,

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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.

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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.

3478:) 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. 3014: 1028:

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.

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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.

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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.

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appears to be incorrect for y=0. While for this equation to be correct it must be sgn(0) = +1, the convention is to set sgn(0)=0. If this is actually a mistake, the formula should be modified or a comment on the re-definition of the sgn-function should be provided.

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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)!

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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

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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.

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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).

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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!

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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.

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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.

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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)

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And I'm not just talking about which symbol (phi or theta) is used for which angle. For the purpose of this post, theta will be the vertical angle (the one off of the XY plane) and phi will be the horizontal angle (the one in the XY plane).

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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.

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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,

884:) 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.-- 4861:-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) 825:

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.

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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.

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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

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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?

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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.

1941:, 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 478:

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

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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,

2187:, 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: 5254: 2093:; 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. 1703:

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

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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.

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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

2886: 1416:"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°. 1188: 1135: 2742: 1486:

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.

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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. –

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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.

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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. --

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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.

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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

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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.

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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?

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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}.

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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

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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.

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The inclination variant is convenient for emphasising symmetry about the pole. Conversely, the elevation is convenient for emphasising symmetry about the reference plane.

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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?

4433: 4379: 3799: 2048:"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. 989:

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?

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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. –

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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.

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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

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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 "

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As always, the wikipedia standard is to use published conventions from reliable publications and document same with references (which this article fails to do).

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I first encountered elevation, in undergraduate studies. Only later did I encounter instances of the inclination variant. I would summarise them as follows:

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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

48: 5319: 607:) for polar coordinates, cylindrical and spherical coordinates are obtained by simply making the coordinate pair a coordinate triple (with the caveat that 154: 4239:{\displaystyle dxdy=r^{2}\cos \theta \sin \theta \left(\cos ^{2}\phi -\sin ^{2}\phi \right)d\theta d\phi \neq r^{2}\cos \theta \sin \theta d\theta d\phi } 3855:, as sort of originally shown. One can reverse two of these three to get the same results, not all three, AFAIK. It should be <-r, -polar, azimuth: --> 3095:

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.

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geometry. People have been and are doing some pretty advanced 3D geometry, including speherical coordinates, without reference to any of those concepts.

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If we were defining spherical coordinates *relative to a previously defined Cartesian coordinate system*, it would be natural to use the unit vectors

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Section on Unique representation had its symbols/positions for Azimuth and Polar angle mixed up: it wasn't following the convention of this article.

17: 5299: 3498:= 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. 130: 3657:

redundant, it does not show the coordinates proper (only their isosurfaces for one point), and requires a much longer caption to make sense.

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So please keep the original simple, direct, algebra-free, non-circular definition; there was nothing wrong with it. Please. All the best, --

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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, --

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parallel if they lie on the same plane but never meet. The existence of parallel lines is an axiom (much disputed until Lobachevski/Riemman).

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

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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

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I added the image at right. However, I don't know if I'm measuring the angles from the correct directions. Could someone fix the caption?

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Notice that where you previously had cos(theta) you now have sin(theta), and where you previously had sin(theta) you now have cos(theta).

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Cartesian ones. Linear algebra (vectors, dot product, cross product, etc.) are definitely NOT "the foundation of 3D geometry", only of

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for succinctness in references to it. There is a reference azimuth direction, also, more precisely a unit vector, which may be called

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and in this article (in the introduction, and right below the formulas in question). In both articles the azimuth angle is denoted by

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convention is not universal (if I am not mistaken, on planets other than the Earth longitudes are measured in the opposite sense). --

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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:

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confusing. Worse is the image on the right which shows the maths convention first. But it isn't the one followed up in the article.

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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

3035: 2874:{\displaystyle \phi =\arg({\hat {x}}\cdot {\hat {r}}+j({\hat {x}}\times {\hat {r}}\cdot {\hat {z}})),\,j={\sqrt {-1}}{\text{, or}}} 5075: 4703: 1343:

article uses phi for elevation and theta for azimuthal. Is there a wikipedia standard for these we could apply to all articles?

3885:("The spherical coordinates (r, θ, φ) of a point can be obtained from its Cartesian coordinates (x, y, z) by the formulas .... ") 3853:

Also, the idea of equating two points: with all 3 values in one point reversed is incorrect. That is, <r, polar, azimuth: -->

3372:

Right, I was talking about 3D rotation about an arbitrary axis and the derivation of the new coordinates of an arbitrary point (

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coordinates for each point; and, even then, there are several choices in common use. This issue is discussed in a later section.

1052:

Conventional spherical coordinates as described in this article are a combination of rotations along the z and y axes. However,

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

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Using the stated formulas for converting cartesian to spherical coordinate returns results I believe are not consistent with

1762:

Someone has introduced the name "directrix" for the reference direction on the equatorial plane (azimuth zero). However the

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anyway, just replace dA in one system with dA in another system. one dV with another dV. no other adjustment is required.

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Perhaps there should be different diagrams for the different coordinate systems. Unfortunately, someone has to draw them.

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

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PS. I added a note on the "position vector". However, note that it too is a concept of linear algebra that is useful

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is the "parity" of the corresponding column subset. List these minors in some canonical order, to get a vector with

1325:

The naming of the coordinates is a matter of convention. Different authors/systems may use different conventions. --

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True. This other coordinate system is mentioned in the third paragraph of the article. It is usually called the

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Agreed. This page is a mine-field of various conventions to the point of making the whole page utterly confusing.

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

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as its unit vectors, and which are just as universally mapped to the spherical coordinate system as I suggested.

5086: 4301: 3534:; which is the volume of the parallelotope whose sides are the given vectors, with a sign that depends on their 1194: 630: 3717: 3638: 5265: 5155: 4922: 3893:, θ and φ might adhere to a different convention with respect to their relation to x,y,y than this article. 1471: 1315: 1242: 219: 197: 4850: 4435:'s in this article. Adding the description of the order of the 3D variables would be, furthermore, useful: 1748: 1497: 1446: 1260: 4872: 1537: 1000: 881: 830: 809: 186:

I printed out this page and the images came out as black squares. get new images, these ones are garbage!

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

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in the Cartesian formulas was latitude? It is actually inclination, that is, 90° minus the latitude.) --

1674: 1403: 1355: 1034: 411:, that's OK, but please don't use uppercase - the convention is to use lowercase (as in the diagram). -- 63: 2079:

Moreover, the concept of "vector" is actually much harder to define than a spherical coordinate system.

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The transformation to Cartesian coordinates, which is specified in the wiki article, is as follows.

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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A globe showing the radial distance, polar angle and azimuth angle of a point P. In this image,

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Maybe he's referring to the fact that the equations for theta and phi appear to be swapped? See

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This article itself states that the international recommended standard for spherical polars is

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Phi is the angle from the x axis in the XY plane and can go from either 0 to 2pi or -pi to pi

5018:

Phi is the angle from the x axis in the XY plane and can go from either 0 to 2pi or -pi to pi

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is the number of column swaps that one must perform to bring the chosen columns to columns 1..

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Indeed, I am a bit unhappy with the current "Definition" section, because it uses the letter

924:

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

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The errors are everywhere! I just fixed a handful of them. Somebody destroyed this page.

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

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The version of spherical coordinates talked about here in Knowledge works like this:

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More generally, we must avoid putting things in the definition (such as the letters

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The main article would be improved considerably by changing the images to pairs of

2620:{\displaystyle x=OP\cdot {\hat {x}},\,y=OP\cdot {\hat {y}},\,z=OP\cdot {\hat {z}}} 1563:; the latter is spherical inclination or elevation. Thus the conversion preserves 1253: 1082:

Conversely, Cartesian coordinates may be retrieved from spherical coordinates by:

5129: 4971: 4077:{\displaystyle dy=r\cos \theta \sin \phi d\theta +r\sin \theta \cos \phi d\phi } 3993:{\displaystyle dx=r\cos \theta \cos \phi d\theta -r\sin \theta \sin \phi d\phi } 3900: 3020: 1880: 1637: 885: 791: 316: 126: 4838:

http://mathworld.wolfram.com/images/equations/SphericalCoordinates/Inline39.gif

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

4828:

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

211: 103: 3801:, then we are identifying the north pole of the unit sphere for any value of 4956: 3890: 3880: 3559:--- all abstractly. (But you don't get handedness yet; for that you need an 2062:"; and, for instance, we understand that this sentence is defining not only 1763: 1591:; it's fixed now.) Also, the spherical radial coordinate is denoted here by 3749:

The section on unique representations is not quite correct. For example,

3488:, without changing their relative order or that of the remaining columns. 3434:

where each row is one of the vectors, in the order given. Then take the

16:

Previous discussions pertaining to the old article have been archived to

1435:φ is referred to as the azimuth and θ is referred to elevation, right? 1382:

Methinks the coordinate system on this page needs to be standardized. --

1883:," and it should be called exactly that, and may be impartially named 3124:

You are assuming that one can or should define spherical coordinates

1875:

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

5249:{\displaystyle \phi =sgn(y)\arccos {\frac {x}{\sqrt {x^{2}+y^{2}}}}} 1378:

http://tutorial.math.lamar.edu/Classes/CalcIII/SphericalCoords.aspx

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Theta is the angle from the XY plane, and goes from -pi/2 to pi/2

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Could somebody with knowledge please check and align if required?

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to represent the reference directions and (b) the position vector

1949:, relative to which the coordinates of a point, represented by a 3508:=2, one gets the formula for rotating a 2D vector by 90 degrees. 2510:{\displaystyle {\hat {\theta }}={\hat {\phi }}\times {\hat {r}}} 4946:

spherical coordinate system. It should not have the stuff from

4808:{\displaystyle \varphi =\tan ^{-1}\left({\frac {y}{x}}\right)} 3088:{\displaystyle {\hat {r}},\,{\hat {\theta }},\,{\hat {\phi }}} 740:

with both cylindrical or polar, i.e., the projection onto the

28: 4750:{\displaystyle \theta =\cos ^{-1}\left({\frac {z}{r}}\right)} 3277:

Dot product, cross product, handedness; abstract and concrete

5052:

The transformation to Cartesian coordinates is as follows.

4320:

Distinguishment between inclination and elevation variables

3883:

or more precisely the spacial dimensions in "Dimensions".

2730:{\displaystyle \theta =\arccos({\hat {r}}\cdot {\hat {z}})} 2121:

My reason for the introduction and use of (a) unit vectors

5015:

Theta is the angle from the z axis, and goes from 0 to pi

3129:

that is not a reasonable assumption for Knowledge readers.

1599:, here and in the other article. Am I missing something? 1183:{\displaystyle {y}=r\,\sin \varphi \,\sin \theta \quad } 1130:{\displaystyle {x}=r\,\sin \varphi \,\cos \theta \quad } 5003: 3699:

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

3028:

Cartesian system is the reuse of two of its coordinate

4820:

http://mathworld.wolfram.com/SphericalCoordinates.html

2242:{\displaystyle {\hat {y}}={\hat {z}}\times {\hat {x}}} 5183: 4764: 4706: 4642: 4495: 4441: 4419: 4387: 4365: 4330: 4270: 4092: 4007: 3923: 3807: 3781: 3755: 3233: 3204: 3175: 3038: 2889: 2745: 2681: 2635: 2525: 2464: 2321: 2257: 2196: 2156: 2127: 2021: 1992: 1963: 1918: 1889: 1806:

No "offense" at all; your contributions were helpful.

1296: 1197: 1144: 1091: 746: 683: 633: 613: 581: 549: 511: 485: 395: 373: 351: 239:(From cylindrical article) Line and volume elements 125:, a collaborative effort to improve the coverage of 3410:Cross products can be generalized to any dimension 5248: 4807: 4749: 4691: 4538: 4481: 4427: 4405: 4373: 4351: 4288: 4238: 4076: 3992: 3854:is not equal to <-r, -polar, azimuth + 180: --> 3813: 3793: 3767: 3522:-vector that is orthogonal to the given ones. If 3248: 3219: 3190: 3097:Also, as I've shown above, two orthogonal vectors 3087: 3008: 2873: 2729: 2665: 2619: 2509: 2448: 2305: 2241: 2171: 2142: 2050:You and I are used to 'algebraic prose' like "let 2036: 2007: 1978: 1933: 1904: 1302: 1227: 1182: 1129: 755: 732: 669: 619: 599: 567: 535: 497: 403: 381: 359: 3594:Inclination or elevation, which should come first 733:{\displaystyle \rho ={\sqrt {x^{2}+y^{2}+z^{2}}}} 389:. If you want to use the other TeX lowercase phi 505:), cylindrical coordinates would be of the form 5305:Knowledge level-5 vital articles in Mathematics 5004:There's actually 2 spherical coordinate systems 4324:The distinguishment would be achieved by using 1870:Linear algebra, analytic geometry, or geometry? 1254:https://keisan.casio.com/exec/system/1359534351 248:The volume element is dV = r\,dr\,d\theta\,dz. 1310:. At least my math classes nomiated them so. - 4692:{\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}}} 4601:Incorrect Formula from Cartesian to Spherical 4539:{\displaystyle \ (r,\ \theta _{el},\ \phi ,)} 3538:-dimensional handedness (in the given order). 3530:one gets a single number, the determinant of 2306:{\displaystyle {\hat {r}}={\frac {OP}{|OP|}}} 1595:, while the cylindrical radial coordinate is 8: 5085:, and is the subject of a separate article. 4938:Need to deal in the main with the main topic 4482:{\displaystyle \ (r,\ \phi ,\ \theta _{el})} 3139:Eucliean geometry). You cannot escape that. 2077:; but I will not use that here, either 8-). 4561:This article tries to cover too much ground 1228:{\displaystyle {z}=r\,\cos \varphi .\quad } 670:{\displaystyle \rho ={\sqrt {x^{2}+y^{2}}}} 4246:anyone knows the reason why like this??? 1461: 71: 5237: 5224: 5214: 5182: 4791: 4775: 4763: 4733: 4717: 4705: 4681: 4668: 4655: 4649: 4641: 4513: 4494: 4467: 4440: 4418: 4394: 4386: 4364: 4337: 4329: 4269: 4200: 4164: 4145: 4112: 4091: 4006: 3922: 3806: 3780: 3754: 3630:...Feel free to quote (and cite) me. :-) 3442:minors, that is, the determinants of all 3235: 3234: 3232: 3206: 3205: 3203: 3177: 3176: 3174: 3074: 3073: 3057: 3056: 3040: 3039: 3037: 2995: 2994: 2980: 2979: 2965: 2964: 2927: 2926: 2912: 2911: 2888: 2866: 2856: 2828: 2827: 2813: 2812: 2798: 2797: 2777: 2776: 2762: 2761: 2744: 2713: 2712: 2698: 2697: 2680: 2652: 2651: 2634: 2606: 2605: 2574: 2573: 2542: 2541: 2524: 2496: 2495: 2481: 2480: 2466: 2465: 2463: 2441: 2430: 2429: 2415: 2414: 2409: 2401: 2390: 2389: 2375: 2374: 2369: 2356: 2355: 2341: 2340: 2337: 2323: 2322: 2320: 2295: 2284: 2273: 2259: 2258: 2256: 2228: 2227: 2213: 2212: 2198: 2197: 2195: 2158: 2157: 2155: 2129: 2128: 2126: 2023: 2022: 2020: 1994: 1993: 1991: 1965: 1964: 1962: 1920: 1919: 1917: 1891: 1890: 1888: 1295: 1252:I think this is the case. Take a look at 1198: 1196: 1145: 1143: 1092: 1090: 921:I don't think this statement is correct: 745: 722: 709: 696: 690: 682: 659: 646: 640: 632: 612: 580: 548: 510: 484: 394: 372: 350: 5170:I may be entirely mistaken here but the 4489:in a right-handed coordinate system and 3679:Unique spherical coordinates for origin? 18:Talk:Spherical coordinate system/Archive 5295:Knowledge vital articles in Mathematics 4366: 4331: 3071: 3054: 2941: 2848: 2588: 2556: 1209: 1167: 1156: 1114: 1103: 399: 377: 355: 73: 32: 4617:) of a point can be obtained from its 3282:Two comments you made still bother me: 5310:B-Class vital articles in Mathematics 3703:The range of the angles bothers me. 1551:Please check the conventions used in 959:Radians should be used, not degrees. 543:. Whereas starting with the notation 7: 5165:Sign-flaw in current cartesian-: --> 5049:Rho is the distance from the origin 5021:Rho is the distance from the origin 4546:in a left-handed coordinate system. 3563:-ary "mixed product" operation, the 2666:{\displaystyle r=OP\cdot {\hat {r}}} 1269:Red the introduction. Don't "think". 345:But TeX isn't showing a capital phi 119:This article is within the scope of 3518:−1, as you noted, one gets another 3422:of vectors as follows: assemble an 62:It is of interest to the following 5320:High-priority mathematics articles 4818:Wolfram disagrees with this here: 3019:All it takes is the definition of 352: 258:I have added surface and volume. – 14: 3220:{\displaystyle {\hat {\varphi }}} 1575:, of vice versa. (There was one 536:{\displaystyle (\rho ,z,\theta )} 459:think we have to live with it. -- 139:Knowledge:WikiProject Mathematics 5290:Knowledge level-5 vital articles 5058:y = rho * cos(theta) * sin(phi) 5055:x = rho * cos(theta) * cos(phi) 5030:y = rho * sin(theta) * sin(phi) 5027:x = rho * sin(theta) * cos(phi) 3462:, each multiplied by (−1) where 3450:submatrices (formed by choosing 3249:{\displaystyle {\hat {\theta }}} 142:Template:WikiProject Mathematics 106: 96: 75: 42: 33: 4352:{\displaystyle \,\theta _{inc}} 3859:I have made these corrections. 3032:. The reason why we cannot use 1508:Please point out the errors. -- 600:{\displaystyle (\rho ,\theta )} 367:, it's showing a lowercase phi 159:This article has been rated as 5300:B-Class level-5 vital articles 5205: 5199: 5083:geographical coordinate system 4952:geographical coordinate system 4533: 4498: 4476: 4444: 4406:{\displaystyle \ \theta _{el}} 3240: 3211: 3182: 3079: 3062: 3045: 3000: 2985: 2970: 2932: 2917: 2842: 2839: 2833: 2818: 2803: 2794: 2782: 2767: 2758: 2724: 2718: 2703: 2694: 2657: 2611: 2579: 2547: 2501: 2486: 2471: 2435: 2420: 2402: 2395: 2380: 2370: 2361: 2346: 2328: 2296: 2285: 2264: 2233: 2218: 2203: 2163: 2134: 2028: 1999: 1970: 1925: 1896: 1374:Yeah, they look swapped to me. 874:prolate spheroidal coordinates 795:21:32, 12 September 2007 (UTC) 594: 582: 562: 550: 530: 512: 25:01:25, 18 September 2006 (UTC) 1: 5097:12:39, 29 December 2016 (UTC) 5076:08:10, 29 December 2016 (UTC) 4927:16:09, 28 December 2013 (UTC) 4855:21:34, 5 September 2011 (UTC) 3875:Inconsistency with Dimensions 3846:07:07, 2 September 2010 (UTC) 3831:01:28, 2 September 2010 (UTC) 3674:21:30, 14 February 2010 (UTC) 3354:What you ask is a problem of 1718:18:48, 24 November 2009 (UTC) 1650:15:27, 12 February 2010 (UTC) 1553:cylindrical coordinate system 1518:18:49, 24 November 2009 (UTC) 1426:23:06, 4 September 2011 (UTC) 1335:18:53, 24 November 2009 (UTC) 1223: 1178: 1125: 1022:20:12, 24 November 2009 (UTC) 1012:may start with any angle. -- 1005:11:30, 19 February 2009 (UTC) 979:20:12, 24 November 2009 (UTC) 909:20:45, 14 February 2010 (UTC) 894:19:56, 12 February 2010 (UTC) 814:11:38, 27 November 2008 (UTC) 498:{\displaystyle \phi ,\theta } 298:10:15, 16 November 2006 (UTC) 133:and see a list of open tasks. 5315:B-Class mathematics articles 5271:01:37, 13 January 2023 (UTC) 5160:17:48, 25 October 2022 (UTC) 4899:17:22, 23 October 2012 (UTC) 4877:12:35, 23 October 2012 (UTC) 3909:20:50, 13 October 2010 (UTC) 3745:Uniqueness of Representation 3582:22:54, 9 December 2009 (UTC) 3550:22:17, 9 December 2009 (UTC) 3406:21:48, 9 December 2009 (UTC) 3368:19:36, 9 December 2009 (UTC) 3347:18:30, 9 December 2009 (UTC) 3328:21:48, 9 December 2009 (UTC) 3267:16:03, 9 December 2009 (UTC) 3151:15:29, 9 December 2009 (UTC) 3112:08:56, 9 December 2009 (UTC) 2110:21:26, 8 December 2009 (UTC) 2054:be a direction on the plane 1862:16:32, 8 December 2009 (UTC) 1847:15:28, 8 December 2009 (UTC) 1802:08:24, 8 December 2009 (UTC) 1776:05:53, 8 December 2009 (UTC) 1695:Error in Cartesian formulas? 1612:03:25, 8 December 2009 (UTC) 1542:00:57, 8 December 2009 (UTC) 1070:10:50, 4 December 2007 (UTC) 939:19:01, 18 January 2007 (UTC) 858:15:59, 16 January 2010 (UTC) 835:04:27, 16 January 2010 (UTC) 320:18:04, 4 February 2007 (UTC) 309:12:37, 11 January 2007 (UTC) 254:Even mathworld has dS...... 5141:21:16, 10 August 2018 (UTC) 4984:Agreed. This main topic is 4948:celestial coordinate system 4942:The main topic here is the 4605:The spherical coordinates ( 4289:{\displaystyle dx\wedge dy} 3899:Thanks, it is appreciated. 3739:23:29, 11 August 2010 (UTC) 3722:23:16, 11 August 2010 (UTC) 3643:07:53, 29 August 2011 (UTC) 3358:geometry. All the best, -- 1753:17:08, 10 August 2010 (UTC) 1502:22:58, 4 October 2009 (UTC) 568:{\displaystyle (r,\theta )} 5336: 4980:20:13, 25 March 2012 (UTC) 4965:20:08, 25 March 2012 (UTC) 4413:instead of using the same 4312:14:54, 28 March 2011 (UTC) 4260:13:45, 28 March 2011 (UTC) 3694:04:33, 17 March 2010 (UTC) 3191:{\displaystyle {\hat {r}}} 2172:{\displaystyle {\hat {x}}} 2143:{\displaystyle {\hat {z}}} 2037:{\displaystyle {\hat {z}}} 2008:{\displaystyle {\hat {y}}} 1979:{\displaystyle {\hat {x}}} 1934:{\displaystyle {\hat {x}}} 1905:{\displaystyle {\hat {z}}} 1431:θ is referred to elevation 1364:19:54, 18 March 2010 (UTC) 949:11:37, 10 April 2007 (UTC) 778:made clear on the article. 464:11:43, 10 April 2007 (UTC) 438:11:37, 10 April 2007 (UTC) 416:11:26, 10 April 2007 (UTC) 404:{\displaystyle \varphi \,} 333:11:13, 10 April 2007 (UTC) 263:11:13, 10 April 2007 (UTC) 4580:07:44, 19 July 2011 (UTC) 4428:{\displaystyle \ \theta } 4374:{\displaystyle \,\theta } 3794:{\displaystyle \theta =0} 3729:cover the sphere twice.-- 3614:07:58, 19 July 2011 (UTC) 3296:The Ambidextrous Universe 2075:oriented projective space 1734:08:22, 19 July 2011 (UTC) 1679:08:51, 19 July 2011 (UTC) 1476:13:19, 5 March 2018 (UTC) 1408:08:57, 19 July 2011 (UTC) 1279:16:39, 9 April 2022 (UTC) 1265:07:11, 8 April 2022 (UTC) 1247:18:34, 9 March 2008 (UTC) 1039:09:05, 19 July 2011 (UTC) 304:That is very confusing. ( 224:22:14, 15 June 2009 (UTC) 158: 91: 70: 4556:11:07, 21 May 2011 (UTC) 2119:graphics that are there. 1451:13:01, 22 May 2008 (UTC) 1392:20:56, 18 May 2011 (UTC) 1320:18:23, 13 May 2008 (UTC) 964:18:55, 9 July 2007 (UTC) 772:18:44, 9 July 2007 (UTC) 165:project's priority scale 5270:spherical_formula": --> 4998:17:05, 3 May 2023 (UTC) 4595:18:30, 3 May 2023 (UTC) 3869:23:25, 4 May 2023 (UTC) 3572:helps. All the best, -- 1583:which should have been 1303:{\displaystyle \theta } 882:Bispherical coordinates 382:{\displaystyle \phi \,} 360:{\displaystyle \Phi \,} 202:08:26, 5 May 2008 (UTC) 122:WikiProject Mathematics 5285:B-Class vital articles 5250: 5124: 4809: 4751: 4693: 4540: 4483: 4429: 4407: 4375: 4353: 4290: 4240: 4078: 3994: 3815: 3795: 3769: 3250: 3221: 3192: 3089: 3010: 2875: 2731: 2667: 2621: 2511: 2450: 2307: 2243: 2173: 2144: 2038: 2009: 1980: 1935: 1906: 1571:must be computed from 1304: 1229: 1184: 1131: 1075:Angle symbols swapped? 757: 734: 671: 621: 601: 569: 537: 499: 405: 383: 361: 5251: 5110: 5061:z = rho * sin(theta) 5033:z = rho * cos(theta) 4810: 4752: 4694: 4619:Cartesian coordinates 4541: 4484: 4430: 4408: 4376: 4354: 4291: 4241: 4079: 3995: 3816: 3814:{\displaystyle \phi } 3796: 3770: 3251: 3222: 3193: 3090: 3011: 2876: 2732: 2668: 2622: 2512: 2451: 2308: 2244: 2174: 2145: 2039: 2010: 1981: 1936: 1907: 1305: 1230: 1185: 1132: 758: 735: 672: 622: 620:{\displaystyle \rho } 602: 570: 538: 500: 406: 384: 362: 49:level-5 vital article 5181: 5174:conversion formula: 4762: 4704: 4640: 4493: 4439: 4417: 4385: 4363: 4328: 4268: 4090: 4005: 3921: 3805: 3779: 3753: 3231: 3202: 3173: 3036: 2887: 2743: 2679: 2633: 2523: 2462: 2319: 2255: 2194: 2154: 2125: 2019: 1990: 1961: 1916: 1887: 1294: 1195: 1142: 1089: 870:elliptic coordinates 744: 681: 631: 611: 579: 547: 509: 483: 393: 371: 349: 145:mathematics articles 3768:{\displaystyle r=1} 3706:0<r< 0<θ 878:Bipolar coordinates 269:American convention 5246: 5125: 4805: 4747: 4689: 4633:) by the formulae 4536: 4479: 4425: 4403: 4371: 4367: 4349: 4332: 4286: 4236: 4074: 3990: 3811: 3791: 3765: 3490:In particular, if 3246: 3217: 3188: 3085: 3072: 3055: 3006: 2942: 2871: 2849: 2727: 2663: 2617: 2589: 2557: 2507: 2446: 2303: 2239: 2169: 2140: 2034: 2005: 1976: 1947:frame of reference 1931: 1902: 1300: 1290:be invariant with 1225: 1224: 1210: 1180: 1179: 1168: 1157: 1127: 1126: 1115: 1104: 756:{\displaystyle xy} 753: 730: 667: 617: 597: 565: 533: 495: 401: 400: 379: 378: 357: 356: 114:Mathematics portal 58:content assessment 5244: 5243: 5166:spherical formula 5119:equals 360°, and 4917:comment added by 4894: 4867:comment added by 4845:comment added by 4799: 4741: 4687: 4526: 4508: 4497: 4462: 4454: 4443: 4421: 4389: 4298:differential form 3712:comment added by 3540:All the best, -- 3243: 3214: 3185: 3082: 3065: 3048: 3003: 2988: 2973: 2935: 2920: 2869: 2864: 2836: 2821: 2806: 2785: 2770: 2721: 2706: 2660: 2614: 2582: 2550: 2504: 2489: 2474: 2444: 2438: 2423: 2412: 2411:, unit vector in 2407: 2398: 2383: 2364: 2349: 2331: 2301: 2267: 2236: 2221: 2206: 2166: 2137: 2031: 2002: 1973: 1943:orthonormal basis 1928: 1899: 1653: 1636:comment added by 1532:comment added by 1492:comment added by 1478: 1466:comment added by 1453: 1441:comment added by 1367: 1350:comment added by 1058:Talk:Euler angles 995:comment added by 728: 665: 204: 192:comment added by 179: 178: 175: 174: 171: 170: 5327: 5255: 5253: 5252: 5247: 5245: 5242: 5241: 5229: 5228: 5219: 5215: 5139: 5137: 5132: 5122: 5118: 5114: 5089: 4929: 4896: 4892: 4879: 4857: 4814: 4812: 4811: 4806: 4804: 4800: 4792: 4783: 4782: 4756: 4754: 4753: 4748: 4746: 4742: 4734: 4725: 4724: 4698: 4696: 4695: 4690: 4688: 4686: 4685: 4673: 4672: 4660: 4659: 4650: 4545: 4543: 4542: 4537: 4525: 4521: 4520: 4507: 4496: 4488: 4486: 4485: 4480: 4475: 4474: 4461: 4453: 4442: 4434: 4432: 4431: 4426: 4420: 4412: 4410: 4409: 4404: 4402: 4401: 4388: 4380: 4378: 4377: 4372: 4358: 4356: 4355: 4350: 4348: 4347: 4304: 4295: 4293: 4292: 4287: 4245: 4243: 4242: 4237: 4205: 4204: 4180: 4176: 4169: 4168: 4150: 4149: 4117: 4116: 4083: 4081: 4080: 4075: 3999: 3997: 3996: 3991: 3820: 3818: 3817: 3812: 3800: 3798: 3797: 3792: 3774: 3772: 3771: 3766: 3724: 3664:All the best, -- 3255: 3253: 3252: 3247: 3245: 3244: 3236: 3226: 3224: 3223: 3218: 3216: 3215: 3207: 3197: 3195: 3194: 3189: 3187: 3186: 3178: 3094: 3092: 3091: 3086: 3084: 3083: 3075: 3067: 3066: 3058: 3050: 3049: 3041: 3015: 3013: 3012: 3007: 3005: 3004: 2996: 2990: 2989: 2981: 2975: 2974: 2966: 2937: 2936: 2928: 2922: 2921: 2913: 2880: 2878: 2877: 2872: 2870: 2867: 2865: 2857: 2838: 2837: 2829: 2823: 2822: 2814: 2808: 2807: 2799: 2787: 2786: 2778: 2772: 2771: 2763: 2736: 2734: 2733: 2728: 2723: 2722: 2714: 2708: 2707: 2699: 2672: 2670: 2669: 2664: 2662: 2661: 2653: 2626: 2624: 2623: 2618: 2616: 2615: 2607: 2584: 2583: 2575: 2552: 2551: 2543: 2516: 2514: 2513: 2508: 2506: 2505: 2497: 2491: 2490: 2482: 2476: 2475: 2467: 2455: 2453: 2452: 2447: 2445: 2442: 2440: 2439: 2431: 2425: 2424: 2416: 2413: 2410: 2408: 2406: 2405: 2400: 2399: 2391: 2385: 2384: 2376: 2373: 2367: 2366: 2365: 2357: 2351: 2350: 2342: 2338: 2333: 2332: 2324: 2312: 2310: 2309: 2304: 2302: 2300: 2299: 2288: 2282: 2274: 2269: 2268: 2260: 2248: 2246: 2245: 2240: 2238: 2237: 2229: 2223: 2222: 2214: 2208: 2207: 2199: 2178: 2176: 2175: 2170: 2168: 2167: 2159: 2149: 2147: 2146: 2141: 2139: 2138: 2130: 2043: 2041: 2040: 2035: 2033: 2032: 2024: 2014: 2012: 2011: 2006: 2004: 2003: 1995: 1985: 1983: 1982: 1977: 1975: 1974: 1966: 1940: 1938: 1937: 1932: 1930: 1929: 1921: 1911: 1909: 1908: 1903: 1901: 1900: 1892: 1837:All the best, -- 1652: 1630: 1544: 1504: 1482:Immediate Action 1436: 1366: 1344: 1309: 1307: 1306: 1301: 1234: 1232: 1231: 1226: 1202: 1189: 1187: 1186: 1181: 1149: 1136: 1134: 1133: 1128: 1096: 1045:Comparsion with 1007: 985:Range of azimuth 762: 760: 759: 754: 739: 737: 736: 731: 729: 727: 726: 714: 713: 701: 700: 691: 676: 674: 673: 668: 666: 664: 663: 651: 650: 641: 626: 624: 623: 618: 606: 604: 603: 598: 574: 572: 571: 566: 542: 540: 539: 534: 504: 502: 501: 496: 410: 408: 407: 402: 388: 386: 385: 380: 366: 364: 363: 358: 187: 147: 146: 143: 140: 137: 116: 111: 110: 100: 93: 92: 87: 79: 72: 55: 46: 45: 38: 37: 29: 5335: 5334: 5330: 5329: 5328: 5326: 5325: 5324: 5275: 5274: 5233: 5220: 5179: 5178: 5168: 5148: 5135: 5130: 5128: 5120: 5116: 5112: 5105: 5087: 5006: 4940: 4912: 4889: 4862: 4840: 4787: 4771: 4760: 4759: 4729: 4713: 4702: 4701: 4677: 4664: 4651: 4638: 4637: 4603: 4563: 4509: 4491: 4490: 4463: 4437: 4436: 4415: 4414: 4390: 4383: 4382: 4361: 4360: 4333: 4326: 4325: 4322: 4302: 4266: 4265: 4196: 4160: 4141: 4140: 4136: 4108: 4088: 4087: 4003: 4002: 3919: 3918: 3916: 3877: 3803: 3802: 3777: 3776: 3751: 3750: 3747: 3707: 3704: 3701: 3681: 3654: 3596: 3414:and any number 3279: 3229: 3228: 3200: 3199: 3171: 3170: 3034: 3033: 2885: 2884: 2741: 2740: 2677: 2676: 2631: 2630: 2521: 2520: 2460: 2459: 2368: 2339: 2317: 2316: 2283: 2275: 2253: 2252: 2192: 2191: 2185:from the outset 2152: 2151: 2123: 2122: 2017: 2016: 1988: 1987: 1959: 1958: 1951:position vector 1914: 1913: 1885: 1884: 1877: 1872: 1760: 1699:Anonymous user 1697: 1631: 1527: 1487: 1484: 1433: 1345: 1341:triple integral 1292: 1291: 1193: 1192: 1140: 1139: 1087: 1086: 1077: 1050: 990: 987: 957: 919: 742: 741: 718: 705: 692: 679: 678: 655: 642: 629: 628: 609: 608: 577: 576: 545: 544: 507: 506: 481: 480: 391: 390: 369: 368: 347: 346: 271: 234: 184: 144: 141: 138: 135: 134: 112: 105: 85: 56:on Knowledge's 53: 43: 12: 11: 5: 5333: 5331: 5323: 5322: 5317: 5312: 5307: 5302: 5297: 5292: 5287: 5277: 5276: 5262:141.14.162.194 5257: 5256: 5240: 5236: 5232: 5227: 5223: 5218: 5213: 5210: 5207: 5204: 5201: 5198: 5195: 5192: 5189: 5186: 5167: 5163: 5152:144.48.134.241 5147: 5144: 5104: 5101: 5100: 5099: 5088:Sławomir Biały 5036: 5005: 5002: 5001: 5000: 4982: 4939: 4936: 4935: 4934: 4933: 4932: 4931: 4930: 4919:71.237.117.249 4904: 4903: 4902: 4901: 4881: 4880: 4816: 4815: 4803: 4798: 4795: 4790: 4786: 4781: 4778: 4774: 4770: 4767: 4757: 4745: 4740: 4737: 4732: 4728: 4723: 4720: 4716: 4712: 4709: 4699: 4684: 4680: 4676: 4671: 4667: 4663: 4658: 4654: 4648: 4645: 4602: 4599: 4598: 4597: 4562: 4559: 4535: 4532: 4529: 4524: 4519: 4516: 4512: 4506: 4503: 4500: 4478: 4473: 4470: 4466: 4460: 4457: 4452: 4449: 4446: 4424: 4400: 4397: 4393: 4370: 4346: 4343: 4340: 4336: 4321: 4318: 4317: 4316: 4315: 4314: 4303:Sławomir Biały 4285: 4282: 4279: 4276: 4273: 4235: 4232: 4229: 4226: 4223: 4220: 4217: 4214: 4211: 4208: 4203: 4199: 4195: 4192: 4189: 4186: 4183: 4179: 4175: 4172: 4167: 4163: 4159: 4156: 4153: 4148: 4144: 4139: 4135: 4132: 4129: 4126: 4123: 4120: 4115: 4111: 4107: 4104: 4101: 4098: 4095: 4073: 4070: 4067: 4064: 4061: 4058: 4055: 4052: 4049: 4046: 4043: 4040: 4037: 4034: 4031: 4028: 4025: 4022: 4019: 4016: 4013: 4010: 3989: 3986: 3983: 3980: 3977: 3974: 3971: 3968: 3965: 3962: 3959: 3956: 3953: 3950: 3947: 3944: 3941: 3938: 3935: 3932: 3929: 3926: 3915: 3912: 3876: 3873: 3872: 3871: 3857: 3851: 3848: 3810: 3790: 3787: 3784: 3764: 3761: 3758: 3746: 3743: 3742: 3741: 3714:168.18.148.126 3702: 3700: 3697: 3680: 3677: 3653: 3650: 3649: 3648: 3647: 3646: 3635:138.194.11.244 3631: 3628: 3623: 3617: 3616: 3595: 3592: 3591: 3590: 3589: 3588: 3587: 3586: 3585: 3584: 3552: 3539: 3509: 3499: 3489: 3479: 3353: 3337: 3334: 3333: 3332: 3331: 3330: 3316: 3310: 3309: 3308: 3307: 3300: 3299: 3292: 3284: 3283: 3278: 3275: 3274: 3273: 3272: 3271: 3270: 3269: 3242: 3239: 3213: 3210: 3184: 3181: 3158: 3157: 3156: 3155: 3154: 3153: 3140: 3137: 3130: 3117: 3116: 3115: 3114: 3096: 3081: 3078: 3070: 3064: 3061: 3053: 3047: 3044: 3017: 3016: 3002: 2999: 2993: 2987: 2984: 2978: 2972: 2969: 2963: 2960: 2957: 2954: 2951: 2948: 2945: 2940: 2934: 2931: 2925: 2919: 2916: 2910: 2907: 2904: 2901: 2898: 2895: 2892: 2882: 2881: 2863: 2860: 2855: 2852: 2847: 2844: 2841: 2835: 2832: 2826: 2820: 2817: 2811: 2805: 2802: 2796: 2793: 2790: 2784: 2781: 2775: 2769: 2766: 2760: 2757: 2754: 2751: 2748: 2738: 2737: 2726: 2720: 2717: 2711: 2705: 2702: 2696: 2693: 2690: 2687: 2684: 2674: 2673: 2659: 2656: 2650: 2647: 2644: 2641: 2638: 2628: 2627: 2613: 2610: 2604: 2601: 2598: 2595: 2592: 2587: 2581: 2578: 2572: 2569: 2566: 2563: 2560: 2555: 2549: 2546: 2540: 2537: 2534: 2531: 2528: 2518: 2517: 2503: 2500: 2494: 2488: 2485: 2479: 2473: 2470: 2457: 2456: 2437: 2434: 2428: 2422: 2419: 2404: 2397: 2394: 2388: 2382: 2379: 2372: 2363: 2360: 2354: 2348: 2345: 2336: 2330: 2327: 2314: 2313: 2298: 2294: 2291: 2287: 2281: 2278: 2272: 2266: 2263: 2250: 2249: 2235: 2232: 2226: 2220: 2217: 2211: 2205: 2202: 2189: 2188: 2165: 2162: 2136: 2133: 2120: 2113: 2112: 2094: 2080: 2078: 2071: 2058:orthogonal to 2049: 2030: 2027: 2001: 1998: 1972: 1969: 1927: 1924: 1898: 1895: 1876: 1873: 1871: 1868: 1867: 1866: 1865: 1864: 1849: 1823: 1813: 1807: 1784: 1783: 1759: 1756: 1737: 1736: 1696: 1693: 1692: 1691: 1690: 1689: 1688: 1687: 1686: 1685: 1684: 1683: 1682: 1681: 1663: 1662: 1655: 1654: 1619: 1618: 1617: 1616: 1615: 1614: 1600: 1546: 1545: 1521: 1520: 1483: 1480: 1468:93.143.203.169 1459: 1458: 1432: 1429: 1413: 1412: 1411: 1410: 1395: 1394: 1380: 1375: 1369: 1368: 1337: 1312:67.171.122.139 1299: 1284: 1283: 1282: 1281: 1239:137.205.76.233 1236: 1235: 1222: 1219: 1216: 1213: 1208: 1205: 1201: 1190: 1177: 1174: 1171: 1166: 1163: 1160: 1155: 1152: 1148: 1137: 1124: 1121: 1118: 1113: 1110: 1107: 1102: 1099: 1095: 1076: 1073: 1049: 1043: 1042: 1041: 1025: 1024: 986: 983: 982: 981: 956: 953: 952: 951: 918: 915: 914: 913: 912: 911: 865: 864: 863: 862: 861: 860: 840: 839: 838: 837: 819: 818: 817: 816: 798: 797: 786: 785: 780: 779: 752: 749: 725: 721: 717: 712: 708: 704: 699: 695: 689: 686: 662: 658: 654: 649: 645: 639: 636: 616: 596: 593: 590: 587: 584: 564: 561: 558: 555: 552: 532: 529: 526: 523: 520: 517: 514: 494: 491: 488: 477: 475: 474: 473: 472: 471: 470: 469: 468: 467: 466: 447: 446: 445: 444: 443: 442: 441: 440: 423: 422: 421: 420: 419: 418: 398: 376: 354: 338: 337: 336: 335: 323: 322: 301: 300: 270: 267: 266: 265: 233: 230: 229: 228: 227: 226: 216:216.99.219.151 194:62.136.133.147 183: 180: 177: 176: 173: 172: 169: 168: 157: 151: 150: 148: 131:the discussion 118: 117: 101: 89: 88: 80: 68: 67: 61: 39: 13: 10: 9: 6: 4: 3: 2: 5332: 5321: 5318: 5316: 5313: 5311: 5308: 5306: 5303: 5301: 5298: 5296: 5293: 5291: 5288: 5286: 5283: 5282: 5280: 5273: 5272: 5267: 5263: 5238: 5234: 5230: 5225: 5221: 5216: 5211: 5208: 5202: 5196: 5193: 5190: 5187: 5184: 5177: 5176: 5175: 5173: 5164: 5162: 5161: 5157: 5153: 5145: 5143: 5142: 5138: 5133: 5109: 5102: 5098: 5094: 5090: 5084: 5080: 5079: 5078: 5077: 5073: 5069: 5065: 5062: 5059: 5056: 5053: 5050: 5047: 5044: 5041: 5037: 5034: 5031: 5028: 5025: 5022: 5019: 5016: 5013: 5010: 4999: 4995: 4991: 4987: 4986:distressingly 4983: 4981: 4977: 4973: 4969: 4968: 4967: 4966: 4962: 4958: 4953: 4949: 4945: 4937: 4928: 4924: 4920: 4916: 4910: 4909: 4908: 4907: 4906: 4905: 4900: 4897: 4895: 4885: 4884: 4883: 4882: 4878: 4874: 4870: 4866: 4860: 4859: 4858: 4856: 4852: 4848: 4847:129.116.33.62 4844: 4839: 4835: 4834: 4830: 4829: 4825: 4822: 4821: 4801: 4796: 4793: 4788: 4784: 4779: 4776: 4772: 4768: 4765: 4758: 4743: 4738: 4735: 4730: 4726: 4721: 4718: 4714: 4710: 4707: 4700: 4682: 4678: 4674: 4669: 4665: 4661: 4656: 4652: 4646: 4643: 4636: 4635: 4634: 4632: 4628: 4624: 4620: 4616: 4612: 4608: 4600: 4596: 4592: 4588: 4584: 4583: 4582: 4581: 4577: 4573: 4567: 4560: 4558: 4557: 4553: 4549: 4530: 4527: 4522: 4517: 4514: 4510: 4504: 4501: 4471: 4468: 4464: 4458: 4455: 4450: 4447: 4422: 4398: 4395: 4391: 4368: 4344: 4341: 4338: 4334: 4319: 4313: 4309: 4305: 4299: 4283: 4280: 4277: 4274: 4271: 4263: 4262: 4261: 4257: 4253: 4249: 4248: 4247: 4233: 4230: 4227: 4224: 4221: 4218: 4215: 4212: 4209: 4206: 4201: 4197: 4193: 4190: 4187: 4184: 4181: 4177: 4173: 4170: 4165: 4161: 4157: 4154: 4151: 4146: 4142: 4137: 4133: 4130: 4127: 4124: 4121: 4118: 4113: 4109: 4105: 4102: 4099: 4096: 4093: 4084: 4071: 4068: 4065: 4062: 4059: 4056: 4053: 4050: 4047: 4044: 4041: 4038: 4035: 4032: 4029: 4026: 4023: 4020: 4017: 4014: 4011: 4008: 4000: 3987: 3984: 3981: 3978: 3975: 3972: 3969: 3966: 3963: 3960: 3957: 3954: 3951: 3948: 3945: 3942: 3939: 3936: 3933: 3930: 3927: 3924: 3913: 3911: 3910: 3906: 3902: 3897: 3894: 3892: 3887: 3886: 3882: 3874: 3870: 3866: 3862: 3858: 3852: 3849: 3847: 3843: 3839: 3836:I fixed it.-- 3835: 3834: 3833: 3832: 3828: 3824: 3808: 3788: 3785: 3782: 3762: 3759: 3756: 3744: 3740: 3736: 3732: 3727: 3726: 3725: 3723: 3719: 3715: 3711: 3698: 3696: 3695: 3691: 3687: 3678: 3676: 3675: 3671: 3667: 3662: 3658: 3651: 3644: 3640: 3636: 3632: 3629: 3627: 3624: 3621: 3620: 3619: 3618: 3615: 3611: 3607: 3602: 3601: 3600: 3593: 3583: 3579: 3575: 3570: 3566: 3562: 3558: 3553: 3551: 3547: 3543: 3537: 3533: 3529: 3525: 3521: 3517: 3513: 3507: 3503: 3497: 3493: 3487: 3483: 3477: 3473: 3469: 3465: 3461: 3457: 3453: 3449: 3445: 3441: 3437: 3433: 3429: 3425: 3421: 3417: 3413: 3409: 3408: 3407: 3403: 3399: 3395: 3391: 3387: 3383: 3379: 3375: 3371: 3370: 3369: 3365: 3361: 3357: 3350: 3349: 3348: 3344: 3340: 3335: 3329: 3325: 3321: 3314: 3313: 3312: 3311: 3304: 3303: 3302: 3301: 3297: 3290: 3289: 3286: 3285: 3281: 3280: 3276: 3268: 3264: 3260: 3237: 3208: 3179: 3168: 3164: 3163: 3162: 3161: 3160: 3159: 3152: 3148: 3144: 3135: 3127: 3123: 3122: 3121: 3120: 3119: 3118: 3113: 3109: 3105: 3100: 3076: 3068: 3059: 3051: 3042: 3031: 3026: 3025:cross product 3022: 3018: 2997: 2991: 2982: 2976: 2967: 2961: 2958: 2955: 2952: 2949: 2946: 2943: 2938: 2929: 2923: 2914: 2908: 2905: 2902: 2899: 2896: 2893: 2890: 2883: 2861: 2858: 2853: 2850: 2845: 2830: 2824: 2815: 2809: 2800: 2791: 2788: 2779: 2773: 2764: 2755: 2752: 2749: 2746: 2739: 2715: 2709: 2700: 2691: 2688: 2685: 2682: 2675: 2654: 2648: 2645: 2642: 2639: 2636: 2629: 2608: 2602: 2599: 2596: 2593: 2590: 2585: 2576: 2570: 2567: 2564: 2561: 2558: 2553: 2544: 2538: 2535: 2532: 2529: 2526: 2519: 2498: 2492: 2483: 2477: 2468: 2458: 2432: 2426: 2417: 2392: 2386: 2377: 2358: 2352: 2343: 2334: 2325: 2315: 2292: 2289: 2279: 2276: 2270: 2261: 2251: 2230: 2224: 2215: 2209: 2200: 2190: 2186: 2182: 2160: 2131: 2117: 2116: 2115: 2114: 2111: 2107: 2103: 2099: 2092: 2088: 2084: 2076: 2069: 2065: 2061: 2057: 2053: 2047: 2046: 2045: 2025: 1996: 1967: 1955: 1952: 1948: 1944: 1922: 1893: 1882: 1874: 1869: 1863: 1859: 1855: 1850: 1848: 1844: 1840: 1835: 1831: 1827: 1821: 1817: 1811: 1805: 1804: 1803: 1799: 1795: 1791: 1786: 1785: 1780: 1779: 1778: 1777: 1773: 1769: 1765: 1757: 1755: 1754: 1750: 1746: 1745:mitch_feaster 1742: 1735: 1731: 1727: 1722: 1721: 1720: 1719: 1715: 1711: 1707: 1702: 1694: 1680: 1676: 1672: 1667: 1666: 1665: 1664: 1659: 1658: 1657: 1656: 1651: 1647: 1643: 1639: 1635: 1627: 1626: 1625: 1624: 1623: 1622: 1621: 1620: 1613: 1609: 1605: 1598: 1594: 1590: 1586: 1582: 1578: 1574: 1570: 1566: 1562: 1558: 1554: 1550: 1549: 1548: 1547: 1543: 1539: 1535: 1531: 1525: 1524: 1523: 1522: 1519: 1515: 1511: 1507: 1506: 1505: 1503: 1499: 1495: 1494:98.179.13.179 1491: 1481: 1479: 1477: 1473: 1469: 1465: 1456: 1455: 1454: 1452: 1448: 1444: 1443:131.180.34.78 1440: 1430: 1428: 1427: 1423: 1419: 1409: 1405: 1401: 1397: 1396: 1393: 1389: 1385: 1381: 1379: 1376: 1373: 1372: 1371: 1370: 1365: 1361: 1357: 1353: 1349: 1342: 1338: 1336: 1332: 1328: 1324: 1323: 1322: 1321: 1317: 1313: 1297: 1289: 1280: 1276: 1272: 1268: 1267: 1266: 1262: 1258: 1257:203.69.59.118 1255: 1251: 1250: 1249: 1248: 1244: 1240: 1220: 1217: 1214: 1211: 1206: 1203: 1199: 1191: 1175: 1172: 1169: 1164: 1161: 1158: 1153: 1150: 1146: 1138: 1122: 1119: 1116: 1111: 1108: 1105: 1100: 1097: 1093: 1085: 1084: 1083: 1080: 1074: 1072: 1071: 1067: 1063: 1059: 1055: 1048: 1044: 1040: 1036: 1032: 1027: 1026: 1023: 1019: 1015: 1010: 1009: 1008: 1006: 1002: 998: 994: 984: 980: 976: 972: 968: 967: 966: 965: 962: 954: 950: 947: 943: 942: 941: 940: 937: 936:DaraParsavand 931: 928: 925: 922: 916: 910: 906: 902: 897: 896: 895: 891: 887: 883: 879: 875: 871: 867: 866: 859: 855: 851: 846: 845: 844: 843: 842: 841: 836: 832: 828: 827:naturalnumber 823: 822: 821: 820: 815: 811: 807: 802: 801: 800: 799: 796: 793: 788: 787: 782: 781: 776: 775: 774: 773: 770: 764: 750: 747: 723: 719: 715: 710: 706: 702: 697: 693: 687: 684: 660: 656: 652: 647: 643: 637: 634: 614: 591: 588: 585: 559: 556: 553: 527: 524: 521: 518: 515: 492: 489: 486: 465: 462: 457: 456: 455: 454: 453: 452: 451: 450: 449: 448: 439: 436: 431: 430: 429: 428: 427: 426: 425: 424: 417: 414: 396: 374: 344: 343: 342: 341: 340: 339: 334: 331: 327: 326: 325: 324: 321: 318: 314: 313: 312: 310: 307: 299: 296: 292: 291: 290: 287: 284: 280: 276: 268: 264: 261: 257: 256: 255: 252: 249: 246: 243: 240: 237: 231: 225: 221: 217: 213: 209: 208: 207: 206: 205: 203: 199: 195: 191: 181: 166: 162: 161:High-priority 156: 153: 152: 149: 132: 128: 124: 123: 115: 109: 104: 102: 99: 95: 94: 90: 86:High‑priority 84: 81: 78: 74: 69: 65: 59: 51: 50: 40: 36: 31: 30: 27: 26: 23: 19: 5258: 5171: 5169: 5150:Suggestions 5149: 5126: 5115:equals 5/6, 5103:Sphere image 5066: 5063: 5060: 5057: 5054: 5051: 5048: 5045: 5042: 5038: 5035: 5032: 5029: 5026: 5023: 5020: 5017: 5014: 5011: 5007: 4985: 4944:mathematical 4943: 4941: 4913:— Preceding 4890: 4869:99.137.50.90 4863:— Preceding 4841:— Preceding 4836: 4831: 4826: 4823: 4817: 4630: 4626: 4622: 4614: 4610: 4606: 4604: 4568: 4564: 4323: 4085: 4001: 3917: 3898: 3895: 3888: 3884: 3878: 3748: 3705: 3682: 3666:Jorge Stolfi 3663: 3659: 3655: 3652:Figure order 3625: 3597: 3574:Jorge Stolfi 3568: 3564: 3560: 3556: 3542:Jorge Stolfi 3535: 3531: 3527: 3523: 3519: 3515: 3511: 3505: 3501: 3495: 3491: 3485: 3481: 3475: 3471: 3467: 3463: 3459: 3458:columns) of 3455: 3451: 3447: 3443: 3439: 3435: 3431: 3427: 3423: 3419: 3415: 3411: 3393: 3389: 3385: 3381: 3377: 3373: 3360:Jorge Stolfi 3355: 3295: 3259:Jorge Stolfi 3166: 3143:Jorge Stolfi 3133: 3125: 3098: 3029: 2184: 2180: 2102:Jorge Stolfi 2097: 2090: 2086: 2082: 2067: 2063: 2059: 2055: 2051: 1953: 1878: 1854:Jorge Stolfi 1839:Jorge Stolfi 1833: 1829: 1825: 1819: 1815: 1809: 1792:. Regards.-- 1768:Jorge Stolfi 1761: 1738: 1710:Jorge Stolfi 1705: 1701:146.87.52.54 1698: 1604:Jorge Stolfi 1596: 1592: 1588: 1584: 1580: 1576: 1572: 1568: 1564: 1560: 1556: 1534:129.65.149.8 1510:Jorge Stolfi 1485: 1462:— Preceding 1460: 1434: 1414: 1327:Jorge Stolfi 1287: 1285: 1271:Cuzkatzimhut 1237: 1081: 1078: 1054:Euler angles 1051: 1047:Euler angles 1014:Jorge Stolfi 997:132.67.97.20 988: 971:Jorge Stolfi 958: 932: 929: 926: 923: 920: 901:Jorge Stolfi 850:Jorge Stolfi 806:78.91.38.146 765: 677:rather than 476: 302: 288: 282: 278: 274: 272: 253: 250: 247: 244: 241: 238: 235: 214:3D images. 185: 160: 120: 64:WikiProjects 47: 15: 5123:equals 30°. 4572:Dlw20070716 3708:—Preceding 3606:Dlw20070716 3480:The parity 3454:out of the 3021:dot product 1881:unit vector 1726:Dlw20070716 1671:Dlw20070716 1632:—Preceding 1528:—Preceding 1488:—Preceding 1437:—Preceding 1400:Dlw20070716 1352:Ban Bridges 1346:—Preceding 1031:Dlw20070716 991:—Preceding 188:—Preceding 136:Mathematics 127:mathematics 83:Mathematics 5279:Categories 5040:follows. 3891:dimensions 3881:dimensions 3823:Austinmohr 3134:analytical 1758:Directrix? 876:; or from 212:stereoptic 4970:Agreed. 2443:direction 2066:but also 1790:User Page 1764:directrix 1741:Mathworld 1062:Han-Kwang 917:Longitude 763:-plane). 627:would be 52:is rated 4990:Lakhindr 4915:unsigned 4865:unsigned 4843:unsigned 4824:Images: 4587:Lakhindr 4548:Kkddkkdd 3861:Lakhindr 3710:unsigned 3686:Gwideman 3494:= 2 and 3356:analytic 2096:vector ( 2015:, & 1646:contribs 1634:unsigned 1567:, while 1530:unsigned 1490:unsigned 1464:unsigned 1439:unsigned 1360:contribs 1348:unsigned 993:unsigned 306:RolandRo 232:Elements 190:unsigned 182:Printing 5146:Physics 5068:Benhut1 4296:. See 4252:Jackzhp 3838:Patrick 3504:=1 and 3430:matrix 3398:Toolnut 3384:), or ( 3339:Toolnut 3320:Toolnut 3315:Thanks! 3104:Toolnut 1794:Toolnut 1418:L e cox 1384:Secruss 961:Jarino1 769:Jarino1 461:Zundark 413:Zundark 295:Shinobu 163:on the 54:B-class 5209:arccos 5131:SharkD 4972:a13ean 4950:] and 4893:HHIPPO 4887:do. — 4381:) and 3901:Aporio 3633:—DIV ( 3468:choose 2689:arccos 2150:& 2089:, and 1810:unique 1638:Gonfer 1559:, not 955:Angles 886:Gonfer 792:Spoon! 784:issue. 317:Rursus 60:scale. 4264:It's 4086:then 3731:Salix 3030:names 946:Pomte 435:Pomte 330:Pomte 260:Pomte 41:This 5266:talk 5156:talk 5093:talk 5072:talk 4994:talk 4976:talk 4961:talk 4957:Dmcq 4923:talk 4873:talk 4851:talk 4591:talk 4576:talk 4552:talk 4359:(or 4308:talk 4256:talk 3914:dxdy 3905:talk 3865:talk 3842:talk 3827:talk 3775:and 3735:talk 3718:talk 3690:talk 3670:talk 3639:talk 3610:talk 3578:talk 3546:talk 3402:talk 3364:talk 3343:talk 3324:talk 3263:talk 3167:only 3147:talk 3126:only 3108:talk 3023:and 2868:, or 2106:talk 1945:, a 1858:talk 1843:talk 1818:and 1798:talk 1772:talk 1749:talk 1730:talk 1714:talk 1675:talk 1642:talk 1608:talk 1538:talk 1514:talk 1498:talk 1472:talk 1447:talk 1422:talk 1404:talk 1388:talk 1356:talk 1339:The 1331:talk 1316:talk 1288:must 1275:talk 1261:talk 1243:talk 1035:talk 1018:talk 1001:talk 975:talk 905:talk 890:talk 854:talk 831:talk 810:talk 575:(or 220:talk 198:talk 155:High 22:Carl 20:. -- 4773:tan 4715:cos 4300:. 4216:sin 4207:cos 4162:sin 4143:cos 4128:sin 4119:cos 4060:cos 4051:sin 4030:sin 4021:cos 3976:sin 3967:sin 3946:cos 3937:cos 3821:. 3737:): 3567:by 3510:If 3500:If 3446:by 3438:by 3426:by 2953:sin 2944:sin 2900:cos 2891:sin 2753:arg 1212:cos 1170:sin 1159:sin 1117:cos 1106:sin 880:to 872:to 5281:: 5268:) 5212:⁡ 5185:ϕ 5158:) 5136:☎ 5095:) 5074:) 4996:) 4978:) 4963:) 4925:) 4875:) 4853:) 4785:⁡ 4777:− 4766:φ 4727:⁡ 4719:− 4708:θ 4629:, 4625:, 4613:, 4609:, 4593:) 4578:) 4554:) 4528:ϕ 4511:θ 4465:θ 4456:ϕ 4423:θ 4392:θ 4369:θ 4335:θ 4310:) 4278:∧ 4258:) 4234:ϕ 4228:θ 4222:θ 4219:⁡ 4213:θ 4210:⁡ 4194:≠ 4191:ϕ 4185:θ 4174:ϕ 4171:⁡ 4158:− 4155:ϕ 4152:⁡ 4134:θ 4131:⁡ 4125:θ 4122:⁡ 4072:ϕ 4066:ϕ 4063:⁡ 4057:θ 4054:⁡ 4042:θ 4036:ϕ 4033:⁡ 4027:θ 4024:⁡ 3988:ϕ 3982:ϕ 3979:⁡ 3973:θ 3970:⁡ 3961:− 3958:θ 3952:ϕ 3949:⁡ 3943:θ 3940:⁡ 3907:) 3867:) 3844:) 3829:) 3809:ϕ 3783:θ 3720:) 3692:) 3672:) 3641:) 3612:) 3580:) 3548:) 3404:) 3392:, 3388:, 3380:, 3376:, 3366:) 3345:) 3326:) 3265:) 3257:-- 3241:^ 3238:θ 3227:, 3212:^ 3209:φ 3198:, 3183:^ 3149:) 3110:) 3099:do 3080:^ 3077:ϕ 3063:^ 3060:θ 3046:^ 3001:^ 2992:⋅ 2986:^ 2977:× 2971:^ 2959:ϕ 2956:⁡ 2950:θ 2947:⁡ 2933:^ 2924:⋅ 2918:^ 2906:ϕ 2903:⁡ 2897:θ 2894:⁡ 2859:− 2834:^ 2825:⋅ 2819:^ 2810:× 2804:^ 2783:^ 2774:⋅ 2768:^ 2756:⁡ 2747:ϕ 2719:^ 2710:⋅ 2704:^ 2692:⁡ 2683:θ 2658:^ 2649:⋅ 2612:^ 2603:⋅ 2580:^ 2571:⋅ 2548:^ 2539:⋅ 2502:^ 2493:× 2487:^ 2484:ϕ 2472:^ 2469:θ 2436:^ 2427:× 2421:^ 2396:^ 2387:× 2381:^ 2362:^ 2353:× 2347:^ 2329:^ 2326:ϕ 2265:^ 2234:^ 2225:× 2219:^ 2204:^ 2183:, 2181:OP 2164:^ 2135:^ 2108:) 2085:, 2029:^ 2000:^ 1986:, 1971:^ 1954:OP 1926:^ 1897:^ 1860:) 1845:) 1832:, 1828:, 1800:) 1774:) 1751:) 1743:. 1732:) 1716:) 1677:) 1648:) 1644:• 1610:) 1602:-- 1587:= 1579:= 1540:) 1516:) 1500:) 1474:) 1449:) 1424:) 1406:) 1390:) 1362:) 1358:• 1333:) 1318:) 1298:θ 1277:) 1263:) 1245:) 1218:φ 1215:⁡ 1176:θ 1173:⁡ 1165:φ 1162:⁡ 1123:θ 1120:⁡ 1112:φ 1109:⁡ 1068:) 1060:. 1037:) 1020:) 1003:) 977:) 907:) 892:) 856:) 833:) 812:) 790:-- 685:ρ 635:ρ 615:ρ 592:θ 586:ρ 560:θ 528:θ 516:ρ 493:θ 487:ϕ 397:φ 375:ϕ 353:Φ 311:) 281:, 277:, 222:) 200:) 5264:( 5239:2 5235:y 5231:+ 5226:2 5222:x 5217:x 5206:) 5203:y 5200:( 5197:n 5194:g 5191:s 5188:= 5172:φ 5154:( 5121:φ 5117:θ 5113:r 5091:( 5070:( 4992:( 4974:( 4959:( 4921:( 4891:H 4871:( 4849:( 4802:) 4797:x 4794:y 4789:( 4780:1 4769:= 4744:) 4739:r 4736:z 4731:( 4722:1 4711:= 4683:2 4679:z 4675:+ 4670:2 4666:y 4662:+ 4657:2 4653:x 4647:= 4644:r 4631:z 4627:y 4623:x 4621:( 4615:φ 4611:θ 4607:r 4589:( 4574:( 4550:( 4534:) 4531:, 4523:, 4518:l 4515:e 4505:, 4502:r 4499:( 4477:) 4472:l 4469:e 4459:, 4451:, 4448:r 4445:( 4399:l 4396:e 4345:c 4342:n 4339:i 4306:( 4284:y 4281:d 4275:x 4272:d 4254:( 4231:d 4225:d 4202:2 4198:r 4188:d 4182:d 4178:) 4166:2 4147:2 4138:( 4114:2 4110:r 4106:= 4103:y 4100:d 4097:x 4094:d 4069:d 4048:r 4045:+ 4039:d 4018:r 4015:= 4012:y 4009:d 3985:d 3964:r 3955:d 3934:r 3931:= 3928:x 3925:d 3903:( 3863:( 3856:. 3840:( 3825:( 3789:0 3786:= 3763:1 3760:= 3757:r 3733:( 3716:( 3688:( 3668:( 3645:) 3637:( 3608:( 3576:( 3569:n 3565:n 3561:n 3557:n 3544:( 3536:n 3532:M 3528:n 3526:= 3524:m 3520:n 3516:n 3514:= 3512:m 3506:n 3502:m 3496:n 3492:m 3486:m 3482:p 3476:m 3474:, 3472:n 3470:( 3464:p 3460:M 3456:n 3452:m 3448:m 3444:m 3440:m 3436:m 3432:M 3428:n 3424:m 3420:n 3418:≤ 3416:m 3412:n 3400:( 3394:φ 3390:θ 3386:r 3382:z 3378:y 3374:x 3362:( 3341:( 3322:( 3261:( 3180:r 3145:( 3106:( 3069:, 3052:, 3043:r 2998:z 2983:r 2968:x 2962:= 2939:, 2930:r 2915:x 2909:= 2862:1 2854:= 2851:j 2846:, 2843:) 2840:) 2831:z 2816:r 2801:x 2795:( 2792:j 2789:+ 2780:r 2765:x 2759:( 2750:= 2725:) 2716:z 2701:r 2695:( 2686:= 2655:r 2646:P 2643:O 2640:= 2637:r 2609:z 2600:P 2597:O 2594:= 2591:z 2586:, 2577:y 2568:P 2565:O 2562:= 2559:y 2554:, 2545:x 2536:P 2533:O 2530:= 2527:x 2499:r 2478:= 2433:r 2418:z 2403:| 2393:r 2378:z 2371:| 2359:r 2344:z 2335:= 2297:| 2293:P 2290:O 2286:| 2280:P 2277:O 2271:= 2262:r 2231:x 2216:z 2210:= 2201:y 2161:x 2132:z 2104:( 2098:y 2091:z 2087:y 2083:x 2068:P 2064:x 2060:z 2056:P 2052:x 2026:z 1997:y 1968:x 1923:x 1894:z 1856:( 1841:( 1834:φ 1830:θ 1826:r 1820:x 1816:z 1796:( 1770:( 1747:( 1728:( 1712:( 1706:θ 1673:( 1640:( 1606:( 1597:ρ 1593:r 1589:φ 1585:φ 1581:θ 1577:θ 1573:z 1569:θ 1565:φ 1561:θ 1557:φ 1536:( 1512:( 1496:( 1470:( 1445:( 1420:( 1402:( 1386:( 1354:( 1329:( 1314:( 1273:( 1259:( 1241:( 1221:. 1207:r 1204:= 1200:z 1154:r 1151:= 1147:y 1101:r 1098:= 1094:x 1066:t 1064:( 1033:( 1016:( 999:( 973:( 903:( 888:( 852:( 829:( 808:( 751:y 748:x 724:2 720:z 716:+ 711:2 707:y 703:+ 698:2 694:x 688:= 661:2 657:y 653:+ 648:2 644:x 638:= 595:) 589:, 583:( 563:) 557:, 554:r 551:( 531:) 525:, 522:z 519:, 513:( 490:, 433:– 283:φ 279:ϑ 275:r 218:( 196:( 167:. 66::

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