Talk:Coordinate system/Archive 1

Integrate with coordinates
just integrated a few lines from coordinates to this page and did a redirect from that page to here Boud 23:10, 27 Dec 2003 (UTC)

republican coordinate system
Dummy me. I must have been in a rush to not realize the stupid joke. Either way, it does not belong here. Oleg Alexandrov 5 July 2005 16:42 (UTC)
 * Goes to the right, points to the left? It was pretty funny, I thought. BJAODN linas 6 July 2005 00:14 (UTC)

Link broken
The link to the external site is broken. -- Nos
 * Just deleted. There are too many external links on Wikipedia anyway. Oleg Alexandrov 17:55, 28 September 2005 (UTC)

This article desperately needs to be rewritten
Looks like this must be a holdover from the very early days of Wikipedia, because in the past few years a buncha mathematical literates have arrived who have been greatly increasing the sophistication and improving the organization of the Wikipedia.

Anyway, I agree that this article should provide an overview, but in present form it completely fails to clarify numerous issues, such as the level of structure. Also, as an organizing article, it should link extensively to more specialized articles, which may well already exist, but because of the effectively broken search feature at WP, who knows? (This is one reason why organizing articles can be so useful when they exist: they can reduce the load on the database servers.)

For example, the notion of coordinate chart is crucial for topological manifold and smooth manifold, although in applications like physics, the level of Riemannian manifold or Lorentzian manifold is more important. Indeed, I came here today looking for a suitable category in which to place some articles I'd like to write on some of the most useful charts for E3, three-dimensional euclidean space: Some specific things I'd like to discuss include My immediate motivation, incidently, is that I'd like to write articles on multipole moments, including their use to describe the shape of the surface of a body or the shape of a Newtonian gravitational field, and then use these to write articles on Weyl vacuums in general relativity, relativistic multipole moments, and so forth.
 * Cartesian coordinates,
 * cylindrical coordinates,
 * rational and trigonometric form of polar spherical coordinates,
 * rational and trigonometric form of prolate spheroidal coordinates,
 * rational and trigonometric form of oblate spheroidal coordinates,
 * rational and trigonometric form of toroidal coordinates,
 * and just for fun, cassini bipolar coordinates and cassini toroidal coordinates.
 * using exterior calculus to compute the Laplace-Beltrami operator in each chart (even if you have no prior exposure to exterior calculus, if you see a few examples, you can probably "fake it", and this stuff is so useful I think students should use it even if they don't completely understand it, since one of the nice things about exterior calculus as a practical tool is that, unlike most powerful tools, it is unlikely that even someone who is "faking it" will come up with wrong answers if they just follow the pattern of a similar worked problem),
 * solving the result by separation of variables in terms of the appropriate special functions (which afford much nice opportunity to illustrate features of function spaces and Sturm-Liouville theory).

I'd also like to write similar articles for some coordinate charts for H3 and for S3 and more. These are very useful for zillions of things like de Sitter models and anti-de Sitter models.---CH &#91;&#91;User_talk:Hillman&#124;(talk)]] 20:45, 20 November 2005 (UTC)
 * upper half space,
 * stereographic (Poincare model),
 * central project (Klein model),
 * cylindrical (analog of Hopf toroidal)
 * stereographic,
 * Hopf toroidal (just the thing for the fascinating Clifford frame),


 * I took a look at this article. If you ask me, it does not look too bad, I like the introduction especially, although improvements are always possible.


 * You say that some things are more important than charts. But things like Riemannian manifold or Lorentzian manifold which you would prefer to see are I think too complicated for the introduction.


 * That is to say, you are more than welcome to edit this article, but I would like to make some points:


 * The first part of this article should not be more complicated than what it is, it is always good to start simple and then complicate things as they go along.


 * From my experience grand schemes at making great rewrites of lousy articles into awesome articles fail. I would suggest that the development of this article be more evolutionary. That is, it is good if we work step-by-step with modest and clearly defined tasks at every step. That is to say, I would suggest you write on the talk page and wait for opinions before making big changes.


 * Let us see how it goes. Oleg Alexandrov (talk) 00:27, 21 November 2005 (UTC)


 * How did it go? More than a year on, essentially nothing has happened, and CH is sadly no longer an editor. Although I don't agree with all of his suggestions about how to rewrite this article, I believe that it is deeply flawed, and the current Mathematics Grading underestimates this. I will start a new talk thread, and tag the article, which has now, in my opinion, become a laundry list of coordinate systems, mostly in Euclidean and spherical geometry. Geometry guy 23:53, 19 February 2007 (UTC)

This article is still in a mess
I beleive this article is flawed on many levels. At first, I hoped some minor edits could fix some of these points, but there is little chance of that, so I have tagged it for one of its main flaws: more than 30% of the article is a list of coordinate systems in Euclidean geometry or astronomy.

The article begins with a redirect to coordinates (mathematics) for a more elementary introduction, so one might expect the article itself to have a broader or more clearly defined scope. The introduction, although confusing and wrong on several points, at least promises something like this, but the body of the text does not deliver even a definition. After a trivial example, a wrong discussion of parametrization, and a lamentable description of change of coordinates by a bijection and its inverse, it reduces to a series of lists of coordinate systems. Most of this would be better placed in (for instance) Euclidean coordinate systems and Astronomical coordinate systems (which redirects here).

The little meat that there is here (mostly in the introduction) is confused or wrong.
 * First a coordinate system assigns a tuple to a point in an n-dimensional space (why not an n-tuple?), then it is allowed to assign more than one such tuple to every point (to accommodate the origin in polar coordinates).
 * The article claims a single coordinate system may not exist if a space is curved. Curvature has nothing to do with it: the northern hemisphere of the earth is curved, but has a coordinate system; the circle (as the space of angles R/2πZ) is flat, but does not have a global coordinate system.
 * Combining these two points, if coordinate singularities, such as the origin in polar coordinates, are allowed, then the whose surface of the earth has global coordinates, despite being the main motivating example of a surface which cannot be covered by one coordinate chart.
 * The article coordinate redirects here, but there is only a suggestion of what a coordinate actually is.
 * Even as a professional geometer, I can barely guess the meaning of the long-standing sentence:
 * When the space has some additional algebraic structure, then the coordinates will also transform under rings or groups; a particularly famous example in this case are the Lie groups.
 * For the novice, this sentence is almost worthless. It refers to coordinates transforming without defining them. I do not know what transformation under a ring means, nor why Lie groups provide a famous example.

To fix this article, a rewrite incorporating many points of view is required. Here are some observations that this geometer can provide.
 * A coordinate is a (scalar valued) function on a space (where the scalars form a ring or field, e.g. the real numbers). The functions allowed as coordinates are adapted to the type of space: linear function(als) on linear spaces, rational functions on algebraic varieties, smooth functions on smooth manifolds, continous functions on topological spaces, measurable functions on measure spaces.
 * Coordinates exist independent of the notion of a coordinate system, and form a ring, called the coordinate ring (which currently redirects to algebraic variety). They transform under the natural automorphisms of the space (by pullback or pushforward).
 * A coordinate system on a space is a collection of coordinates which provide an isomorphism of an open subset of the space with an open subset of Rn where R is the ring of scalars. This can be made precise in specific contexts, e.g., for manifolds, coordinate systems provide diffeomorphisms: this means in particular that the coordinates in a coordinate system must be independent, i.e., have independent differentials.
 * Coordinate systems sometimes extend over coordinate singularities: these are places where either the coordinates are not all uniquely defined, or the system does not have an inverse. For example, coordinates on manifolds can become dependent at certain points: they do not form a genuine coordinate systems around these points.

Please add your own comments. Geometry guy 01:02, 20 February 2007 (UTC)


 * You raise valid points. I see no reason not to be bold if you have the time. Please use your favorite delistifier with extreme prejudice. --MarSch 12:14, 11 March 2007 (UTC)

I made a start since the above diatribe, so the introduction, at least, answers some of the above issues. I will come back to this when I have time, but welcome other editors. Another outstanding problem is to relate coordinate systems to parameterizations (which essentially redirect here). Parameterizations are, in a sense, inverse to coordinate systems (i.e., they are maps from a coordinate space Rn into the geometrical space of interest). In particular, polar coordinates work better from this point of view:
 * (r,&theta;) &rarr; (r cos &theta;, r sin &theta;)

is a parameterization of the plane. Geometry guy 00:00, 12 March 2007 (UTC)


 * Hmm, I tend to view coordinate systems as isomorphisms between a space you know the coordinates of, like Rn, and a space you want to map, like for example Sn. In this view "parameterization" and "coordinatization" are synonyms. I'm not convinced that it is useful or even meaningful to make a distinction.--MarSch 11:44, 12 March 2007 (UTC)

Informally, yes, but more formally it is vital to distinguish between a map and its inverse (e.g. differentiation and integration are inverse isomorphism on suitable spaces of functions, but are not synonymous). I agree that a coordinate system can refer to the pair consisting of coordinate chart (or system of coordinate functions) and the inverse parameterization. However, coordinates and parameterizations are not synonymous. There is no isomorphism between Sn and Rn. Lattitude is a coordinate on S2, not a parameterization. Fibre bundles have fibre coordinates and base coordinates. A map from an open subset of R2 into R3 can be a parameterization of a surface, even though it is not a diffeomorphism. I hope these examples convince you! Geometry guy 22:38, 12 March 2007 (UTC)


 * Lattitude and longitude are coordinates for part of S2 and they also parameterize that part. The diffeomorphism is not between Sn and Rn, but between the domain and codomain of the maps. Lattitude is also not a coordinatization of S2. A map from an open subset of R2 into R3 which is a parameterization of a surface would have to be isomorphic (in some category) to its image, unless you want to allow self-intersections in which case I am convinced. --MarSch 12:44, 13 March 2007 (UTC)


 * Agreed, and sometimes self-intersections are allowed (e.g. immersions). I guess my point is that it is sometimes useful to consider components, or extend codomains (whereas one cannot usually extend domains) - hence it is important to know whether you are talking about a chart or its inverse. Geometry guy 13:12, 13 March 2007 (UTC)

I am new to Wikipedia (as an editor) and new to this article. It is clear from this talk page that this article has been recognized as needing major revision for some time. I concur. There is so much to be done, but I would like to make a few observations in the hopes that it will lead to focused discussion on a serious revamp of the article: (1) Most of what I know, need to know, or can readily find about the formal notions of coordinate systems relates to differentiable manifolds. Indeed there is a substantial established framework of concepts and mechanisms within Differential Geometry which would supply most of the material that, I suspect, most people would want to discover when exploring this article. Consequently, one task is to figure out how to incorporate those established results without being simply redundant. (2) Having made the first observation, it is my personal additional observation that a lot of Differential Geometry is too abstract for people whose mathematical needs in this domain are more practical. My favorite example would be the otherwise practical need to understand and actually do coordinate transformations. This is only glibly alluded to now and yet a good encyclopedia, I think anyway, should actually explain enough to equip the reader to either recognize an existing trasformation or create their own as needed -- within practical limits that don't push this all the way toward being a tutorial. (3) Its not enough to say that a coordinate system maps points to n-tuples or vice-a-versa. There are important notions of topology and metrics that need to be carefully exposed without creating a Tome on Differential Geometry. (4) Its a modern digital world -- so let's not ignore some really obvious practical things not found in the typical Differential Geometry text. Like the fact that there are discrete manifolds in need of their own "coordinate systems" and that not every manifold needs to be differentiable. It took me no time at all to Google and find papers and articles on discrete spaces with coordinate systems. I don't mean to suggest we have to get involved with ongoing research, but any reader looking at their computer screen will appreciate that it can be described as a collection of pixels organized into a 2-d space with a natural discrete coordinate system. This is not esoteric and not cutting edge -- and it draws into question wither dimensionality can be defined for these purposes in the absence of continuity -- of course it can! (5) I came to this article because I was seeking a nice review (for the umpteenth time!) of things that would allow me to check my thinking on constructing transformations between a Euclidian system and a "novel" nonorthogonal system. Never mind curved spaces (another opportunity to get carried away, but with justification) -- just reliably creating the right transformations for a novel nonorthogonal coordinate system in flat space can be vexing and error prone 'unless' the basic ideas are made very clear and yet very general. It is all too common to see elaborate derivations of ideas arbitrarily limited to systems -- like orthonormal ones -- which fail to provide a rich enough foundation to encompass the situations that are truly challenging unless the simple things are also defined in a truly general way. Does anyone challenge the goal of having the article satisfy this need? Thanks! scanyon 05:49, 7 May 2007 (UTC)

I don't want to deliver a dump of my stream of consciousness about coordinate systems, but after thinking about my own comments above, it occurs to me that what might be going on is that the idea of a coordinate system is close to being primitive and therefore intuitive and yet the subtleties of topology and geometry have many consequences which make particular spaces, particular systems of coordinates, and the transformations among them difficult to organize into a cogent presentation. Moreover, as a practical matter, while we could sterilize the topic and render a glib explanation of coordinate systems per se, anyone contemplating a choice of coordinate system has a context and the interaction of that context with the coordinate system is where much of the complexity comes from. Specifically, in scientific applications, most problems entail tensors (with simple vectors being the most common -- but I'd like to stay general without sounding pedantic) and the choice of coordinate system is entangled with the choice of basis. In the foregoing post I was so used to worrying about basis issues as my context for thinking about coordinate systems that it was easy not to mention this explicitly. I think I can forgive myself and make a useful point by suggesting that most coordinate systems induce natural basis choices and that problem solving within the underlying space proceeds in a way that keeps these things tied together. If I want to change coordinate systems my desire is to take advantage of a symmetry of the problem or conform more naturally to the geometry of the boundary, but in the end I want to know what tensors and tensor operators look like in that coordinate system in a basis appropriately chosen to reflect that coordinate system. Its not so much that coordinate systems are hard to describe or explain, it is how they relate to each other and how they affect basis choices and the representation of tensors and operators in the underlying space. Oh, and a retraction of sorts. # (4) above is not quite right. I am not a geometer, but it later occurred to me that the pixel space on my screen had a 2-D geometry because of the topology of the space that the pixels were imbedded in. It remains true that we can partition a continuous space and develop a discrete coordinate system to identify the partions, but that dimensionality in that case was inherited from the space. I suppose that we can define neighborhoods of discrete points and accomplish a result that is dimensional in character, but this is not what I said and I do not feel qualified to pursue this idea logically in a way that fits neatly into an explanatory scheme. What remains true, however, is that "coordinate system" can be constructed to address discrete spaces and that the exposition should encompass this possibility as smoothly as possible without becoming esoteric. I forsee no danger of indulging in research or speculation. There is plenty of established thought and useage to document, but most of what I have read at the level of text books in inadequately generalized and narrowly focused. I will await comments and then try my hand at a new opening for the article. The goal then will be to create an example that applies the foregoing ideas, but I think we need to agree on an outline before seriously recreating the article. Thanks! --scanyon 19:22, 7 May 2007 (UTC)


 * I find it hard to extract the actual things you are proposing from what you wrote. Perhaps it will become clear if you make some edits to the article. Vector_field contains examples about transformation of scalars and vectors. --MarSch 12:39, 8 May 2007 (UTC)


 * By the way, i examined Vector_field which you cite above. It gives support to a challenge below to the existing use of the word scalar in the present coordinate system article. I also disagree with some details of the explanation given. However, it is far indeed from an explanation of coordinate system transformations. I do think that coordinate system transformations is a topic in need of its own article (there are some) and I also still think that it is inextricably tied up with basis changes.  For the coordinate systems article here I am attempting to at least point this out with some explanation since it bears directly on an understanding both of the impact of coordinate system choice upon basis choice AND the general relationships among coordinate systems as such. --scanyon 00:48, 11 May 2007 (UTC)

You are right. There are no concrete proposals as of yet -- just an attempt to discover the essence of the problem. Indeed the point of submitting an actual rewrite of some of it will be a test of whether I understand enough about what's wrong to actually fix it. I did not and still do not think that its wise to just jump in and fix specific technical things. Something more fundamental seems to be missing. Let's see if I can help find it and demonstrate that I have done so using a concrete example. Unhappiness with this article has persisted for a long time for some reason .. and yet it is rated as having high importance. scanyon 17:28, 8 May 2007 (UTC)

Two small challenges as proposed major revision progresses.
I am making good progress on major revisions, but along the way begin to notice a few things that should be briefly debated:
 * Individual coordinates are now characterized as scalars. I agree that they are one-component objects, but because the term scalar has come to mean slightly different things in different contexts I question the helpfulness of this characterization.  One of the common abuses of the scalar characterization is when tensor components are described as scalars.  In that context scalar implies invariance under change of coordinate system which defies the very idea of what a component is. Here we are claiming that something which is fixed by a choice of coordinate system is nevertheless a scalar.  Yes, I understand the subtle distinctions, but invariance under coordinate system change remains a reasonable association with the term and it has the potential of being misleading here.  If all we are saying is that individual coordinates are not themselves multi-component, then why not avoid the issue by just saying that?--scanyon 18:08, 10 May 2007 (UTC)


 * A coordinate system is a set of partial functions from a space to a coordinate space (a ring), which together form an isomorphism in the appropriate category between part of the space and (part of) the coordinate space. A scalar is simply a function from the space to the coordinate space. It follows that each individual coordinate is a scalar in the strictest meaning of scalar. Scalars can multiply tensors. The components/coordinates of a tensor in a particluar coordinate system are exactly the multiples of the basis tensors associated to the chosen coordinate system. Therefore they also are scalars. The components/coordinates of a tensor are not scalars, since they are functions from the product of the space and the space of coordinate systems to the coordinate space. But if the tensor is a scalar then its component is a function which doesn't depend on its coordinate system argument and we blur the distinction. Thus a scalar is its own component and by specialization a coordinate is its own component. The number of components of a coordinate is trivially one, because coordinates and components are both scalars. Component is short for _scalar_ component.
 * Maybe this should be put in the article. --MarSch 11:35, 11 May 2007 (UTC)


 * Maybe.. but I'd wait a bit. For one thing, a partial function is defined such that elements of the space are mapped to at most one and only one element of the coordinate set.  Since this is violated at the origin designated for a polar (or spherical) coordinate system, a defintion based on partial functions will fail. It is not the job of an encyclopedia to overrule common usage in favor of mathematical elegance.  Its late and so I will come back later and give a proper response to your arguments in defense of the term scalar.  Please see my response below to your continued insistence that the coordinate space is naturally a ring. --scanyon 06:03, 12 May 2007 (UTC)


 * The origin is simply not mapped by polar or spherical coordinates. But we don't need to stop there. By all means mention that the origin is often included. What we shouldn't be doing is providing erroneous mathematical definitions, at least not while pretending to discuss the technical meaning of the term.--MarSch 10:54, 12 May 2007 (UTC)


 * Hmmm.. What I keep seeing in this discussion is that we are struggling to formalize in terms of available modern constructs what is essentially a more intuitive and perhaps even ad hoc scheme that works in practice and has for a long time. I insert this broad observation here becuase of your stated belief that "the origin is simply not mapped by polar or spherical coordinates."  Is this correct?  I started off thinkng that it was indeed mapped in that direction, but ambiguously, rather than not mapped at all.  Maybe I have been deceiving myself, but I have believed for a long time that when I used polar or spherical coordinates I got to address the origin, but if I nevertheless tried to do certain things there involving angles that things fell apart.  If I have a problem for which the angles are ignorable (a valid term in lagrangian mechanics) this coordinate system is perfectly behaved at zero.  What this shows is that there is a lot of practical functionality in well known coordinate systems that defies being easily fit into glib formalism.  Why, by the way, MUST we define coodinate system by a map from the space to the coordinates (yes, I know we can do it) when it might be more helpful to define them as a map from the coordinate space to the a subspace (possibly all) of the points?  Both have inverse mappings, but the latter is closer to intuition -- or better -- closer to how coordinate systems are used in practice?  Okay, that's another one my naive questions, but seriously, I keep seeing that a simple idea like coordinate systems is difficult to formalize and explain comprehensively because we are trying to cram a broad and flexible technique into a limited set of formal pigeon holes.  P.S. I'm still thinking about scalars... and I still don't like using the term in the way we do in the article.  Before turning this mole hill into a mountain I want to think hard about why this bothers me and whether it is worthwhile to bother others about it.  In the meantime thank your your attempt to defend the current use. --scanyon 07:38, 14 May 2007 (UTC)


 * Back to scalars! I've been distracted for the past week, but not thinking about something consciously often leads to new insights when the topic is placed back into the stream of consciousness. So look MarSch, your analysis above breaks down early with the statement "A scalar is simply a function from the space to the coordinate space." Let's suppose I have a coordinate system consisting of two angles used to locate points on the surface of a sphere.  I want to map some measurable on the sphere -- say surface temperature.  So, I have a function from the points of the sphere to the temperature scale.  Well, the temperature scale 'might' be coincidentally "scaled" to match the units of one of the angles, but we know this is bogus.  It is not true, in this case, that the "scalar" function of the points on the sphere are literally mapped to the coordinates.  Never mind range of values, which coordinate is it?  It just breaks down.(cont'd)


 * Other things which follow in your argument have merit, but I suspect we have missed the point. The term "scalar" usually occurs in a context.  In tensor analyis it corresponds to a rank zero tensor which has but one component whose value is independent of the coordinate system.  What we are focusing on now are elements of the coordinate tuple and upon noticing that they are typically just individual real numbers that they must also be scalars.  My problem with this from the beginning has been that this is a context, like zillions of others, in which dignifiying a single number as being a scalar constitutes a useless distinction -- unless maybe you are discussing a programming language.  Our abstract mathematical world is full of single numbers, but we only designate some of them to be scalars to distinguish them from other objects in some similar or same category in that context according to how their representation changes under certain transformations. My argument is that telling a reader that individual coordinates within a tuple are scalars is a useless display of vernacular even if it is somehow defensible. (cont'd)


 * So one last thought about this scalar thing. I began my participation in this topic by observing that typically we are concerned about coordinate systems which locate points in differentiable manifolds.  This is because in applications, (both math and physics) we tend to like to do calculus on objects living in some sort of space.  But a coordinate system in its most general form is just some kind of device for locating points in a space by associating it with a set of objects.  There are lots of kinds of spaces and lots of kinds of objects that can be collected into spaces and made to serve as coordinates.   The point is that whether it is helpful in doing calculus or not, I 'can' invent some kind of coordinate system, maybe useful for something, made up of objects that simply don't remind us of scalars. (Yes, I know, we can always find a one-component set that is isomorphic to our coordinate set -- but we digress.)  Okay I've said more than enough.  I am not enclined to designate coordinates as scalars, but I will listen to any rejoinder before indulging this.--scanyon 22:25, 21 May 2007 (UTC)


 * I think temperature is a function from your space to a space of temperatures (something one-dimensional) or equivalently a section of the trivial bundle with base your space and fiber the temperature space. Your temperature scale (e.g. Kelvin) would be a coordinate on the temperature space. In the presence of this new kind of object I might choose to extend what I call scalars. I might call every non-tensor, such as this temperature, a scalar. But it might be best to scrap the scalar terminology. --MarSch 11:00, 22 May 2007 (UTC)


 * Okay, this time we really are in agreement. Thanks for your patience in hashing this out. Please do not infer any hostility in the vigor with which I debate small issues.  My appreciation for your participation is sincere. Concensus is often hard won.  --scanyon 22:18, 22 May 2007 (UTC)


 * I followed the link that was supposed to support the claim that the coordinates of a space form a coordinate ring. I found the use of coordinate ring in the article, but was unable to see that it justified the claim.  Maybe I'm just wrong about this, but whether your believe the claim is meant to refer to the relationship among n-tuples or among sets of n-tuples corresponding to range of possible coordinate systems, the bottom line is that for n>1 closure under multiplication will be difficult to defend.  This should be further complicated by the fact that some coordinates, like angles, are bounded and the set of them are not closed under ordinary mulitplication.--scanyon 18:08, 10 May 2007 (UTC)


 * As the article says, coordinates are functions of some space to a ring. Therefore the set of scalars forms a ring too, probably one with zerodivisors. I think this is the coordinate ring, though my not so fluent algebraic geometry is a bit rusty. I'm also not sure this term is customary in this context. Ring of scalars would be a better term. --MarSch 11:35, 11 May 2007 (UTC)


 * First off that's not exactly what the article now says. It actually comes closer to simply declaring that the set of components is per se a commutative ring.  What you say amounts to a defintion of a coordinate system as being a mapping to a ring, but if so I challenge your right to define coordinate systems that way.  Both you and the article claim that it is conventional to refer to the set of coordinates as a coordinate ring, but when you follow that link you find a target article on algebraic varieties and the section which introduces the term coordinate ring is limited to functions from a particular kind of space and not spaces in general.  I have further challenged the basic notion that coordinate sets naturally constitute a ring given the enormous variety of possible ways to usefully define a coordinate system -- which include using angles which are elements of a bounded set not closed under ordinary multiplication.  That does not automatically imply there is no way to define some kind of addition and multiplication on the set of n-tuples corresponding to each and every conceivable coordinate system, but it creates doubt and the proof that this can be done has so far not been brought to my attention.  Again, maybe you are right, but my doubts have not been resolved. --scanyon 06:03, 12 May 2007 (UTC)


 * If you look at my previous reply you will see that we are mostly in agreement here. I'm saying that what is meant by coordinate ring is probably the ring structure of the set of scalars. I agree that the set of coordinates itself probably doesn't have a ring structure.--MarSch 10:46, 12 May 2007 (UTC)


 * I do see that we are approaching agreement, but not that we are there yet. I do see that, for what its worth, some coordinates of some systems are rings under ordinary arithmetic -- just because they are E1!  What I see that is more fundamental, however, is that each coordinate in any system is related to a geometrical construct within the space and that some of these are "cyclic" which complicates any effort to generalize on the structure of ALL coordinates reflected in the n-tuple spade or, as we have been discussing, to generalize on the structure of all possible n-spaces.  If we dig deeper we see that the essence of the problem is geometry, not algebra... which is as far as I have gotten so far.  If we don't wear out I think we will get this right ... and without having to invent new concepts and thus violate the prime directive of our encyclopedic work.  Hey, to lighten up a bit, if thinking clearly were so easy everybody would do it! --scanyon 07:38, 14 May 2007 (UTC)

Perhaps even the opening sentence is too narrow
Hi guys. I'm about ready to post a rewrite of the opening to the article. Before doing so I have one more curiosity. The opening statement claims that coordinate systems for n-dimensional spaces are mapped by n-tuples. I might have written this same statement myself a month ago -- now I think its too narrow -- for several reasons. First consider one of the examples given: Plücker coordinates. By the very description provided it violates the initial claim. It is also possible to purposely address only a subspace -- though in that case the dimensionality of the subspace is what's assumed relevant. But more generally, I might want to address a subset of the space that is not a proper subspace. I found one useage of the term coordinate map which defines it as the specific case of the most general one in which the dimensionality does match the number of items in the n-tuple and the coordinate set is isomorphic to the space, but that implies that, as I fear, the general case is much more broad. I intend to exploit all this. Any comments?--scanyon 16:42, 1 June 2007 (UTC)

Reference frame and coordinate system
I found no mention to the expression "reference frame" in this article.

I use the two expressions "reference frame" and "coordinate system" as synonyms. Physicists tend to use the expression "reference frame", rather than "coordinate system", probably because Newton used the first rather than the second, but I just cannot see the difference. In physics, you always need a coordinate system (with axes and units) to define any quantity. This coordinate system can be inertial or not... Indeed, a coordinate system is a system of reference...

I see the need of a separate article about the adjective "inertial". But I cannot see the reason why there should be a separate article about reference frames. Paolo.dL 13:02, 24 August 2007 (UTC)

I found no mention to the expression "reference frame" in this article.
 * However, the article contains (and contained at least since August 2007) the following


 * the space itself is considered to exist independently of any particular choice of coordinates. [...]
 * When the space has additional structure, one restricts attention to the functions which are compatible with this structure.


 * Surely it is understood that " this structure " itself would also exist independently of any particular choice of coordinates, and that it exists as a particular relation between certain (subsets of) elements of " the space " under consideration.


 * In mathematics, such a structure might be the topology of that space (which is manifestly independent of any particular coordinate assignment), or its metric (as far as it is expressed independently of a particular coordinate assignment).


 * In physics and physical geometry such a structure is called "reference frame"; representing measured geometrical relations.

In physics, you always need a coordinate system (with axes and units) to define any quantity.


 * On the contrary. Physical real quantities (such as ratios between durations, as the basic notion for establishing a reference frame in the sense of Synge's Chronometric geometry, for instance) must and surely can be found independently of any particular coordinate assignment.
 * It is therefore a genuine question (as indicated in the article), whether some particular coordinate assignment is "compatible" ("continuous", "smooth", "affine") to "the structure" (which therefore had to be identified separately), or not.

I see the need of a separate article about the adjective "inertial".


 * I certainly agree that Wikipedia at present is a poor reference on the (physics-) topic of how to define/measure "(absence of) force", or "(absence of) acceleration" or "geodesic motion"; apart from pointing out a certain equivalence of these notions.
 * (This issue is however quite involved, as illustrated by publications such as U. Schelb, "Characterizability of Free Motion in Special Relativity", Found. Phys. 30, 867, 2000;
 * and it seems also far from settled, as the mentioned reference, for instance, apparently depends on a priori knowledge about which coordinate assignments to elements of a particular worldline are "smooth" to the duration between these elements, and which are not.)
 * Furthermore my preference is an encyclopedia which represent this. Frank W ~@) R 19:43, 5 March 2008 (UTC)

Room for improvement
There several issues with the article and related articles that should be addressed. Some of these have already been mentioned above. --RDBury (talk) 16:02, 9 May 2010 (UTC)
 * 1) The definition in the lead section states that a coordinate system uses n coordinates for a space of dimension n; this excludes systems such as homogeneous coordinates where redundant coordinates are used. The definition also excludes coordinates for elements other than points such as line coordinates.
 * 2) In general, this article seems to confuse coordinate systems with coordinate charts. The concepts are related and there should be cross references, but material specific to coordinate chart belongs in that article.
 * 3) This article, Coordinates (mathematics) and Algebraic geometry cover the same ground multiple times and there should be two articles instead of three. 'Coordinates (mathematics)' could easily be merged with the other two articles. There may be something to be said for having a nontechnical introductory article for this material, but any such material should be in 'Algebraic geometry'.
 * 4) The template for the list of orthogonal coordinate systems creates a list section in multiple articles. List sections should be considered carefully in general and in this article it adds extraneous links. I'd propose that the template be turned into a stand-alone list and a single link used instead of the template.
 * 5) There is a merge tag for geographical coordinate systems with no corresponding discussion thread. It seems clear that the merge should be done.
 * 6) There are other sections for which a separate article exists; redundant material should be merged and links to the articles should be added.
 * Update: I just completed the merge with Coordinates (mathematics). I've dealt with the first two issues as well.--RDBury (talk) 20:39, 26 May 2010 (UTC)

Assessment comment
Substituted at 14:34, 1 May 2016 (UTC)