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they are used for and how they are related? The tan equations here are used to relate the geocentric latitude, θ, and the parametric latitude, β, to the geodetic latitude,φ. We could define a scale factor σ=b/a=√(1-e²) for the ellipsoid. I found that tanθ=σtanβ so that checks with tanθ=σ²tanφ. My interest was spurred by a desire check the accuracy of the GPS location provided by my cell phone and converting the spread of observations into a distance measure. The primary question is how do we determine vertical? The astronomical definition is it the direction provided by the plumb line. For geographic purposes it's the normal to the ellipsoid at a given point.
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equilibrium) the rotational and ellipsoidal effects are of the same order (because the flattening of the earth is due to rotation). Thus the extent to which the field lines deviate from the hyperboloid (due to rotation) is of the same order as the extent to which they deviate from a cone (due to ellipsoidal + rotational effects).
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the equator and the pole and is just a parameter associated with the point. It appears to be related to Kepler's eccentric anomaly, E. θ_p is more descriptive but we need to be concerned with the ease of typesetting. θ' might be better if one wants to use the angle as a subscript as one would for a derivative.
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satisfies too conditions (1) it's an orthogonal system so that laplacian operator is sane and (2) the boundary condition, that the potential is constant on the reference ellipsoid, is aligned with the coordinate system. Nevertheless, the article is somewhat misleading here because solving for the potential
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looking diagrams. I.e. for people with less literacy in mathematics and mathematical figures. (Since the way mathematics is taught is using its own language that you have to use day in, day out to remain fluent in.) If there's something specific about the animations you don't like, maybe I can address it. ➧
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We can use a unit vector to indicate a point on a sphere but there is no unit vector for a point on an ellipsoid. The best we could do in to include a term involving the eccentricity in the z component. So in the parametric representation of a point on an ellipsoid the θ above is not an angle between
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I don't really find any particular choice of symbols to be especially confusing as long as they are clearly defined and consistently used (at least within each article). I looked around at a few more sources and there doesn’t seem to be any kind of strong convention. If anyone wants to change this I
3188:, who is (according to his user page) Charles Karney, author of the paper you linked. (He's also not very active on Knowledge any more.) Before that edit the page used psi. I do feel that this is a confusing use for theta, and I'd prefer we go back to psi, but I don't have that strong of an opinion.
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By the way, I've updated the section on numerical comparisons to use the parameters for the WGS84 ellipsoid and to plot the signed difference. It seems that my change to the figure will take a while to percolate through to the article page. (It was too easy for people to look at the old figure and
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Let me address just the first point here. The criterion "looked a lot more like hyperbolas than cones" is a rather weak one. My contention is that the actual normal gravity for the earth is "as close" to straight lines as to hyperbolas. By "as close" I mean that the deviation from true gravity is
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You're reading altogether too much in the choice of coordinate system in this case. It's not a question of "importance" or "relevance" but the more utilitarian issue of making the partial differential equation for the potential easy to solve. In this instance the choice of ellipsoidal coordinates
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Let me finish by saying that the concept of "surface of latitude" is not standard in this field. People are thoroughly used to working with "geodetic coordinates" (latitude, longitude, and normal height). And, of course, it is possible to define a surface on which the latitude is constant (a cone
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I was thinking in terms of simplifying the math required to understand
Geodesy, Knowledge article and others and improving overall coherence. More specifically, how do we improve this article for those who are unfamiliar with the various definitions of latitude. Can we provide more context on what
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For a point (x,y,z) on the surface of the ellipsoid one can compute the angles for spherical coordinates. One can also use (x,y,z) to compute the the parametric latitude so given one latitude one can find the other. My use of θ was presumptuous and let to some confusion. The transformation between
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The geodetic latitude is defined with respect to the ellipsoid surface normal. The astronomical latitude is said to be defined with respect to a plumb line. However, on an ellipsoid in gravitational-rotational equilibrium the surface normal and the plumb line should be the same, shouldn't they? I
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My guess is that most (lay) people, if asked to try to define latitude themselves, would come up with the geocentric latitude, or at least I know I would have before I learned that there are (at least) two definitions in use. When we use the passive voice here—"is assumed," "is used"—it begs the
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However the caption to figure d "Within the confines of the reference ellipsoid the field lines very nearly form a truncated hyperboloid" is true but misleading. The field lines equally well form a truncated cone. The reason is that for the earth (and other bodies under approximate hydrostatic
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The concept being illustrated is essentially a 2d one. Using a 4d image (3d + time) disguises the key points. I realize that there's some modest skill involved understanding that the 2d image represents a rotationally symmetric concept. But the reader is brought on board by some of the early
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The cone is only a measure of the plumb line at the body's surface. Deviation from the cone is to be expected. The graphs you showed me on Stack
Exchange looked a lot more like hyperbolas than cones, too. And, the animations are meant for people who may not be able to understand the more formal
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Why the "rabbit out of a hat" aspect for your animations. They start off all (more or less) the same and the user has to wait for the "reveal". Perhaps using a cut-away would help but would not overcome my other objections. See Jahnke + Emde, Tables of
Functions, for some nice examples of
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It’s all about the official locations of things. Surveying measures geodetic latitudes; traditional techniques have no way of measuring geocentric latitude. Every national system of mapping uses geodetic latitudes for that reason. Hence all official documentation, worldwide, is in geodetic
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about how
Geodetic Latitude behaves above and below the surface of the reference ellipsoid. I'm not sure whether this article answers this question fully, and would appreciate some input. (Maybe we can then expand the article a little bit with the new information afterward.) Thanks.
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in this case) or, indeed, a "surface of longitude" (a meridian plane). However, I've never seen this idea adopted in the literature and
Knowledge shouldn't be used to promote it. (Indeed the most usual surface associated with latitude is the plane parallel to the equator.)
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which has a uniform density. I now give a figure showing the internal field lines in this case too. The field lines are curved in the opposite direction (relative to a straight lines) compared with hyperbolas. This reinforces the idea that it's best to stick with cones.
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The standard definition of the (geographic) latitude for an arbitary point is to drop a perpendicular onto the ellipsoid and to take the latitude of the point of intersection. The resulting triplet, (latitude, longitude, and normal height), defines "geodetic coordinates".
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The article currently states: " is the definition assumed when the word latitude is used without qualification." This statement is repeated (nearly verbatim) elsewhere across related
Knowledge articles. I believe it could use some clarification and possibly sourcing.
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Without getting into the interpretation of your math, it looks to me like you’ve computed the parametric latitude. My official texts are packed away for awhile, so I’m just going to suggest searching on geocentric latitude for credible references on the Web, such as
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1429:(Much later.) It's "important" in the sense that property lines/borders will (eventually) need to be determined beneath the Earth's surface at some point in the future. Though if the depth is less than a few kilometers it might not make much of a difference. ➧
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1362:"My contention is that the actual normal gravity for the earth is "as close" to straight lines as to hyperbolas. By "as close" I mean that the deviation from true gravity is comparable. In this case, the simpler solution (straight lines) is to be prefered."
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In the absence of specifying, “eccentricity” means “first eccentricity” in all of the literature I have ever read. We can clarify that in the article, of course, though that is clear from its relationship to flattening as given in the article.
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Check the article
Ellipsoid, Parameterization for the equation a point, P, on the surface of an ellipsoid using spherical coordinates and angles measured from the equator. The radial distance, r, from the center, C, to the point, P, is
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However I don't see that your animation (cool though it is) is any more helpful than the similar 2-d figure in the "geodetic coordinates" section of this article. In fact, I find the 3d rendering to be less clear and the motion to be
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I assumed that by calling them a "natural choice" it implied they are somehow important and relevant. I mean, if there is no causal relationship or link at some fundamental level, why use ellipsoidal coordinates in the first place?
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geodetic latitude to the same point. Nevertheless, each would be a geodetic latitude. My library in storage for awhile, so I can’t just pull up the usual texts. Poking around online should confirm what I’m saying here, though.
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Thus if you wanted to compare two different definitions, these could be shown on a single 2d figure with the differences readily apparent. With the 4d images, you have to show two distinct animations making the comparison less
2121:{\displaystyle {\vec {\textbf {v}}}\equiv {\vec {\textbf {g}}}_{\lambda }\times {\vec {\textbf {g}}}_{\theta }=a\cos \theta {\begin{pmatrix}b\cos \theta \cos \lambda \\b\cos \theta \sin \lambda \\a\sin \theta \end{pmatrix}}}
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Latitude and longitude are used to label coordinates on the surface of the ellipsoid. In most geodetic applications the third coordinate is the normal height, so a surface of constant latitude is a cone (your figure
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I fear that the major impression of the animations is that there are 4 plugs popping out of a ball with little understanding of the main point (i.e., the small differences in the shapes of the plugs).
1372:(both highly exaggerated cases) that the difference as you approach the equatorial plane is considerable. The thickness of the latter near the center is much greater for the same degree of latitude.
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If necessary, it's easy to take a measurement from a 2d figure (print it out and measure with a ruler). This is much less simple with a rendered 3d image and nigh impossible with an animation.
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assume that corrections for atmospheric correction, height of observer etc need to be made for precision measurements and thus should not affect the relation between the two latitude types.
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1945:{\displaystyle d{\vec {\textbf {r}}}={\frac {\partial {\vec {\textbf {r}}}}{\partial \lambda }}d\lambda ={\vec {\textbf {g}}}_{\theta }d\theta +{\vec {\textbf {g}}}_{\lambda }d\lambda }
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The plumb line may also be affected by the thickness of the Earth's crust at a particular point. Mountains versus oceans for instance. The difference may be inconsequential however. ➧
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I'm still not quite sure what you are aiming for in this discussion though. Are you proposing a change to this article, or just asking for someone to look at your algebra? –
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Geographic latitude must be used with care. Some authors use it as a synonym for geodetic latitude whilst others use it as an alternative to the astronomical latitude.
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Can we specify some examples of specific contexts in which each one is most commonly used? When is geocentric latitude used? E.g. Google Maps uses geodetic latitude.
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There may be some confusion among alternative definitions of eccentricity. The article needs clarification. See
Eccentricity (mathematics), Alternative names.
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Turning to the issue of animated renderings and, for the sake of this argument, let's say we want to compare various definitions of the "surface of latitude".
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However, there's a more important problem. The gravity internal to the earth is unknown (to the accuracy required to resolve this issue). My first figure on
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coordinates, which means the latitude is geodetic. However, the datum varies from locale to locale, which means that two different surveys do not assign the
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satisfies the equation for an ellipsoid, x²/a²+y²/a²+z²/b²=1, but my θ is not the θ used for spherical coordinates as I had initially assumed.
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I checked this formula numerically using small angular separations to approximate the normal vectors and determine their angles and it appears
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and consequently the vectors g_θ and g_φ will determine the tangent plane at the chosen point. A normal to the plane at this point is given by
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1807:{\displaystyle {\vec {\textbf {r}}}={\begin{pmatrix}a\cos \theta \cos \lambda \\a\cos \theta \sin \lambda \\b\sin \theta \end{pmatrix}}}
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to mean the geocentric latitude, which again is, I believe, a somewhat standard symbol. For example Karney adopts these symbols,
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assume that the auxiliary latitude were bigger can the geodetic latitude -- although the text did try to address this issue.)
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Can we source this statement of geodetic as the norm, and/or these specific contexts in which it is or is not the norm? E.g.
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3239:) is one of the world's foremost experts on this topic. Here at Knowledge he e.g. wrote and illustrated the excellent page
2227:{\displaystyle {\hat {\textbf {e}}}_{w}\equiv {\hat {\textbf {e}}}_{x}\cos \lambda +{\hat {\textbf {e}}}_{y}\sin \lambda }
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So geographic latitude does not describe points above and below the surface at all? Or, what does "normal height" mean? ➧
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in the parametric expression for a point on the surface of an ellipsoid to avoid confusion. Could we do a RFC on this?
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a uniform ellipsoid is relatively easy without using ellipsoidal coordinates. I'll look into editing this sentence.
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would recommend doing a more serious literature survey, especially focusing on the most popular recent textbooks. –
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But if you have changes you want to see to this article, can you be a little more specific about what they are? –
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was used for both isometric latitude and geocentric latitude, which is incredibly confusing. Note that Karney (
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shows the field for the unrealistic case with all the mass of the earth concentrated on an equatorial disc.
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Animations are distracting when trying to read the text. Multiple this by 4 for 4 synchronized animations.
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for the geocentric latitude. This is, I beliveve, a non-standard usage and it conflicts with the use of
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your figures needs updating; perhaps you can take care of this? Let me know if there are any issues.
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I can see how this would be the case when the flatting of the Earth is only 1/300. But you can see in
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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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on Knowledge. If you would like to participate, please visit the project page, where you can join
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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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on Knowledge. If you would like to participate, please visit the project page, where you can join
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Note that geodesists use 2 definitions of eccentricity. See Torge & Müller, Geodesy, p. 91.
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To specify a point on an ellipsoid of revolution, typically people use the geodetic latitude
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The angle β for the parametric latitude appears to be related to Kepler's eccentric anomaly.
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2451:{\displaystyle \tan \phi ={\frac {a\sin \theta }{b\cos \theta }}={\frac {a}{b}}\tan \theta }
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I've come to the conclusion that my θ above is the parametric latitude. The formula for
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comparable. In this case, the simpler solution (straight lines) is to be prefered.
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which I believe is a somewhat standard symbol for it in geodesy. This article uses
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We could change the symbol for the geocentric latitude to something other than
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for that quantity (with the prime used to connote the co-quantity). Comments?
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1691:{\displaystyle \theta (\phi )=\tan ^{-1}\left({\frac {b}{a}}\tan \phi \right)}
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whether animated renderings of the ellipsoid help illustrate this concept.
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In A R Clarke, Geodesy on p. 103 it is stated that a tan(u) = b tan(φ).
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as the colatitude for spherical coordinates; so I further propose using
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2523:{\displaystyle \theta =\tan ^{-1}\left({\frac {b}{a}}\tan \phi \right)}
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whether it's useful to define a "surface of latitude" as a hyperboloid.
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Actually, I was wrong about where theta was introduced: it was in
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Normal height means the perpendicular height above the ellipsoid.
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A more realistic (but still not great) model of the earth is the
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Similar remarks apply to your figure d with a hyperbolic surface.
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somewhere on Knowledge. Does anyone object to its contents? ➧
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hasn't been active in two years, so good luck asking him.
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If you are curious you can read the Knowledge pages about
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A formula derived using Differential Geometry indicates
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Then, ignoring the common factor, the normal vector is
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High-importance maritime transport task force articles
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which defines a point on the surface of the ellipsoid
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The section "Geodetic and geocentric latitudes" says:
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1920:
1890:
1855:
1832:
1713:
23:for discussing improvements to the
3537:Unknown-importance PSP SA articles
3487:High-importance Transport articles
3452:High-importance geography articles
3354:article and the section on WGS84.
1867:
1848:
995:Knowledge:WikiProject South Africa
14:
3547:WikiProject South Africa articles
3542:Knowledge Primary School articles
3517:Mid-priority mathematics articles
2373:relative to the equatorial plane
998:Template:WikiProject South Africa
884:Knowledge:WikiProject Mathematics
563:Geographical coordinates articles
312:WikiProject Geography To-do list:
3472:High-importance history articles
3422:Knowledge level-4 vital articles
2825:{\displaystyle {\overline {CP}}}
1553:Geodetic and astronomic latitude
1040:Knowledge Primary School project
962:
952:
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887:Template:WikiProject Mathematics
851:
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685:
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347:Missing articles about Locations
321:
242:
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178:
169:
138:
45:Click here to start a new topic.
3328:Usage of geodetic vs geocentric
1015:This article has been rated as
904:This article has been rated as
775:This article has been rated as
755:Knowledge:WikiProject Transport
664:This article has been rated as
295:This article has been rated as
275:Knowledge:WikiProject Geography
3507:WikiProject Transport articles
3457:WikiProject Geography articles
3432:B-Class level-4 vital articles
2863:
2357:
2305:
2269:
2251:
2203:
2170:
2146:
2013:
1989:
1971:
1924:
1894:
1859:
1836:
1717:
1637:
1631:
758:Template:WikiProject Transport
547:, which encourages the use of
278:Template:WikiProject Geography
198:It is of interest to multiple
1:
3522:B-Class South Africa articles
2996:and comments that Adams uses
1037:This article is supported by
989:and see a list of open tasks.
878:and see a list of open tasks.
797:This article is supported by
749:and see a list of open tasks.
644:Knowledge:WikiProject History
638:and see a list of open tasks.
369:Unassessed geography articles
269:and see a list of open tasks.
42:Put new text under old text.
3512:B-Class mathematics articles
3477:WikiProject History articles
2817:
1609:05:07, 5 February 2021 (UTC)
1589:03:59, 27 January 2021 (UTC)
1574:21:29, 26 January 2021 (UTC)
1457:Currently, the article uses
1443:05:04, 5 February 2021 (UTC)
647:Template:WikiProject History
2928:{\displaystyle \theta _{p}}
1545:18:01, 30 August 2018 (UTC)
1519:14:33, 29 August 2018 (UTC)
1504:13:25, 29 August 2018 (UTC)
1417:13:35, 29 August 2018 (UTC)
1394:02:47, 29 August 2018 (UTC)
1355:14:38, 30 August 2018 (UTC)
1306:14:26, 28 August 2018 (UTC)
1254:03:32, 27 August 2018 (UTC)
1235:14:45, 26 August 2018 (UTC)
1203:22:11, 24 August 2018 (UTC)
1173:02:10, 24 August 2018 (UTC)
1143:23:57, 23 August 2018 (UTC)
1121:23:54, 23 August 2018 (UTC)
1099:20:22, 23 August 2018 (UTC)
1083:20:15, 23 August 2018 (UTC)
50:New to Knowledge? Welcome!
3563:
3482:B-Class Transport articles
3447:B-Class geography articles
3402:02:21, 2 August 2023 (UTC)
3385:21:23, 1 August 2023 (UTC)
3364:19:57, 1 August 2023 (UTC)
3065:The user who introduced it
1066:question on Stack Exchange
1021:project's importance scale
781:project's importance scale
670:project's importance scale
301:project's importance scale
3348:Google Maps documentation
3309:02:50, 19 June 2023 (UTC)
3294:23:23, 18 June 2023 (UTC)
3282:22:57, 18 June 2023 (UTC)
3268:22:39, 18 June 2023 (UTC)
3251:23:27, 18 June 2023 (UTC)
3241:Geodesics on an ellipsoid
3211:23:22, 18 June 2023 (UTC)
3198:22:20, 18 June 2023 (UTC)
3176:20:23, 18 June 2023 (UTC)
3117:{\displaystyle \lambda .}
3077:20:22, 18 June 2023 (UTC)
3036:{\displaystyle \phi _{c}}
2989:{\displaystyle \phi _{g}}
2945:19:43, 18 June 2023 (UTC)
2904:20:16, 17 June 2023 (UTC)
2889:19:20, 17 June 2023 (UTC)
2842:07:25, 17 June 2023 (UTC)
2793:14:42, 17 June 2023 (UTC)
2758:05:53, 17 June 2023 (UTC)
2744:05:45, 17 June 2023 (UTC)
2725:05:33, 17 June 2023 (UTC)
2702:04:04, 17 June 2023 (UTC)
2686:03:30, 17 June 2023 (UTC)
2667:00:58, 17 June 2023 (UTC)
2652:16:55, 16 June 2023 (UTC)
2617:04:46, 15 June 2023 (UTC)
2602:05:13, 14 June 2023 (UTC)
1036:
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80:Be welcoming to newcomers
3467:B-Class history articles
2342:Since φ is the angle of
1318:figures in this article.
1185:I am thinking of adding
978:WikiProject South Africa
910:project's priority scale
554:Geographical coordinates
549:geographical coordinates
519:Geographical coordinates
444:Knowledge requested maps
409:Geographical coordinates
3532:B-Class PSP SA articles
3286:What does this mean? ––
3157:{\displaystyle \theta }
3056:{\displaystyle \theta }
2962:{\displaystyle \theta }
2780:eccentricity § ellipses
1059:
867:WikiProject Mathematics
800:the Maritime task force
3417:B-Class vital articles
3229:
3215:Previous to that edit
3182:special:diff/857259930
3158:
3138:
3137:{\displaystyle \beta }
3118:
3095:
3057:
3043:. Not sure why we use
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2990:
2963:
2929:
2873:
2826:
2671:Torge (1980) is here,
2584:
2524:
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2367:
2333:
2228:
2122:
1946:
1808:
1692:
1263:There are two issues:
1033:
793:
75:avoid personal attacks
3390:Here's an explanation
3352:World Geodetic System
3230:
3228:{\displaystyle \psi }
3159:
3139:
3119:
3096:
3094:{\displaystyle \phi }
3058:
3038:
3011:
3009:{\displaystyle \psi }
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738:WikiProject Transport
363:WikiProject Geography
258:WikiProject Geography
185:level-4 vital article
132:Auto-archiving period
100:Neutral point of view
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890:mathematics articles
105:No original research
3337:question, by whom?
1339:I recommend Tufte,
970:South Africa portal
627:WikiProject History
3225:
3184:, made in 2018 by
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1287:Maclaurin spheroid
1034:
859:Mathematics portal
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761:Transport articles
539:is of interest to
281:geography articles
194:content assessment
86:dispute resolution
47:
3340:So two thoughts:
2866:
2861:
2820:
2768:eccentric anomaly
2654:
2638:comment added by
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1700:Given the vector
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66:Assume good faith
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1615:Is θ(φ) correct?
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395:Deletion sorting
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2460:And therefore,
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777:High-importance
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3101:and longitude
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2969:. Snyder uses
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1159:Okay thanks. ➧
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1017:Low-importance
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942:Low‑importance
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1509:Yes, please.
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803:(assessed as
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110:Verifiability
108:
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52:Learn to edit
49:
46:
41:
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37:
36:
32:
26:
22:
18:
17:
3371:
3345:
3342:
3339:
3335:
3331:
2772:mean anomaly
2764:true anomaly
2634:— Preceding
2591:
2534:
2531:
2462:
2459:
2375:
2341:
2235:
2129:
1956:
1953:
1818:
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1345:
1340:
1338:
1310:
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1296:
1292:
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1277:
1273:
1262:
1216:distracting.
1064:I started a
1063:
1056:
1038:
1016:
992:South Africa
983:South Africa
976:
933:South Africa
906:Mid-priority
905:
865:
831:Mid‑priority
798:
776:
736:
665:
625:
541:
540:
536:
477:
476:
463:
462:
449:
448:
435:
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421:
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379:
378:
352:
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334:
333:
296:
256:
200:WikiProjects
183:
135:
121:
94:
19:This is the
881:Mathematics
872:mathematics
828:Mathematics
31:not a forum
3411:Categories
3301:Jbergquist
3274:Jbergquist
3260:Jbergquist
2937:Jbergquist
2896:Jbergquist
2881:Jbergquist
2834:Jbergquist
2776:flattening
2750:Jbergquist
2736:Jbergquist
2694:Jbergquist
2659:Jbergquist
2609:Jbergquist
2594:Jbergquist
1329:cut-aways.
451:Notability
400:Listed at
3288:jacobolus
3245:jacobolus
3237:user:cffk
3205:jacobolus
3186:user:cffk
3170:jacobolus
2787:jacobolus
2680:jacobolus
1599:datumizer
1433:datumizer
1384:datumizer
1244:datumizer
1193:datumizer
1163:datumizer
1111:datumizer
1073:datumizer
752:Transport
743:Transport
693:Transport
272:Geography
263:geography
219:Geography
188:is rated
88:if needed
71:Be polite
21:talk page
3394:Apocheir
3190:Apocheir
3069:Apocheir
2648:contribs
2636:unsigned
2630:this one
2626:this one
698:Maritime
537:Latitude
136:730 days
123:Archives
56:get help
29:This is
27:article.
25:Latitude
3350:or the
1527:Done.
1360:Quote,
1019:on the
908:on the
779:on the
668:on the
641:History
632:History
588:History
423:Infobox
381:Cleanup
299:on the
190:B-class
3377:Strebe
3063:here.
2782:, etc.
2717:Strebe
2640:Strebe
1581:Strebe
1511:Strebe
1405:inside
938:PSP SA
354:Assess
196:scale.
3311:sure.
1322:easy.
1311:LATER
479:Stubs
465:Photo
177:This
84:Seek
3398:talk
3381:talk
3372:same
3360:talk
3305:talk
3278:talk
3264:talk
3194:talk
3073:talk
2941:talk
2900:talk
2885:talk
2838:talk
2754:talk
2740:talk
2721:talk
2698:talk
2663:talk
2644:talk
2613:talk
2598:talk
1585:talk
1570:talk
1541:talk
1537:cffk
1515:talk
1500:talk
1496:cffk
1413:talk
1409:cffk
1368:and
1351:talk
1347:cffk
1302:talk
1298:cffk
1231:talk
1227:cffk
1187:this
1139:talk
1135:cffk
1095:talk
1091:cffk
771:High
660:High
484:See
470:See
456:See
442:See
428:See
414:See
386:See
371:and
345:and
341:See
291:High
73:and
3291:(t)
3248:(t)
3243:. –
3208:(t)
3173:(t)
2790:(t)
2683:(t)
2628:or
2556:tan
2545:tan
2507:tan
2477:tan
2440:tan
2415:cos
2401:sin
2383:tan
2321:sin
2285:cos
2216:sin
2183:cos
2130:If
2102:sin
2086:sin
2077:cos
2061:cos
2052:cos
2032:cos
1788:sin
1772:sin
1763:cos
1747:cos
1738:cos
1675:tan
1645:tan
1212:c).
1011:Low
900:Mid
437:Map
3413::
3400:)
3383:)
3362:)
3307:)
3280:)
3266:)
3223:ψ
3196:)
3152:θ
3132:β
3109:λ
3089:ϕ
3075:)
3051:θ
3025:ϕ
3004:ψ
2978:ϕ
2957:θ
2943:)
2917:θ
2902:)
2887:)
2864:→
2840:)
2818:¯
2778:,
2774:,
2770:,
2766:,
2756:)
2742:)
2723:)
2700:)
2665:)
2650:)
2646:•
2632:.
2615:)
2600:)
2562:ϕ
2559:
2551:θ
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2510:
2489:
2481:−
2470:θ
2446:θ
2443:
2421:θ
2418:
2407:θ
2404:
2389:ϕ
2386:
2358:→
2339:.
2327:θ
2324:
2306:^
2291:θ
2288:
2270:^
2252:→
2222:λ
2219:
2204:^
2189:λ
2186:
2171:^
2159:≡
2147:^
2108:θ
2105:
2092:λ
2089:
2083:θ
2080:
2067:λ
2064:
2058:θ
2055:
2038:θ
2035:
2021:θ
2014:→
2002:×
1997:λ
1990:→
1978:≡
1972:→
1940:λ
1932:λ
1925:→
1910:θ
1902:θ
1895:→
1880:λ
1871:λ
1868:∂
1860:→
1849:∂
1837:→
1794:θ
1791:
1778:λ
1775:
1769:θ
1766:
1753:λ
1750:
1744:θ
1741:
1718:→
1681:ϕ
1678:
1657:
1649:−
1635:ϕ
1629:θ
1604:☎
1587:)
1572:)
1543:)
1517:)
1502:)
1438:☎
1415:)
1389:☎
1353:)
1304:)
1249:☎
1233:)
1198:☎
1168:☎
1141:)
1116:☎
1097:)
1078:☎
936::
807:).
696::
365:}}
361:{{
134::
54:;
3396:(
3379:(
3358:(
3303:(
3276:(
3262:(
3192:(
3168:–
3112:.
3071:(
3029:c
2982:g
2939:(
2921:p
2898:(
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2860:r
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2811:C
2752:(
2738:(
2719:(
2696:(
2675:.
2661:(
2642:(
2611:(
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2576:a
2573:b
2568:=
2517:)
2502:a
2499:b
2493:(
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2473:=
2435:b
2432:a
2427:=
2412:b
2398:a
2392:=
2354:v
2318:a
2313:z
2302:e
2294:+
2282:b
2277:w
2266:e
2258:=
2248:v
2211:y
2200:e
2192:+
2178:x
2167:e
2154:w
2143:e
2114:)
2099:a
2074:b
2049:b
2043:(
2029:a
2026:=
2010:g
1986:g
1968:v
1937:d
1921:g
1913:+
1907:d
1891:g
1883:=
1877:d
1856:r
1843:=
1833:r
1826:d
1800:)
1785:b
1760:a
1735:a
1729:(
1724:=
1714:r
1685:)
1670:a
1667:b
1661:(
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1641:=
1638:)
1632:(
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1539:(
1533::
1529:@
1513:(
1498:(
1486:′
1484:θ
1478:θ
1472:θ
1466:ψ
1460:ψ
1411:(
1380:➧
1349:(
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1229:(
1137:(
1093:(
1069:➧
1043:.
1023:.
912:.
783:.
672:.
482::
468::
454::
440::
426::
412::
398::
384::
375:.
357::
339::
303:.
202:.
128:1
125::
58:.
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