Talk:Latitude

Latitude above and below the Earth's surface
I started a question on Stack Exchange 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. &#x27A7; datumizer   &#9742;  20:15, 23 August 2018 (UTC)


 * 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". cffk (talk) 20:22, 23 August 2018 (UTC)


 * So geographic latitude does not describe points above and below the surface at all? Or, what does "normal height" mean? &#x27A7; datumizer   &#9742;  23:54, 23 August 2018 (UTC)


 * Normal height means the perpendicular height above the ellipsoid. cffk (talk) 23:57, 23 August 2018 (UTC)


 * Okay thanks. &#x27A7; datumizer   &#9742;  02:10, 24 August 2018 (UTC)


 * I am thinking of adding this somewhere on Wikipedia. Does anyone object to its contents? &#x27A7; datumizer   &#9742;  22:11, 24 August 2018 (UTC)


 * 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 c).
 * 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 distracting.
 * Similar remarks apply to your figure d with a hyperbolic surface.
 * 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 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).
 * cffk (talk) 14:45, 26 August 2018 (UTC)
 * 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 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. &#x27A7; datumizer   &#9742;  03:32, 27 August 2018 (UTC)

There are two issues:


 * whether it's useful to define a "surface of latitude" as a hyperboloid.
 * whether animated renderings of the ellipsoid help illustrate this concept.

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 comparable. In this case, the simpler solution (straight lines) is to be prefered.

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 Stack Exchange shows the field for the unrealistic case with all the mass of the earth concentrated on an equatorial disc.

A more realistic (but still not great) model of the earth is the Maclaurin spheroid 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.

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 in this case) or, indeed, a "surface of longitude" (a meridian plane). However, I've never seen this idea adopted in the literature and Wikipedia shouldn't be used to promote it. (Indeed the most usual surface associated with latitude is the plane parallel to the equator.)

cffk (talk) 14:26, 28 August 2018 (UTC)

LATER 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". I recommend Tufte, The Visual Display of Quantitative Information (1983), for useful advice (including keeping your graphics as simple as possible).
 * 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 figures in this article.
 * 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 easy.
 * 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.
 * 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 cut-aways.
 * Animations are distracting when trying to read the text. Multiple this by 4 for 4 synchronized animations.
 * 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).

cffk (talk) 14:38, 30 August 2018 (UTC)


 * Quote, "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." I can see how this would be the case when the flatting of the Earth is only 1/300. But you can see in File:Surface of latitude ellipsoid cone.gif and File:Surface of latitude ellipsoid hyperboloid.gif (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.
 * Anyway, I think the article threw me when it said, "These coordinates are the natural choice in models of the gravity field for a uniform distribution of mass bounded by the reference ellipsoid." 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? &#x27A7; datumizer   &#9742;  02:47, 29 August 2018 (UTC)


 * 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 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 inside a uniform ellipsoid is relatively easy without using ellipsoidal coordinates.  I'll look into editing this sentence. cffk (talk) 13:35, 29 August 2018 (UTC)


 * (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. &#x27A7; datumizer   &#9742;  05:04, 5 February 2021 (UTC)

Nomenclature for geocentric latitude
Currently, the article uses $ψ$ for the geocentric latitude. This is, I beliveve, a non-standard usage and it conflicts with the use of $ψ$ for isometric latitude. I propose to use $θ$ instead. There is now a conflict with the use of $θ$ as the colatitude for spherical coordinates; so I further propose using $θ′$ for that quantity (with the prime used to connote the co-quantity). Comments?

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 assume that the auxiliary latitude were bigger can the geodetic latitude -- although the text did try to address this issue.)

cffk (talk) 13:25, 29 August 2018 (UTC)


 * Yes, please. Strebe (talk) 14:33, 29 August 2018 (UTC)


 * Done.  your figures needs updating; perhaps you can take care of this?  Let me know if there are any issues. cffk (talk) 18:01, 30 August 2018 (UTC)

Geodetic and astronomic latitude
The section "Geodetic and geocentric latitudes" says:

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.

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 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. 150.227.15.253 (talk) 21:29, 26 January 2021 (UTC)


 * See the section “Astronomical latitude”. Strebe (talk) 03:59, 27 January 2021 (UTC)


 * 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. &#x27A7; datumizer   &#9742;  05:07, 5 February 2021 (UTC)

Is θ(φ) correct?
A formula derived using Differential Geometry indicates

$$\theta(\phi) = \tan^{-1}\left(\frac{b}{a}\tan\phi\right)$$

Given the vector

$$\vec{\textbf{r}}=\begin{pmatrix}a\cos\theta\cos\lambda\\a\cos\theta\sin\lambda\\b\sin\theta\end{pmatrix}$$

which defines a point on the surface of the ellipsoid

$$ d\vec{\textbf{r}} = \frac{\partial\vec{\textbf{r}}} {\partial\lambda}d\lambda = \vec{\textbf{g}}_\theta d\theta + \vec{\textbf{g}}_\lambda d\lambda$$

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

$$\vec{\textbf{v}}\equiv \vec{\textbf{g}}_\lambda \times \vec{\textbf{g}}_\theta=a\cos\theta\begin{pmatrix}b\cos\theta\cos\lambda\\b\cos\theta\sin\lambda\\a\sin\theta\end{pmatrix}$$

If $$\hat{\textbf{e}}_w\equiv\hat{\textbf{e}}_x\cos\lambda + \hat{\textbf{e}}_y\sin\lambda$$

Then, ignoring the common factor, the normal vector is $$\vec{\textbf{v}}=\hat{\textbf{e}}_w b\cos\theta + \hat{\textbf{e}}_z a\sin\theta$$.

Since φ is the angle of $$\vec{\textbf{v}}$$ relative to the equatorial plane

$$\tan\phi=\frac{a\sin\theta}{b\cos\theta}=\frac{a}{b}\tan\theta$$

And therefore,

$$\theta=\tan^{-1}\left(\frac{b}{a}\tan\phi\right)$$

I checked this formula numerically using small angular separations to approximate the normal vectors and determine their angles and it appears

$$\frac{\tan\theta}{\tan\phi} = \frac{b}{a}$$

is consistently true to the desired accuracy. Jbergquist (talk) 05:13, 14 June 2023 (UTC)


 * In A R Clarke, Geodesy on p. 103 it is stated that a tan(u) = b tan(φ). Jbergquist (talk) 04:46, 15 June 2023 (UTC)


 * 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 this one or this one. — Preceding unsigned comment added by Strebe (talk • contribs) 16:55, 16 June 2023 (UTC)
 * Note that geodesists use 2 definitions of eccentricity. See Torge & Müller, Geodesy, p. 91. Jbergquist (talk) 00:58, 17 June 2023 (UTC)
 * Torge (1980) is here, https://archive.org/details/geodesyintroduct0000torg/page/48/.
 * 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? –jacobolus (t) 03:30, 17 June 2023 (UTC)
 * There may be some confusion among alternative definitions of eccentricity. The article needs clarification. See Eccentricity (mathematics), Alternative names. Jbergquist (talk) 04:04, 17 June 2023 (UTC)


 * 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. Strebe (talk) 05:33, 17 June 2023 (UTC)
 * A R Clarke uses the first eccentricity which is
 * e=√(1-b²/a²). So his b/a=√(1-e²). Jbergquist (talk) 05:45, 17 June 2023 (UTC)
 * The angle β for the parametric latitude appears to be related to Kepler's eccentric anomaly. Jbergquist (talk) 05:53, 17 June 2023 (UTC)
 * If you are curious you can read the Wikipedia pages about true anomaly, eccentric anomaly, mean anomaly, flattening, eccentricity § ellipses, etc.
 * But if you have changes you want to see to this article, can you be a little more specific about what they are? –jacobolus (t) 14:42, 17 June 2023 (UTC)
 * 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 $$\overline{CP}$$. For the eccentric anomaly the radial distance is measured from a focus of the ellipse. Jbergquist (talk) 07:25, 17 June 2023 (UTC)
 * I've come to the conclusion that my θ above is the parametric latitude. The formula for $$\vec{\textbf{r}}$$ 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. Jbergquist (talk) 19:20, 17 June 2023 (UTC)
 * 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 coordinates helps clarify the situation. Jbergquist (talk) 20:16, 17 June 2023 (UTC)
 * My recommendation would be to use θ' or $$\theta_p$$ in the parametric expression for a point on the surface of an ellipsoid to avoid confusion. Could we do a RFC on this? Jbergquist (talk) 19:43, 18 June 2023 (UTC)
 * We could change the symbol for the geocentric latitude to something other than $$\theta$$. Snyder uses $$\phi_g$$ and comments that Adams uses $$\psi$$, while the Osborne reference uses $$\phi_c$$. Not sure why we use $$\theta$$ here. The user who introduced it hasn't been active in two years, so good luck asking him. Apocheir (talk) 20:22, 18 June 2023 (UTC)
 * To specify a point on an ellipsoid of revolution, typically people use the geodetic latitude $$\phi$$ and longitude $$\lambda.$$ In this article the parametric latitude is called $$\beta$$ which I believe is a somewhat standard symbol for it in geodesy. This article uses $$\theta$$ to mean the geocentric latitude, which again is, I believe, a somewhat standard symbol. For example Karney adopts these symbols, https://arxiv.org/pdf/2212.05818.pdf –jacobolus (t) 20:23, 18 June 2023 (UTC)
 * Actually, I was wrong about where theta was introduced: it was in special:diff/857259930, made in 2018 by user:cffk, who is (according to his user page) Charles Karney, author of the paper you linked. (He's also not very active on Wikipedia 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. Apocheir (talk) 22:20, 18 June 2023 (UTC)
 * 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 would recommend doing a more serious literature survey, especially focusing on the most popular recent textbooks. –jacobolus (t) 23:22, 18 June 2023 (UTC)
 * Previous to that edit $$\psi$$ was used for both isometric latitude and geocentric latitude, which is incredibly confusing. Note that Karney (user:cffk) is one of the world's foremost experts on this topic. Here at Wikipedia he e.g. wrote and illustrated the excellent page Geodesics on an ellipsoid. –jacobolus (t) 23:27, 18 June 2023 (UTC)
 * 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 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. Jbergquist (talk) 22:39, 18 June 2023 (UTC)
 * We must think in terms of the greater good. Jbergquist (talk) 22:57, 18 June 2023 (UTC)
 * What does this mean? ––jacobolus (t) 23:23, 18 June 2023 (UTC)
 * I was thinking in terms of simplifying the math required to understand Geodesy, Wikipedia 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 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. Jbergquist (talk) 02:50, 19 June 2023 (UTC)sure.

Usage of geodetic vs geocentric
The article currently states: "[Geodetic latitude] is the definition assumed when the word latitude is used without qualification." This statement is repeated (nearly verbatim) elsewhere across related Wikipedia articles. I believe it could use some clarification and possibly sourcing.

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 question, by whom?

So two thoughts:

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.

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. Google Maps documentation or the World Geodetic System article and the section on WGS84. RileyJohnGibbs (talk) 19:57, 1 August 2023 (UTC)


 * 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 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 same 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. Strebe (talk) 21:23, 1 August 2023 (UTC)
 * Here's an explanation, although it's not one we can cite. Everyone just seems to take geodetic latitude for granted, but it's hard to find a reference that says that. Apocheir (talk) 02:21, 2 August 2023 (UTC)