Talk:Latitude

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. ➧datumizer  ☎  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)

Confusion with "altitude"?

user:MakaylaHippo1998 you just added a hatnote pointing to "altitude" here. Is "latitude" really commonly confused with "altitude"? I'm not convinced this is necessary. Remember, there's a slight distraction cost with every extra hatnote we add. –jacobolus (t) 05:37, 23 July 2025 (UTC)

I agree. Altitude is not typically confused with latitude. cffk (talk) 12:13, 23 July 2025 (UTC)

Conformal latitudes on triaxial ellipsoids

In a recent paper H. Hakula, A. Rasila: Laplace–Beltrami Equations and Numerical Conformal Mappings on Surfaces. SIAM Journal on Scientific ComputingVol. 47, Iss. 1 (2025) https://doi.org/10.1137/24M1656840 we computed a Mercator-type map projection for a triaxial ellipsoid corresponding to the parameters of the dwarf planet Haumea. This example is somewhat interesting, because the conformal images of latitudes are not straight lines on the map. I can contribute the images, if there is interest in adding this example. Arasila (talk) 06:01, 25 December 2025 (UTC)

I think that would be interesting and informative. Strebe (talk) 04:18, 27 December 2025 (UTC)