# Talk:Latitude

## Formula

I think I know how to derive authalic latitude. Take the ellipsoid ${\displaystyle E=(\cos \eta \cos \lambda ,\cos \eta \sin \lambda ,b\sin \eta )}$ and the sphere ${\displaystyle S=(\cos \beta \cos \lambda ,\cos \beta \sin \lambda ,\sin \beta )}$. Note that ${\displaystyle \partial E/\partial \eta \cdot \partial E/\partial \lambda =\partial S/\partial \beta \cdot \partial S/\partial \lambda =0}$ so the areal elements are ${\displaystyle |\partial E/\partial \eta ||\partial E/\partial \lambda |}$ and ${\displaystyle |\partial S/\partial \beta ||\partial S/\partial \lambda |}$ so we equate those, throwing in an arbitrary constant factor and a correction factor for the different variables: ${\displaystyle |\partial E/\partial \eta ||\partial E/\partial \lambda |=A|\partial S/\partial \beta ||\partial S/\partial \lambda |d\beta /d\eta }$. Expand to get ${\displaystyle {\sqrt {\sin ^{2}\eta +b^{2}\cos ^{2}\eta }}\cos \eta \,d\eta =A\cos \beta \,d\beta }$, integrate (use ${\displaystyle u=\sin \eta }$), adjust the arbitrary constants so ${\displaystyle \eta =\beta =0}$ and ${\displaystyle \eta =\beta =\pi /2}$ happen, and substitute ${\displaystyle \tan \eta =b\tan \phi }$ and ${\displaystyle b^{2}+e^{2}=1}$. (I was all night getting the right pieces *shame*.) 142.177.169.65 15:41, 12 Aug 2004 (UTC)

The continents along the equator, Africa, South America, and Indonesia are the poorest

Is Indonesia a continent? Strange. --Nk 12:45, 5 Oct 2004 (UTC)

I think it's fair to state that Antarctica, with a GDP of approximately zero, is both a continent, and the poorest. But it ain't quite equatorial!--King Hildebrand 17:13, 8 February 2007 (UTC)

Another strange thing:

Each degree of latitude is further sub-divided into 60 "minutes". In modern navigation, part of a minute may be expressed as a decimal. A fully qualified latitude may be expressed thus; 13° 19.717′ N. Until the 1960s, parts of a minute were normally expressed in seconds; for instance 13° 19′ 42" N. There are 60 seconds in a minute.

This is outright wrong. Deg/min/sec is very common usage today, with fractional seconds for accuracy. The alternative is fractional degrees. Fractional minutes are in fact not so common. -- Egil 06:40, 12 Feb 2005 (UTC) Some GPS systems output into fractional minutes - because I am having a horrible time with using the outputted values. --84.64.183.208 (talk) 14:27, 7 April 2009 (UTC)

## "angular measurement" ?

Article starts with:

Latitude is an angular measurement ranging from 0° at the Equator to 90° at the poles

is Latitude really an angular measurement? Since it's not really an angle but and evenly spaced distance apart? SimonLyall 13:59, 24 Mar 2005 (UTC)

Rectifying latitude is not geographic latitude.--Patrick 14:09, 24 Mar 2005 (UTC)
But is geographic latitude a "angular measurement" ? , sorry I'm not an expert but it just doesn't look right. SimonLyall 20:31, 24 Mar 2005 (UTC)
It is the angle between the equator plane and the line from the center of the Earth to the location.--Patrick 21:48, 24 Mar 2005 (UTC)
No, geocentric is the angle between the equator and a line from the center of the Earth. Geodetic and geographic are the same thing and are the angle between the equator and a normal to the reference spheroid. 2005 June 22, 13:44 EDT

## geodetic latitude is not referred to a plumb line

This section is wrong: In common usage "latitude" refers to geodetic or geographic latitude φ and is the angle between a plumb line and the equatorial plane — because it originated as the angle between horizon and pole star. Because the Earth is slightly flattened by its rotation, cartographers refer to a variety of auxiliary latitudes to precisely adapt spherical projections according to their purpose.

Geodetic or geographic latitude is the angle between the equatorial plane and a line normal to a reference spheroid. Astronomical latitude is the angle between the equatorial plane and the normal to the local geopotential (ie a plumb line). These are not quite the same thing. I'll to figure out better wording to change this. 2005 June 22, 13:44 EDT

## Distance between "Latitutdes

How does one calculate the distance (in meters) between two co-ordinates? I'm most intersted in finding out! --TheSimkin 16:44, July 20, 2005 (UTC)

I suggest a rewording of the sentence: "Reduced or parametric latitude β is the latitude of the same radius on the sphere with the same equator." to the following: "Reduced or parametric latitude β is the latitude of the same radius projected along the minor axis on the sphere with the same equator." Otherwise it could be confused with the Geocentric Latitude. Furthermore, I agree with EDT in referring the geodetical latitude to the (local) normal to the spheroid rather to the plumb line, but it seems to me that the math is congruent with the former, as geodetic and geocentric latitudes are in the correct relationship. [Netsaver]

## Latitude

I can not calculate authalic latitude. Way over my head. Please avoid putting that in.— Preceding unsigned comment added by 65.0.202.213 (talkcontribs) 02:34, 3 December 2005 (UTC)

## Latitude and Wealth

I think it is wrong to state that there is a distinct correlation. IMO there is a general correlation because there are many exceptions to the rule. In Africa the richest countries are South Africa and Nigeria, Nigeria is a clear exception. Also within richer countries (e.g. the UK), the North is poorer than the South. There are way too many factors to state categorically that the closer you are to the equator the poorer you will be. I think this section needs rewording to take this into account. --138.37.219.207 12:01, 10 April 2006 (UTC)

I agree. This discussion seems out of place in this venue. Perhaps this can be moved to its own category? S Schaffter 18:45, 26 July 2006 (UTC)

I agree that the 'correlation' (soundly debunked) should be moved to its own page, or possibly merged with http://en.wikipedia.org/wiki/Charles_de_Secondat%2C_Baron_de_Montesquieu, http://en.wikipedia.org/wiki/Eugenics, or http://en.wikipedia.org/wiki/Industrial_Revolution.

I'm not saying these are perfect places to put it, but they're markedly better than an article on navigation. Sorry, but I don't know how to use wikilinks -- FM.— Preceding unsigned comment added by 203.129.52.30 (talkcontribs) 05:10, 4 August 2006 (UTC)

I agree with moving these sections out. They are not central to the topic and they are fringe theories. —The preceding unsigned comment was added by Futurebird (talkcontribs) 02:26, 13 March 2007 (UTC).

I too, agree to move this out. The statement, which might be misleading if the circumstance is not well defined, is not germane to the topic here. --Natasha2006 14:03, 10 April 2007 (UTC)

In the particular cases named, Nigeria is not one of the wealthiest countries in Africa, being the 36th country over 53 African countries in GDP per capita. About the reccomendations to move the topic to other pages, I find the proposed pages very unappropriate. —Preceding unsigned comment added by 186.48.15.252 (talk) 02:58, 16 November 2010 (UTC)

## Gravity

I've heard that gravity changes depending on your latitude, at 45.5° (and sea level) acceleration due to gravity is 9.80665m/s². How much does that change as you go north or south? Thrawst 02:03, 9 May 2006 (UTC)

Gravity is the attraction between two masses so gravitational force is independent of location on earth (grossly made assumptions about altitude, land vs. water, spherical planet, yada, yada). On the equator the angular velocity is much higher than if you travel away from the equator; since the rotation is wanting to throw you off the planet (think being at the edge of a merry-go-round vs. in the center) then your weight will be less on the equator. Cburnett 05:31, 25 August 2006 (UTC)
The Earth to a first approximation is a sphere, and Cburnett's comments apply. A second approximation notes the flattening of the poles by about 40 km relative to the equator (can't remember offhand whether this figure applies to radius or diameter). Therefore you're closer to the centre of the Earth at the poles, and hence gravity is greater. Then throw in mountains and depressions, each varying your distance from the centre - the lower you go, the greater the acceleration due to gravity becomes. Finally recognise local density variations in the rocks making up the Earth's surface (deeper ones have little effect). If you're standing above a copper mine, you will experience greater gravity than if you're on sedimentary rock or ocean. --King Hildebrand 17:24, 8 February 2007 (UTC)

Getting back to the guy's original question: apparent weight at the equator is 0.5% less than at the poles. Tim Zukas (talk) 22:13, 16 August 2010 (UTC)

Additionally the apparent weight is modified by the centrifugal force of rotation. At the poles this effect is zero, at the equator it makes the preceived gravity smaller by 0.03 m/s2 or 0.34%. −Woodstone (talk) 12:01, 17 August 2010 (UTC)
When I said "apparent weight" I was including that effect. The total difference between the equator and the pole is 0.5%. Tim Zukas (talk) 18:35, 30 August 2010 (UTC)
Also see:
Earth's_gravity#Mathematical_models
-Ac44ck (talk) 01:38, 31 August 2010 (UTC)

## The Earth has no Latitude

The Earth has no Latitude because it is flat. Latitudes are curved but they can't be present b/c our planet is flat. It absolutely is. Because it isn't... think of a ball... ur on a ship and u go over the curved part and you'll be upside down. so that doesnt happen so earth is flat and so earth has no latitude.— Preceding unsigned comment added by 68.88.253.37 (talkcontribs) 17:47, 10 September 2006 (UTC)

I agree I think there is a POV problem with this article. I would like to see some proof that the world is round.— Preceding unsigned comment added by 137.195.15.26 (talkcontribs) 10:43, 27 October 2006 (UTC)
I'm going to commit suicide because of the mental retardation of whoever wrote those. Gravity keeps stuff from "falling down" if its upside down. But wouldn't (if your on the bottom of the round earth) up be the down of the top of the earth? If that doesnt make any sense jsut forget about it and keep thinking the earth is flat.— Preceding unsigned comment added by 71.212.97.221 (talkcontribs) 07:20, 2 December 2006 (UTC)
How do you, proponents of the flat earth, explain that the sun is well above the horizon at some places but, at the very same instant of time, well below the horizon at other places? In case you doubt this happening, just make phone calls to flat-earth proponents on other continents. – Rainald62 (talk) 14:38, 1 July 2009 (UTC)

## Latitude - is it phi or lambda?

I'm studying for a college exam and came across two different opinions....Wikipedia's and Nasa's. If you go to http://www-istp.gsfc.nasa.gov/stargaze/Slatlong.htm they state that latitude is lambda and longitude is phi. Your articles are just the opposite. Thought you might want to verify this and ensure your website is correct.

It depends on what source you use! P=/
In all of the (particularly geodetic) articles and formularies I've seen, latitude = phi and longitude = lambda (though, in a lot of ellipsoidal formularies, lambda is used as the ellipsoidal/auxiliary longitude and either "L" or something else is used for the geodetic/geographic longitude): See Vincenty (PDF) and Borre (also PDF). A basic explanation about these types of discrepancies can be found here (further confusing things!!! P=)  ~Kaimbridge~15:49, 4 November 2006 (UTC)

## Lowercase phi

Isn't latitude always lower case phi? I've never seen upper case used. I've changed the page mgb —Preceding unsigned comment added by 24.87.70.209 (talk) 01:23, 31 May 2009 (UTC)

## Astronomical latitude

Why do you call it "obscure"? To me it seems to be the most natural notion of latitude, the one the most easily to measure.

To me, e.g., reduced, authalic and conformal latitude seem to be much more obscure. Do they have any application or are they just mathematically interesting? --84.159.207.253 12:21, 2 December 2006 (UTC)

## This is a fringe theory

I don't think it makes sense to put it here. It could go in the article on J. Philippe Rushton. futurebird 02:25, 13 March 2007 (UTC)

### Evolutionary explanations

{{NPOV-section}} One controversial explanation currently being advocated by certain evolutionary psychologists, is claimed to be grounded in evolutionary theory. Some have argued that as humans migrated into higher latitudes and encountered colder weather there, the cold weather forced the evolution of higher group intelligence by forcing inhabitants to perform more intellectually demanding tasks, such as building shelter, fires, and clothing, in order to survive (Lynn, 1991).

One study that supports this notion was performed by Beals et al. (1984, p. 309), who found a correlation of 0.62 (p=0.00001) between latitude and cranial capacity in samples worldwide and reported that each degree of latitude was associated with an increase of 2.5 cm³ in cranial volume.

Researchers such as psychologist J. Philippe Rushton have argued that the association of greater brain size with greater latitude is due to the fact that cold weather imposes on its inhabitants more cognitively demanding tasks such as the need to construct shelter, make clothing, and store food.

Nevertheless, these explanations seem to be contradicted by the fact that it was in Africa, at near equatorial latitude, that harsh conditions such as extreme drought have brought our species, homo sapiens, to existence. Another contradicting observation is the high number of advanced civilizations that flourished near the equator -- such as Sumerian, Egyptian, Hindu among many others. The demanding tasks of shelter construction, cloth making and food storing, seem to be less likely to have constrained man's evolution since the invention of agriculture and writing.

## Self-contradiction and confusion

Quote from the article:

Each degree of latitude (111.32 km) is further sub-divided into 60 minutes. One minute of latitude is one nautical mile, defined exactly as 1852 metres (this is approximate due to slight variation with latitude (at sea level) and is because the earth is slightly oblate)

1852 meters * 60 = 111.120 km and not 111.32 km as this article states it does both exactly and approximate in the same sentence. The article is confused. —Pengo 11:00, 14 June 2007 (UTC)

It seems to vary from 110.574 to 111.694 km according to non-wikipedia sources, which makes one minute 1842.9 to 1861.5666... meters... It could be explained a lot better. and perhaps mention that a nautical mile is measured around 45 degrees latitude? I'll leave it to an expert to update the article. :) —Pengo 14:41, 14 June 2007 (UTC)
Yeah, the whole section was poorly expressed——it's now been updated by an expert! P=) ~Kaimbridge~15:18, 15 June 2007 (UTC)
You swapped the degree lengths for the poles and equator. They are defined as (π/180) times the radius of curvature, which is small at the equator and large at the poles due to flattening. Your lengths also do not match the international reference ellipsoid WGS84. Its equatorial radius is a = 6378137 m with a flattening of f = 1/298.257223563. The resulting eccentricity is e = sqrt(2f−f²) = 0.0818191908426. The length of a degree of latitude at the equator (0°) is (π/180)a(1−e²) = 110.574 km. The length of a degree of latitude at the poles (90°) is (π/180)a/sqrt(1−e²) = 111.694 km. The variation between the two is 110.574 + 1.120 sin²φ km, where φ is the latitude.
Historically, the nautical mile was not created to avoid Earth's oblateness. Indeed, it was created long before the true size of the Earth was known, let alone whether it was oblate. Hence each nation and even each writer had their own values for the nautical mile about 1500. The 1529 Treaty of Saragossa states that there are 17.5 leagues per equatorial degree (with three Spanish miles per league). Britain and France soon had their own values. These were merged to give the modern definition for a nautical mile of 1/60 of an equatorial degree. However, I'm not sure whether it was created before or after the ellipsoidal nature of the Earth was recognized, so I'm leaving that part alone for now. It was divorced from from the size of the Earth in 1929 when the nautical mile was defined to be exactly 1852 m. — Joe Kress 07:35, 18 June 2007 (UTC)
You're defining the meridional degree of arc not axis (i.e., a and b). I note in the next sentance that arc varies not only with latitude, but with direction, too.
Actually, it was originally based on a spherical Earth having a circumference of 40000 km, or a radius of 6366.19772367581 km (as it turns out, the average meridional radius is about 6367.447 km). Thus the nautical mile of 1852 m/111.12 km degree of arc equals a radius of 6366.70701949371 km. ~Kaimbridge~15:16, 18 June 2007 (UTC)
I have restored my comment. Do NOT place any part of your response within another's comment—that is unacceptable on Wikipedia because it makes it impossible to determine who said what, especially at a later date by someone reading this talk page for the first time.
As I understand the sentence in question, the meridional degree of arc of geodetic latitude was being described. A degree of arc of geocentric latitude would distort the meaning of a "degree of latitude". I do not understand what you mean by a degree of axis. Of course a and b enter into all equations of an ellipsoid. I do not understand how a degree of latitude can vary with direction because there are only two directions, north and south. It must have the same value whether it is traversed from south to north or north to south. I assume you are not referring to the arc of a non-north-south great circle—that doesn't seem to be called for within an article on latitude. The values I gave were lengths on a tangent to an ellipse, they ignore the fact that the beginning of any degree of latitude cannot be at the same latitude as the end of that same degree. Thus it would be more correct to define the length of a minute, not degree, of geodetic latitude. What are your sources?
The nautical mile was NOT originally based on 40000 km for the circumference of a spherical Earth. That doubly distorts the original metric definition of the metre, which was 1/10,000,000 of a meridional arc from a pole to the equator. When the metre was proposed in 1792, it was well known, especially in France, that the Earth was not spherical, so it was known that the arc being subdivided was not a circular arc. The metre and the kilometre were to replace all kinds of feet and miles, especially English and French and land and sea. The nautical mile was well known long before the metric system was proposed, so none of their original definitions could have relied on the kilometre. — Joe Kress 00:22, 19 June 2007 (UTC)
By "degree of axis", I'm just referring to the degree length equivalent of actual surface-to-center radius (as opposed to radius of arc/curvature). As to geocentric vs. geodetic/geographic latitude, the surface-to-center radius at any point can be expressed by either parametric, geographic or geocentric latitude:
{\displaystyle {\begin{aligned}{\color {white}{\Big |}}{\mbox{Radius}}&={\sqrt {(a\cos(\beta ))^{2}+(b\sin(\beta ))^{2}}},\\&={\sqrt {\frac {a^{4}\cos(\phi )^{2}+b^{4}\sin(\phi )^{2}}{(a\cos(\phi ))^{2}+(b\sin(\phi ))^{2}}}},\\&={\frac {b}{\sqrt {a^{4}\sin(\psi )^{2}+(ab\cos(\psi ))^{2}}}};\end{aligned}}\,\!}
Yes, I mean any arc——north-south (M), east-west (N) and anything in between: If you are standing on the equator facing north-south, the arcradius will equal ${\displaystyle \scriptstyle {\frac {b^{2}}{a}}\,\!}$, while facing east-west it will equal ${\displaystyle a\,\!}$. Why is east-west any less important than north-south when talking about latitude?
As for the origin of the nautical mile, my 40000 km circumference was referring to the "1/10,000,000 of a meridional arc" metre basis (times four): If you look at the history of the nautical mile, it says that it was "historically defined as a minute of arc along a meridian of the Earth", hence my above assertion/reference (I'm not saying I'm right and you're wrong, we just might be experiencing different interpretations of ther same basic facts! P=). BTW, what is your background——geodesist, math enthusiast, student or just hobbyist? I'm strictly an autodidactic amateur geodesist (the "principal problems"——i.e., calculating distance). ~Kaimbridge~16:06, 19 June 2007 (UTC)
I'm adding a section entitled "Degree length" giving the lengths of a degree of both latitude and longitude along with appropriate references, including a calculator from the U.S. government. I may include the length of a degree of a great circle for any direction at any latitude. — Joe Kress (talk) 03:51, 25 December 2007 (UTC)
Ah, okay, but are you sure you know what the "length of a degree of a great circle for any direction at any latitude" is——it's not the "radius of curvature", but "radius of arc": See explanation. If you are talking about a constant, spherical great circle degree value, the best choice isn't the authalic ("surface area"), either: See another discussion. P=) ~Kaimbridge~20:28, 25 December 2007 (UTC)

## Coplanar

"All locations of a given latitude are collectively referred to as a circle of latitude or line of latitude or parallel, because they are coplanar, and all such planes are parallel to the equator." This is a common misperception. Actually, the angle of latitude is measured between a ray that is normal to the surface of the sphere(-oid) and the equator. The revolution of this ray around the axis forms a cone. The intersection of this cone with the sphere(-oid) forms the circle mentioned in the article. If they were really coplanar as the article suggests, then locations of varying altitude would have differing latitudes. This can be easily demonstrated by drawing the cross-section of the sphere(-oid) and determining the intersection of the normal and the line of latitude. If points of varying elevation (i.e., lying along the normal) don't lie along the line of latitude, then they are not coplanar. All locations with the same latitude are only coplanar if they are mapped to a cylinder. SharkD (talk) 07:29, 17 February 2008 (UTC)

## Reduced Latitude

The description of reduced latitude makes no sense to me:

 ...is the latitude of the same radius on the sphere with the same equator.


Same as what? I think that this needs to be reworded. Ty8inf (talk) 16:38, 21 April 2008 (UTC)

Somebody tried to improve it
"On a spheroid, lines of reduced or parametric latitude, , form circles whose radii are the same as the radii of circles formed by the corresponding lines of latitude on a sphere with radius equal to the equatorial radius of the spheroid."
but it's still useless, since we don't know what "corresponding" means. A cross-section drawing is what we need; lacking a drawing, try the explanation that now follows the formulas in this section. Tim Zukas (talk) 16:54, 31 August 2010 (UTC)

## Degree length

I originally adjusted it to how you have it Woodstone, but Rracecarr disagreed.
I added the cosine for the same reason I adjusted the table header in the first place (to clarify an ambiguity), as ${\displaystyle \scriptstyle {\Delta \lambda }=\lambda _{f}-\lambda _{s}\;\!}$ regardless of the latitude, it is the cosine adjusted difference that shrinks to 0 at the poles: One degree of arc at the poles equals the same in any direction, about 111.694 km. Is there a better notational clarification? ~Kaimbridge~ (talk) 11:28, 18 June 2008 (UTC)

Stating "1° of ${\displaystyle \scriptstyle {\cos(\phi )\Delta \lambda }\;\!}$" is not meaningful, because of the presence of the cosine. The 1° applies to the (difference) measurement in degrees. People that have read as far as this place in the artcile can be expected to understand the terms latitude and longitude. In my view there is no need to keep saying E-W or N-S. The situation is different for radius of curvature. For those it makes sense to name them after E-W and N-S. −Woodstone (talk) 13:34, 18 June 2008 (UTC)
I changed the headings again... Rracecarr (talk) 15:26, 18 June 2008 (UTC)

Much better! P=) ~Kaimbridge~ (talk) 19:43, 18 June 2008 (UTC)

## Added notes for geodetic and geocentric latitudes

Intuition seems likely to miss the distinction between geocentric latitude and geographic latitude. And I repeated some info about the strange-looking symbol for the esoteric notion of "angular eccentricity" to help the reader find the definition if they jumped to the middle of the article.

The other types of latitude differ, too; but it may be common to think that geographic latitude (which already has two other names: 'common' and 'geodetic') is the same thing as geocentric latitude. The distinction wasn't clear to me until I found it here. Understanding the difference required reading the description of "common latitude" more than once.

The description of geographic latitude is quite technical. The notion of a line being "normal to" an irregularly curved surface may be unfamiliar to many readers. It could probably use a picture that is less complicated than the one showing a point at some height above the surface.

The placement of the sentence about a Gauss map seems awkward, but I didn't find a better place to put it. - Ac44ck (talk) 02:55, 29 December 2008 (UTC)

## Positive/negative notion ?

Sometimes N/S Latitudes are mentioned in with signs, ie. + for northern, - for southern. Can that be reflected in the article? Article on Longitude contains mention of signs. —Preceding unsigned comment added by 115.64.167.186 (talk) 05:29, 31 January 2009 (UTC)

## Distance between Latitudes and Degree Length

The distance-per-degree value varies continuously from equator to pole. I suspect that the values in the table are "instantaneous" values. A distance-per-degree value between 45 and 46 degrees would be an average. The exact distance between two particular lines of latitude on the reference ellipsoid would be evaluated via an elliptic integral – after converting geodetic latitude to geocentric latitude. The actual distance over real terrain, if it can be defined (re: coastline of England), will vary with topography. 111.132 km / 60 = 1.8522 km. I would not expect the distance between 45 degrees 0 minutes North and 45 degrees 1 minute North to be exactly 1.8522 km because the distance-per-degree value is not constant from 45 to 46 degrees. A formula here may be adequate for your purposes:
Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane
The value of K_1 at 45.5 degrees should be close to "the distance between 45 degrees North and 46 degrees North", though it seems that this formula is not based on the latest version of the reference ellipsoid. Various on-line calculators will report the distance between given coordinates on the latest reference ellipsoid. Google may be reporting coordinates for a particular spot at your address. They are not deciding for you how many of the digits are meaningful in your application. How precise do you need to be? -Ac44ck (talk) 17:01, 16 March 2009 (UTC)
Actual distance from 44 to 45 deg latitude on the GRS80/WGS84 spheroid is 111.122 km; from 45 to 46 deg, 111.142 km.Tim Zukas (talk) 00:23, 30 August 2010 (UTC)

I've recently taken up geo-caching, a hobby which employs latitude and longitude to find hidden "treasure". It made me curious about the readings I was looking at and how they might translate into linear distances. If I have a reading which gives me a location to the hundredth of a second, is it really about a foot from where I’m standing? The answer seems to be, maybe, which is fine. I'm really just trying to get a picture of scale in my head, so knowing that the degrees are about 70 miles apart and the minutes are about a mile apart helps me a lot. Thanks for your response. --ErinHowarth (talk) 16:13, 18 March 2009 (UTC)

The location of the spot for which Google is providing coordinates can be found using the tool here:
http://itouchmap.com/latlong.html
I got the following values for the locations of first and second base in Yankee Stadium by using the tool linked above:
• First base: 40.82677152165525, -73.92815664410591
• Second base: 40.8269491114027, -73.92793670296669
The distance between the two sets of coordinates is 88.8345 feet - 14 inches short of 90 feet - according to:
http://192.206.28.84/Website/KML/icaoGCircleCal.htm
The distance between bases should be exactly 90 feet. My placement of the markers on the photo might have been off by 14 inches. -Ac44ck (talk) 04:47, 6 June 2009 (UTC)

## Map that someone could use

Is there some reason for not including numbers on the lines of latitude on the maps? It's a handy thing, you know, when things are based on numbers, to include the numbers. But, I know, wikipedia policy: maps don't have names, why should they include numbers ....--IP69.226.103.13 | Talk about me. 03:24, 9 January 2010 (UTC)

Oh, the first link - "libproj4: A Comprehensive Library of Cartographic Projection Functions" - on External links is missing or "locked", I am sad to say. --Finn Bjørklid (talk) 12:59, 18 March 2010 (UTC)

Template:Fixed 19:51, 18 March 2010 (UTC)
After some unsuccessful attempts, I found several copies in the wayback machine. I'm adding the latest version to the article. — Joe Kress (talk) 02:04, 20 March 2010 (UTC)

## Notation and angular eccentricty

I have added a small paragraph defining the isometric latitude using the conventional notation which is to be found in most (English) textbooks and papers in cartography. At a very early stage the other latitudes on this page were defined in similar conventional notation. However, at some time or other, an editor has infested all the wiki pages which touch on ellipse or ellipsoid with the notation of angular eccentricity, at the same time implying that is in some way more fundamental: everything is written first in terms of angular eccentricity. This is a complete distortion of the world as defined by contemporary books and papers on geodesy, cartography, mathematics, astronomy. A google search on "angular eccentricity" brings up only the wiki pages (and derivatives) which have been infested in this way. I would maintain that the use of "angular eccentricity" is an anachronism which should be edited away. Interestingly the page angular eccentricity admits the perversity of writing everything in this way. Moreover the notation makes the mathematics inelegant and less transparent. Contrast the formula given on this page for the conformal latitude with the following:

${\displaystyle \chi =2\arctan \left[\left({\frac {1+\sin \phi }{1-\sin \phi }}\right)\left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)^{\!e}\;\right]^{1/2}-{\frac {\pi }{2}}\,\!}$

I am in favour of altering many of the wiki pages. This will clearly upset (at least) one editor. Please express your support or otherwise on this section. Check your bookshelves for any mention of angular eccentricity. Let me know what you find. Peter Mercator (talk) 22:32, 11 October 2010 (UTC)

• No one has commented adversely on my proposal to replace the unconventional use of the angular eccentricity by the conventional notation employed by practising geodicists and cartographers in both research papers and text books. I have now made these changes and made one or two consequential changes to the section structure.   Peter Mercator (talk) 22:00, 5 January 2011 (UTC)

## Proposed major revision

I would like to propose a major revision of the Latitude page. There are a number of points which need attention.

1. There is no precise definition. The definition "latitude is an angular distance" is inadequate for it doesn't specify the angle. Any explanation requires some suitable figures relating to reference surfaces (sphere/ellipsoid). The "lat-lon" template is not a suitable illustration because parallels defined uniquely on the reference surface may appear in many ways on projections: they cannot always be identified with "horizontal" (or nearly so). Wiki should adhere to conventional definitions in terms of the angle between the equatorial plane and the normal to the reference surface. The definition should make clear that there is no unique value for latitude. (See ISO 19111, sec 10.3).
2. The Vermont illustration is amusing but hardly instructive. It should be relocated or even scrapped.
3. The named latitudes (Capricorn etc) should be explained (briefly) with reference to the earth-sun geometry and the inclination of the ecliptic.
4. The table of latitudes is interesting but not fundamental. Perhaps it should be a separate wikilist. There is already a List of cities by latitude. Removing this list will give a cleaner article.
5. The section on subdivisions is excessive. Since this material is covered in Degree (angle) this section could be removed.
6. The section on the effect of latitude should perhaps reduced to a link to Effect of sun angle on climate. The header of the Latitude article does say that this article is concerned with geographic reference systems.
7. The section on elliptic parameters warrants the restoration of conventional notation. The literature of cartography, geodesy and map projections does not use angular eccentricity.
8. Meridian length could be pruned drastically by using references to other pages. The discussion of curvature, other than on the meridian, is very confused and in places simply incorrect. In particular there is no need at all to introduce normal curvature in the prime vertical plane. (This is an article on latitude, not the geometry of the ellipsoid). The length per degree in longitude follows immediately from the radius of a parallel circle. This can be calculated simply (and the value of N is actually usually derived from this radius by Meusnier's theorem). It is a moot point whether the longitude degree intervals should be included in this article.
9. I have already restored conventional notation for the auxiliary latitudes and added the briefest of comments. This section is more mathematical and should probably move to the end of the article. I believe the first few sections of a Wiki should be accessible explanations for the general reader and it's important that articles do not simply become lists of formulae. Further details on the auxiliary latitudes would be best kept for more technical articles on map projections.
10. The comparison of latitudes section could probably be jettisoned and replaced by a simple statement giving the order of magnitude of the differences at their maximum value. The graph is actually misleading because the geodetic latitude is greater than the others and a general reader will read 'difference' as 'positive difference'. The table of differences is too detailed and only the multiples of 15 degrees could be retained (if the table is retained at all). I suspect that neither the the general reader nor the professional has use for such data in a Wiki article, besides, the paper by Adams (see proposed draft linked below) has tables at 1 degree intervals and it is now available on the web. Removing the graph and table would give a leaner and cleaner article.
11. I think this article should stick to latitude and longitude as surface coordinates and only link to discussions of their relation to geocentric Cartesians: this material is already on several Wikis but it does all need tidying up. The extension of surface coordinates to three and four dimensional geodetic systems is already (partially) covered in geodesy articles
12. Astronomical latitudes require more discussion in the context of the measurement of latitude by instruments oriented in the gravity field. This opens up the whole area of modern geodesy which uses mean Earth reference ellipsoids to model the geoid and the gravity field together.
13. Paleolatitude is important but perhaps it should have its own page. It does, however, tie in to the problems of modern time dependent reference systems which take into consideration the motion of the tectonic plates.

It would be good to have some responses to the above points. In the mean time I have commenced a rough draft (User:Peter Mercator/Draft for Latitude) which is incomplete in content and is needing work on the references (other than to wiki pages). Whether or not I shall continue to develop this article will depend on responses to these comments.  Peter Mercator (talk) 16:05, 17 February 2011 (UTC)

In general, I agree with the points above. I differ on a few points:
• I don't find the table of latitudes to be interesting.
• How to slice an angle brings is a religious issue. What is needed is a nomenclature section. While perhaps redundant if include here, something helpful for the following may be included:
Talk:Geographic_coordinate_system#Needs_more_clarity.2C_examples.2C_practical_explanation.
• I think the comparison of latitudes section is worth keeping. I didn't know the difference between geocentric and geodetic latitudes before reading it here. The distinction is usually ignored in calculating great circle distances. Perhaps the accuracy would be better if the conversion from common latitude o geocentric latitude was made before using a great circle formula. The table seems to provide some sense of how significant errors may be if no correction is made for the type of latitude.
• I don't understand the objection in Item 11.
• I hadn't noticed the astronomical latitude before. Is this the latitude that is relevant when using a sextant? If so, that would be interesting to note.
-Ac44ck (talk) 01:22, 19 February 2011 (UTC)
(1) Yes, the table of latitudes is uninteresting. Make it a wikilist. (2) Both decimal degrees and sexagesimal degrees have their uses. (3) All the different latitudes should stay. It's just that the table is too detailed and the graph is misleading. (4) Basically I was implying that this article shouldn't spread to 3dim coordinate systems and therefore the paragraph emphasising that latitude transformations depend on height is not required. (5) Every instrument that is 'levelled' in the gravitational field, telescopes particularly, will not measure geodetic coordinates; they give the the so-called astronomic latitudes.   Peter Mercator (talk) 21:48, 11 March 2011 (UTC)
• What is misleading about the graph?
• I don't know how significant the corrections are for altitude. It's a short section; I would favor keeping it. I probably wouldn't have thought about it if it weren't here. Some example of how much correction is needed at the top of a mountain might be interesting. Just an additional sentence or two. I don't think it needs much "ink" here. Just a reminder for those who are doing calculations with an Earth radius that is "accurate" to a fractional millimeter.
• What is "so-called" about the astronomical latitude? Should it be called something else?
-Ac44ck (talk) 17:09, 13 March 2011 (UTC)
Go for it. A couple of suggestions: (1) Use the standard math notation for arc-trig functions, e.g., ${\displaystyle \sin ^{-1}\phi }$, etc. (2) Use inverse hyperbolic functions to simplify those log expressions; these give more compact formulas; it's easier to check the parity; and they are more easy to compute accurately. cffk (talk) 18:37, 4 August 2011 (UTC)

## Parametric first?

I propose that it would be clearer to describe parametric latitude before geocentric latitude, as follows:

### {{safesubst:#invoke:anchor|main}} Reduced (or parametric) latitude

The name "parametric latitude" arises from the parameterization of the equation of the ellipse describing a meridian section. In terms of Cartesian coordinates p, the distance from the axis, and z, the distance above the equatorial plane, the equation of the ellipse is

${\displaystyle {\frac {p^{2}}{a^{2}}}+{\frac {z^{2}}{b^{2}}}=1.}$

The Cartesian coordinates of the point are parameterized by

${\displaystyle p=a\cos \beta ,\qquad z=b\sin \beta .}$

The derivatives with respect to ${\displaystyle \beta }$ of these equations are

${\displaystyle p'=-a\sin \beta ,\qquad z'=b\cos \beta .}$

So the slope of the ellipse is

${\displaystyle {\frac {z'}{p'}}=-{\frac {b}{a}}\cot \beta .}$

The negative reciprocal of this slope is perpendicular to the ellipse and is the tangent of the geodetic latitude:

${\displaystyle \tan \phi =-{\frac {p'}{z'}}={\frac {a}{b}}\tan \beta }$
${\displaystyle \beta =\arctan \left[{\frac {b}{a}}\tan \phi \right]}$

### Geocentric latitude

The geocentric latitude, denoted here by ${\displaystyle \psi }$, is the angle between the equatorial plane and the radius from the centre to a point on the surface. The relations among the geodetic latitude (${\displaystyle \phi }$) and the parametric (${\displaystyle \beta }$) and geocentric (${\displaystyle \psi }$) latitudes are

${\displaystyle \tan \psi ={\frac {z}{p}}={\frac {b\sin \beta }{a\cos \beta }}={\frac {b}{a}}\tan \beta ={\frac {b^{2}}{a^{2}}}\tan \phi }$
${\displaystyle \psi =\arctan \left[{\frac {b}{a}}\tan \beta \right]=\arctan \left[{\frac {b^{2}}{a^{2}}}\tan \phi \right]}$

The geodetic and geocentric latitudes are equal at the equator and poles. The maximum difference of ${\displaystyle (\phi -\psi )}$ is approximately 11.5 minutes of arc at a geodetic latitude of 45°6′.

— Preceding unsigned comment added by 208.53.195.38 (talkcontribs) 16:21, 11 August 2011 (UTC)

• The contributions above are unsigned. I am not in favour of putting the parametric latitude before the geocentric latitude. Wiki guidelines (see Wikipedia:What Wikipedia is not) suggest that the needs of the general reader should come before those of the specialist. I suspect that such general readers will need only understand the distinction between geodetic and geocentric latitudes and will have no wish to look at all the other latitudes. Furthermore I don't think that the relation between geocentric and parametric latitudes is important: it suffices to relate all the auxiliary latitudes to the geodetic latitude. One could write down relations between all pairs of auxiliary latitudes but to what import? Note that the guidelines discourage mathematical derivations of a text book nature as suggested here. Citations would be much better. Finally note that most workers in the field would write these definitions in terms of the eccentricity, e.
• Whilst on this page, may I say that I shall shortly go ahead with the major revision of the first five sections of this article as suggested above in Section 23 with a preliminary draft here.    Peter Mercator (talk) 21:41, 12 August 2011 (UTC)
I think we agree on putting the needs of the general reader before those of the specialist. That is why I want to show the simple, elegant, and easily demonstrated relationship among the tangents of these three latitudes as multiplication by powers of the ratio b/a, which ratio I dare say the uninitiated, as opposed to workers in the field, can grasp more easily than an expression involving the eccentricity. My reason for swapping the order of parametric and geocentric latitudes is of course that the relationship between geodetic and geocentric latitudes is more easily seen when geodetic and geocentric latitudes are each described in relation to parametric latitude, as I did. — Preceding unsigned comment added by 75.54.94.235 (talk) 18:03, 13 August 2011 (UTC)

## Wrong values in table

The values for authalic latitudes in the table seem to be wrong (maybe others too, haven't checked). They seem to be based on a wrong formula that was previously in the article and probably based on the author's own work. The plot seems to use these values and is thus wrong too, but the difference isn't too much. --92.229.16.35 (talk) 17:25, 8 December 2011 (UTC)

## Major rewrite

1. Some time ago I proposed a major rewrite to address some of the points outlined above. Since no one objected to the proposals I have gone ahead and I will put up the new version immediately after this note. I hope the article requires no further 'justification. It is of course in a slightly unpolished state and I am sure there must be infelicities, typos and errors to correct.
2. Still needed are more references and cross wiki links. I also wish to enlarge the discussion of astronomic latitude and comment on measurement of latitude.
3. It would be good to have another couple of examples showing how the latitude of a (famous) location varies with datum and ellipsoid. Nice if we had three different values for three locations. Can anyone provide a third value for the Eiffel Tower?
4. I would like someone to remove palaeolatitude: it doesn't really fit and it would be much better added to palaeomagnetism with consequential changes to links.
5. I have also removed the 'climate' section. That material is covered in climate articles (for which the attractive figure was originaly prepared).
6. I have incorporated much of the section 'Corrections for altitude' into a section where I distinguish 2D and 3D coordinate systems. I have kept this section short for it is (or should be) covered in other articles. The previous figure was nice but it was over-detailed for the context. No numbers are required to point out that locations with the same geodetic latitude have differing geocentric latitudes.
7. I thinned out the table because no one actually uses numerical values of auxiliary latitudes per se. They are usually buried within more complex mathematical structures.
8. I moved the list of countries by latitude to a page of its own. It is pretty poor compared with the list of cities by latitude.
9. The 'subdivision' section duplicates material elsewhere which is called upon by many other articles. This article is not the place to expand the description of 'degree'.

I look forward to your comments and corrections.   Peter Mercator (talk) 23:18, 18 December 2011 (UTC)

Nice job! I added references to Legendre and Cayley in the section on the reduced latitude.cffk (talk) 15:20, 19 December 2011 (UTC)
"In general the direction of the true vertical at a point on the surface does not coincide with the either the normal to the reference ellipsoid or the normal to the geoid."
If "vertical" isn't perpendicular to the geoid, what does define it? Tim Zukas (talk) 16:45, 22 December 2011 (UTC)
I think my sentence is ok since the Earth's surface (where precision gravity measurements are carried out) does not coincide with the geoid. It is true that at a point on the geoid the vertical defined by the gravity field is perpendicular to the geoid but even this statement needs refining: see Torge section 3.4. As I said above, the section on astronomical latitude requires further work but I am hesitant to get too deeply into geodesy since the required level of discussion is much too technical for present day Wiki articles.  Peter Mercator (talk) 18:04, 30 December 2011 (UTC)
I guess you mean since the plumbline is curved, the straight line thru a surface point that is vertical at the point won't be vertical when it reaches the geoid. True enough; if that is all you mean I'll add a footnote clarifying that. It would help do give some idea what the curvature amounts to, but probably none of us is up to that. Tim Zukas (talk) 20:33, 4 January 2012 (UTC)
Yes, that's right. Suggest we don't get too deeply into the gravity field in this article: a footnote would be fine. My 'bible' for the subject is Torge but it is tough going. I am preparing a figure to ilustrate this section.  Peter Mercator (talk) 10:19, 5 January 2012 (UTC)

## Definitions are useful

When making future changes, please make sure there is a definition of latitude at the very top of the article (as opposed to a joint definition of latitude and longitude). I think the thing most people who check this article want to know is whether latitude is the one that goes north-south or the one that goes east-west (I'm personally guilty of that). Thanks! Gabi Teodoru (talk) 09:13, 25 December 2011 (UTC)

Greetings. Surely the very first sentence of the article did point out that latitude is a north-south measure from the equator and that's what you wanted to know! Your edit is not a definition: it introduces numbers for latitude but this is illogical without first defining precisely the angle that is measured. I shall probably revise your sentence and reconsider the intro further but let me say that I designed this rewrite to include latitude in the context of a two dimensional coordinated system so some mention of longitude seemed advisable. I also drafted the first section so that it might better slide into the intro of Geographic coordinate system. Then, this article could indeed be restricted to a discussion of some of the technical aspects of latitude. Making this article closer to Longitude isn't really advisable because that article needs a lot of attention, particularly its use of the 'latlon' template that I removed from the previous version of this article. (See comment 23 above).  Peter Mercator (talk) 18:54, 30 December 2011 (UTC)

## Inverse auxiliary latitudes formula

Please add inverse formulas for auxiliary latitudes (auxiliary to geodetic). I do not know how to add maths. 27.69.44.223 (talk) 11:01, 22 February 2012 (UTC) P.S. These are available in Map Projections: A Working Manual. 27.69.44.223 (talk) 11:03, 22 February 2012 (UTC)

The wiki guidelines say that we should not be writing a text book. Padding out the article with the formulae for the inverses (and possible derivations of the formulae) would make the article too long and also change its nature; surely the reference to Snyder should suffice. For the life of me I don't understand why non-specialists want to get involved in these transformations.  Peter Mercator (talk) 20:34, 23 February 2012 (UTC)
I have now added links to the inverse series and their derivations.Peter Mercator (talk) 20:07, 25 May 2013 (UTC)

## Contradiction with Geoid

In the section on "Astronomical latitude" this article says "In general the true vertical at a point on the surface does not exactly coincide with either the normal to the reference ellipsoid or the normal to the geoid." Geoid says "the local vertical (plumb line) is always perpendicular to the geoid". Which is correct? JonH (talk) 10:35, 25 May 2013 (UTC)