Talk:Latitude

From Wikipedia, the free encyclopedia

Mathematics Portal

This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.

Mathematics rating:

B Class

Mid Priority

Field: Basics

One of the 500 most frequently viewed mathematics articles.

Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments.
Please add to or update the comments to suggest improvements to the article.
Geometry guy 20:39, 20 May 2007 (UTC)

	This article is part of WikiProject Maritime Trades, a group of editors working to improve Merchant Shipping topics. To learn more or join the project, please visit the project page.
Start	This article has been rated* as Start-Class on the assessment scale.*
???	This article has not yet received an importance rating on the assessment scale.

Latitude is of interest to WikiProject Geographical coordinates, which encourages the use of geographical coordinates in Wikipedia. If you would like to participate, visit the project page.

1 Formula
2 "angular measurement" ?
3 geodetic latitude is not referred to a plumb line
4 Distance between "Latitutdes
5 Latitude
6 Latitude and Wealth
7 Gravity
8 The Earth has no Latitude
9 Latitude - is it phi or lambda?
10 Astronomical latitude
11 This is a fringe theory
- 11.1 Evolutionary explanations
12 Self-contradiction and confusion
13 Coplanar
14 Reduced Latitude

[edit] Formula

I think I know how to derive authalic latitude. Take the ellipsoid $E = (cosηcosλ,cosηsinλ, b sinη)$ and the sphere $S = (cosβcosλ,cosβsinλ,sinβ)$ . Note that $\partial E/\partial\eta\cdot\partial E/\partial\lambda=\partial S/\partial\beta\cdot\partial S/\partial\lambda=0$ so the areal elements are $|\partial E/\partial\eta||\partial E/\partial\lambda|$ and $|\partial S/\partial\beta||\partial S/\partial\lambda|$ so we equate those, throwing in an arbitrary constant factor and a correction factor for the different variables: $|\partial E/\partial\eta||\partial E/\partial\lambda| = A |\partial S/\partial\beta||\partial S/\partial\lambda|d\beta/d\eta$ . Expand to get $\sqrt{\sin^2\eta+b^2\cos^2\eta}\cos\eta\,d\eta=A\cos\beta \,d\beta$ , integrate (use $u = sinη$ ), adjust the arbitrary constants so $η = β = 0$ and $η = β = π / 2$ happen, and substitute $tanη = b tanφ$ and $b 2 + e 2 = 1$ . (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)

[edit] "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

[edit] 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

[edit] 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]

[edit] Latitude

I can not calculate authalic latitude. Way over my head. Please avoid putting that in.

[edit] 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.

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 (talk • contribs) 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)

[edit] 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?

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)

[edit] 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.

I agree I think there is a POV problem with this article. I would like to see some proof that the world is round.

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.

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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.

[edit] 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:

$\begin{align}{\color{white}\Big|}\mbox{Radius} &=\sqrt{(a\cos(\beta))^2+(b\sin(\beta))^2},\\ &=\sqrt{\frac{a^4\cos(\phi)^2+b^4\sin(\phi)^2}{(a\cos(\phi))^2+(b\sin(\phi))^2}},\\ &=\frac{b}{\sqrt{a^4\sin(\psi)^2+(ab\cos(\psi))^2}};\end{align}\,\!$

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 $\scriptstyle{\frac{b^2}{a}}\,\!$ , while facing east-west it will equal $a\,\!$ . 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)

Thanks for the link to Salix alba's page. It was difficult to determine who said what, which is why I absolutely detest anyone splitting another user's comments or anyone splitting mine in order to comment on single phrases or sentences within it. He explicitly states that "radius of arc" is undefined on a non-spherical surface. Hence my confusion when you used of that term in your original edit. I previously saw your discussion in Talk:Earth. Although I commend your attempt to discuss it, it is original research which does not belong in any Wikipedia article unless you can cite a source (it and its results are OK on a talk page). I hesitate to add any section about the radius of curvature (infinitesimal) in any direction other than N-S or E-W because it varies dramatically for all azimuths and latitudes making it difficult to present in an understandable manner and has little practical value. The term "great circle" was a poor choice. The 'radius of curvature' of a great circle on an ellipsoid of revolution is meaningless. An arc of a great circle is the shortest distance between two points but requires elliptic integrals to calculate on an ellipsoid of revolution. Great circle distance already discusses the practical use over a large distance. — Joe Kress (talk) 21:55, 26 December 2007 (UTC)

[edit] 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)

[edit] 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)

ulo ka pak u so much —Preceding unsigned comment added by 125.60.241.184 (talk) 01:12, 13 June 2008 (UTC)

See also ebooksgratis.com: no banners, no cookies, totally FREE.