Talk:Spherical geometry

Angle Sum > 270°?
I am in doubt about the following point of the "Properties"-Section:


 * The angle sum of a triangle is greater than 180° and less than 270°.

I can easily imagine a triangle on a sphere that has an angle sum of 270°: Just image one corner at the North Pole, and two on the equator, one at 0° and one at 90° East. This gives you a triangle with and angle sum of 270°.

From there, it is also possible to further increase the angle sum of sliding the third point along the equator to 179°E, resulting in an angle sum to 359°. Furthermore, if you slide the two points from the equator down along the meridians to almost the south pole, you should be able to get close to an angle sum of 540°.

Maybe there's something I'm missing here though, but it seems to me that this statement is wrong and should instead be:


 * The angle sum of a triangle is greater than 180° and less than 540°.

See Sum_of_angles_of_a_triangle where it says "For a spherical triangle, the sum of the angles is greater than 180° and can be up to 540°". This agrees with you.

Imagine any three points on the equator. Connect them to make a triangle. Each angle is 180 degrees for a total of 540 degrees. — Preceding unsigned comment added by 84.17.48.18 (talk) 12:30, 31 August 2019 (UTC)

Earth picture
Hey, how come the picture on this article shows a section of planet Earth? Does it matter that Earth is an oblate spheroid and not a sphere? Coolsnak3 23:32, 11 May 2007 (UTC)


 * It's not even an oblate spheroid, that's just an approximate model. Modeling it as a sphere is less accurate but more convenient (no need to distinguish geodetic latitude from geocentric latitude, simple formula for the volume, and it can used to illustrate articles like this).  --Vaughan Pratt (talk) 02:01, 1 January 2009 (UTC)

Relation to Euclid's postulates?
In the section on Euclid's postulates, I'm having the following difficulties with the claim that the sphere violates Postulates 1, 3, and 5.

Postulate 5 does not say "there is no point through which a line can be drawn that never intersects a given line," it only says that two lines inclined towards each other (suitably formalized) meet when produced sufficiently far. It does not specify conditions under which they do not meet because Euclid believed he'd proved that Postulates 1-4 covered that situation.

Postulate 3 does not say there are circles of arbitrarily large radius, it merely imposes the requirement that two points determine a circle. One can only infer the existence of arbitrarily large circles when there are arbitrarily long distances.

Postulate 1 does not require that the straight line between any two points be unique. But even if it did it would not fix Euclid's proof of Proposition 16, which fails even in situations where uniqueness obtains. Furthermore, must all spherical models of Euclid's postulates have antipodal points? By symmetry they will for spheres embedded in F3 and centred on the origin, where F is any Euclidean field. However I don't see how you'd show this for spheres constructed as a Riemannian manifold with unit Gaussian curvature, with no reference to 3-space and with all distance measured constructively along geodesics, that is, as arc lengths, so as to be constructible numbers. --Vaughan Pratt (talk) 00:20, 29 August 2013 (UTC)

Unit
I'm not familiar with Wikipedia guidelines on this point (i.e. maybe everything is already fine), but I find odd that angular measure is expressed in degrees and not in radians.

2D vs (n>2)D confusion?
The of the introduction says "Spherical geometry is not elliptic geometry but shares with that geometry the property that a line has no parallels through a given point." Do we have any WP:RS claiming that elliptic geometry is only for $$n\ge3$$ spaces? Or a reference claiming that the n>2 spaces cannot be called spherical geometries? If the answers are "no, no", respectively, then we should replace the sentence by something like:

Spherical geometry usually refers to the 2-dimensional case of elliptic geometry.

Any objections? Boud (talk) 23:38, 15 November 2014 (UTC)
 * Elliptic, flat, and hyperbolic are synonyms for positive, zero, and negative local curvature; and when used as names for types of geometry they also mean the space is homogeneous (a symmetric space). Beyond that I don't think there is much consistency and certainly no convention about dimension. A sphere and real projective space might both be called "elliptic" but "spherical geometry" typically means geometry of the full sphere whereas "elliptic geometry" (when counterposed to hyperbolic and Euclidean geometry and seen as one alternative to the parallel postulate) would mean its quotient, the metrized projective space in which distinct coplanar lines have one point of intersection rather than 2 for cospherical great circles.
 * I think this is part of the confusion in the current article; spherical geometry as non-Euclidean geometry really refers to the spherical construction of elliptic geometry whereas the metric geometry of a sphere is usually considered in the context of 3-dimensional Euclidean space. The two points of intersection technically are a violation of the parallel postulate but a lot of other Euclidean geometry goes out the window with that, and the sphere was familiar as a Euclidean construct, so the (not projectivized) geometry of great circles was not seen as a brave new non-Euclidean world of the kind that Bolyai and Lobachevsky demonstrated. 73.89.25.252 (talk) 21:41, 27 December 2020 (UTC)

greeks
I cut this out because it needs some sort of explanation; Muslims, according to Carra de Vaux, were "unquestionably the inventors of plane and spherical geometry, which did not, strictly speaking, exist among the Greeks". well known greeks were geometers. J8079s (talk) 07:17, 31 December 2014 (UTC)

better source
For the history of trig J8079s (talk) 17:21, 31 December 2014 (UTC)
 * Glen van Brummelen is now listed in this article. He should be cited at spherical trigonometry. — Rgdboer (talk) 18:12, 24 August 2020 (UTC)

Islamic section
"The book contains formulae for right-handed triangles," Is this just a typo? Should be "right-angled triangles? — Preceding unsigned comment added by 87.113.215.232 (talk) 23:57, 12 November 2016 (UTC)

Euler's work
For five years this article has contained in a leading section a list of Latin language works of Leonard Euler on topic. This section does not enhance the article for the likely reader. Ideally an editor familiar with Latin will summarize Euler's contribution and put the works in reference. Failing that, the material should be moved lower. — Rgdboer (talk) 17:39, 24 August 2020 (UTC)

WP:COMMONNAME, and revisionism about "non-Euclidean"
(Update: solved by edit of 4 Jan 2021)

The lede baldly states that spherical geometry is a non-Euclidean geometry and the article proceeds in that vein.

Historically and in its applications until today, this is false. Spherical geometry and trigonometry has predominantly been a body of theorems and formulas of 3-dimensional Euclidean geometry, relating Euclidean measurements on a Euclidean sphere (the locus where Euclidean distance to a center point is constant) in Euclidean space, and often referring to the 3-d model where lines can tunnel through the interior of the sphere and so on. Call this "extrinsic spherical geometry". All this is thoroughly Euclidean.

The nonEuclidean thing is the intrinsic theory of a spherical surface, in which the interior and exterior do not exist, there is no surrounding space (except maybe a higher dimensional sphere), and the meaning of "straight line" has been altered to mean geodesic in a spherical metric or a great circle. This is the sense of spherical geometry that does disobey the parallel postulate but it is a second and arguably the secondary meaning of the term especially to a general audience.

Speaking in historical-revisionist terms about spherical geometry as though it was always non-Euclidean is common but incorrect, and the distinction of intrinsic/extrinsic should be maintained.

More fundamentally the point of view of the article, appropriating the term to mean non-Euclidean geometry, violates WP:COMMON. The article does not cite a single book devoted to spherical geometry or trigonometry. Nearly all such works, including all the references cited at Spherical trigonometry, take the extrinsic point of view as it is more useful and pedagogically accessible, and is simply what the subject was historically and practically outside of some areas of pure mathematics where independence of the parallel postulate or the possibility of working without reference to a particular model are important. The average reader would be surprised to learn that in spherical geometry (in the definition of this article) the sphere-surface has no center, but that is what the article currently implies. 73.89.25.252 (talk) 19:34, 27 December 2020 (UTC)
 * I have overhauled the article to remove the ahistorical implications about non-Euclidean geometry, and better explain what spherical geometry is. 73.89.25.252 (talk) 07:11, 4 January 2021 (UTC)

non-Euclid
The section on non-Euclidean geometries could be improved by better relating elliptic geometry and that of the real projective plane – if indeed these are distinct concepts. —Tamfang (talk) 02:42, 14 March 2023 (UTC)
 * Elliptic geometry is projective geometry with an additional structure; roughly, a metric. However, the statement in Spherical geometry is incorrect, since the projective plane does not have distances and angles, but only cross ratios. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:17, 14 March 2023 (UTC)