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When I was an undergrad, the field of spherical trigonometry was cited as a once-popular area of math that has since died. Is this true? Are the results from spherical trigonometry relevant for contemporary research?

Dave Shulman
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    Like a usual trigonometry, I would say. It is a useful working instrument, but it does not produce new problems itself. – Fedor Petrov May 15 '22 at 12:28
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    Take anything very strongly opinionated about a field of math with a grain of salt. That being said, it is true that some topics are more fashionable at a given period of time. But this does not mean that the other fields are then "dead". Once someone told me that Lie theory was dead. When I "grew up" mathematically, I learned that this was certainly not true. – Malkoun May 15 '22 at 22:38
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    Was your undergraduate lecturer teaching the class some spherical trigonometry, or just making an offhand comment? I'm curious as to why the lecturer thought to mention spherical trigonometry. – Timothy Chow May 18 '22 at 13:54

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It is not. As a proof, I will mention three relatively recent papers where I am a co-author:

M. Bonk and A. Eremenko, Covering properties of meromorphic functions, negative curvature and spherical geometry, Ann of Math. 152 (2000), 551-592.

A. Eremenko, Metrics of positive curvature with conic singularities on the sphere, Proc. AMS, 132 (2004), 11, 3349--3355.

A. Eremenko and A. Gabrielov, The space of Schwarz--Klein spherical triangles, Journal of Mathematical Physics, Analysis and Geometry, 16, 3 (2020) 263-282.

As you see, they are all published in mainstream math journals. All contain some new results on spherical triangles. And I am not the only person who is involved in this business:

Feng Luo, A characterization of spherical polyhedral surfaces, J. Differential Geom. 74(3): 407-424.

Edit. To address one comment: here is a forthcoming conference on spherical geometry

Martin Sleziak
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Alexandre Eremenko
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    Well I stand corrected! I need to let my undergrad lecturer know about this! – Dave Shulman May 15 '22 at 14:43
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    Another thing is surprising me: that spherical geometry is still taught to undergraduates:-) – Alexandre Eremenko May 15 '22 at 17:39
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    Who knows? They may be getting ready to head out to sea as midshipmen. – Lee Mosher May 16 '22 at 03:48
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    I suspect that neither midshipmen nor captains nowadays know any spherical geometry. I've heard that the subject is not taught anymore even in the naval schools. – Alexandre Eremenko May 16 '22 at 08:55
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    @AlexandreEremenko What gave you the impression that spherical geometry is still taught to undergraduates? Making a remark that spherical geometry is dead hardly counts as teaching spherical geometry. – Timothy Chow May 16 '22 at 13:23
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    @AlexandreEremenko The maths of it, probably not. However the principle of Great Circle routes is widely taught for sailing, flying, and general navigation. The fact that a computer can do the sums better than you can, and there's always likely to be a suitable computer around, means the important part is just knowing how to get the computer to give the right results. And even back in the days when you needed to do the sums yourself, midshipmen weren't expected to know how to recalculate the tables of trig or log values which they used. – Graham May 16 '22 at 13:27
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    Perhaps interesting: Navy cadets won't discard their sextants (1998) – Rodrigo de Azevedo May 16 '22 at 13:34
  • @TimothyChow Spherical geometry is taught on certain second-year undergraduate mathematics courses. – Hollis Williams May 16 '22 at 20:39
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    Spherical trigonometry didn't die but was assimilated into the larger fields like "Non-euclidean geometry" and "Differential geometry". People from these fields may write papers on spherical trigonometry from time to time (and a few may specialize into it) but I would not expect spherical trigonometry to be self-sufficient (as in having conferences and journals dedicated to it) – Taladris May 17 '22 at 05:45
  • @Taladris: see the edit of my answer at the end. – Alexandre Eremenko May 17 '22 at 10:58
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    @AlexandreEremenko, you seem to be conflating spherical geometry (which we all agree is active and growing) with spherical trigonometry, which is more limited. –  May 17 '22 at 12:26
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    No, I am not conflating anything. Spherical trigonometry is about spherical triangles. It is a part of spherical geometry, and one of its principal tools. The papers I cited in my reply contain, among other things, new results about TRIANGLES. The conference I referred to is about spherical geometry, where trigonometry plays a prominent role. – Alexandre Eremenko May 17 '22 at 16:23
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    @Timothy Chow: What gave me this impression? The first comment by Dave Shulman. – Alexandre Eremenko May 18 '22 at 10:56
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    @AlexandreEremenko Your suspicion is fortunately incorrect! At least the sea captain students here in Gothenburg learn about spherical trigonometry during their first year. They are expected to learn, among other things, the spherical laws of sine and cosine. By the end of the course, they can solve spherical triangles, and answer questions like "What is the northernmost point on the great circle path between Galway and New York?" or "Where does the path between A and B cross the equator?". I held exercise sessions for this course as a PhD student, and always liked the "hands-on-ness" of it! – EdvinW May 18 '22 at 17:21
  • @EdwinW: I am pleased to receive this interesting information. What is the exact name of this institution? Sjösportskolan? University of Gothenburg? Do they also teach Celestial navigation? – Alexandre Eremenko May 18 '22 at 23:06
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There is a lively site which publishes research questions, some of which essentially involve spherical geometry and trigonometry. Recent examples include:

Jukka Kohonen
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I’d say spherical trigonometry has never been an area of research mathematics. Instead, it is an area of mathematical techniques that have been useful for 1) other areas of mathematical research, and more importantly for 2) astronomy or 3) surveying and navigation. But those mathematical techniques are less known and less taught now than in the past, for good reason.

  1. The other answers here show that spherical trigonometry can be useful in convex geometry and ergodic theory and elsewhere in mathematical research. But spherical trigonometry is not itself the area of research — e.g. in the 2010 MathSciNet classification, the word “trigonometry” appears only in “97G60, plane and spherical trigonometry (educational aspects)”, under the top-level category of “97-XX, mathematics education”.

  2. Spherical trigonometry originated in astronomy, but much of its use there has been replaced with a more vectorial approach. E.g.: How would you calculate the angular distance $\theta$ between two celestial objects with given azimuths $a,a’$ and zenith distances $z,z’$? (Azimuth = compass direction of the closest point on the horizon.) The traditional answer uses the spherical law of cosines: $$\cos \theta = \cos z \cos z’ + \sin z \sin z’ \cos(a-a’),$$ which is efficient for calculating with slide rules and trig tables. Most of us here would find it easier to compute unit vectors, take a dot product, and take the $\arccos$ of that. The calculations may be equivalent, but we don’t refer to the law of cosines in the process.

  3. Surveying and navigation used to be more important, and spherical trigonometry can be useful for both. Nowadays, we have bigger ships and require fewer navigators; we have aerial maps and require fewer surveyors; we have GPS and automated directions, and avoid these topics even more. This decline of surveying and navigation is the reason we teach less spherical trigonometry, which is why fewer mathematicians know it.

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    "Never been an area of research" seems too strong. Someone first came up with the spherical law of sines and spherical law of cosines. – Timothy Chow May 16 '22 at 21:21
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    It was an active area of research in medieval Islamic astronomy! So saying “spherical trigonometry is dead as an area of mathematical research” is parallel to “the prediction of eclipses is dead as an area of mathematical research” — it makes more sense to say that they were areas of astronomical research and that both the astronomy and the resulting mathematical techniques are well-settled by now. –  May 16 '22 at 22:57
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    “take a dot product, and take the $\arccos$” – actually, neither that or the spherical law of cosines is a very good idea, because the arc cosine is numerically unstable due to the discontinuity at small angles and antipodes. (Doesn't matter for most concrete examples, but if you write any piece of computation code based on this then somebody's in for some nasty surprises...) Instead, you should e.g. compute both the dot product and magnitude of the cross product, and take the atan2 of them. (Or derive something stable based on the spherical law of cotangents, if you can be bothered...) – leftaroundabout May 17 '22 at 10:12
  • @leftaroundabout, both the magnitude function and the arc cosine function have infinite square roots at extreme values -- why is one more stable than the other? –  May 17 '22 at 15:58
  • @MattF. the problem is taking the $\arccos$ of a sum of two terms that can have opposite sign (IOW, the $\arccos$ of a difference). Both terms can be relatively large and thus have significant floating-point error, but if they almost cancel the difference then has a large relative error, and that's blown out of proportion by the $\arccos$. By contrast, when computing the magnitude you're only taking the square root of a sum of squares, i.e. of terms that are guaranteed to be all nonnegative, thus the relative error will only be on the order of the relative error of the individual terms. – leftaroundabout May 17 '22 at 16:07
  • (This problem cost me at least a month while I was working on my master's thesis – I was running numerical simulations of planetary magnetospheres, and at some discretisation parameters I inexplicably got very large artifacts. After digging through thousands of lines of code I found the offending ACOS function, written by someone who had learned about how “nice” spherical trigonometry is but not which formulas are good, which ones are bad and which ugly...) – leftaroundabout May 17 '22 at 16:16
  • I'm still confused....why is the arc cosine so problematic for a difference near 0, when its derivative is finite (=-1) for arguments near 0? For the formula in my answer, I would expect some instability for small angles, but then the argument is near 1 and I wouldn't see an issue of signs. –  May 17 '22 at 16:40
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    @MattF. hmright. I may be misremembering the details. But actually for your question regarding why $\arccos$ is more problematic than $\sqrt\cdot$, suffice it to say that the latter has its discontinuity at $0$ where the floating-point accuracy becomes very high, so unless you're passing in a sum of different-sign terms it's generally stable. Whereas $\arccos$ has its discontinuities at $\pm1$, where the floating point error is constant. – leftaroundabout May 17 '22 at 22:11
  • Spherical geometry has other uses also. For example, consider giving a weather report based on a zip code -- spherical geometry provides a useful approximation here for finding distance based on lat/long (post offices and weather stations both have lat/long, zip code identifies post office).

    Also useful in game programming...

     – rdm May 18 '22 at 19:03
  • @rdm, I consider calculation of distances from coordinates included in surveying...but I agree that the applications in gaming and computer vision are new and notable. –  May 18 '22 at 20:26
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    @MattF. See William Kahan's paper How Futile are Mindless Assessments of Roundoff in Floating-Point Computation? §12: Mangled Angles. – user76284 May 20 '22 at 16:52
  • @user76284, that is interesting -- can you also provide a clean version of the formula recommended there? The pdf is mangled with subexpressions like $|x\cdot|y|$. –  May 20 '22 at 17:29
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    @MattF. I believe the recommended formula is $\angle(x, y) = 2 \operatorname{atan2}(x |y| - |x| y, x |y| + |x| y)$, where atan2 is the 2-argument arctangent. – user76284 May 20 '22 at 18:08
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Not sure that it genuinely counts, but some interesting research in dynamical systems considers the dynamics of billiards of various shapes on the sphere. Examples:

Spina, M. E., & Saraceno, M. (2001). Quantum spectra of triangular billiards on the sphere. Journal of Physics A: Mathematical and General, 34(12), 2549.

Spina, M. E., & Saraceno, M. (1999). On the classical dynamics of billiards on the sphere. Journal of Physics A: Mathematical and General, 32(44), 7803.

Blumen, V., Kim, K. Y., Nance, J., & Zharnitsky, V. (2012). Three-period orbits in billiards on the surfaces of constant curvature. International Mathematics Research Notices, 2012(21), 5014-5024.

The reason I'm not sure is that the current research in classical, flat billiards would then entail that euclidean geometry is also not dead as a research field... but hey, I'm going to stand by that.

Alessandro Della Corte
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Other answers give examples of advanced research, but I thought I'd point out a modest result from my somewhat recreational investigations in the area:

If $W$ is area of the "hypotenuse-face" of a spherical tetrahedron with right "leg-faces" of area $X$, $Y$, $Z$, then $$\cos\tfrac12W = \cos\tfrac12X\cos\tfrac12Y\cos\tfrac12Z+\sin\tfrac12X\sin\tfrac12Y\sin\tfrac12Z \tag{$\star$}$$

This is a counterpart of de Gua's theorem (aka, the "Pythagorean Theorem for Euclidean tetrahedra"): $$W^2 = X^2 + Y^2+Z^2$$ Of course, a hyperbolic counterpart exists as well; just append "h"s to all the trig functions in $(\star)$, and change "$+$" to "$-$". There are even associated Laws of Cosines, but I won't go into that here.

I will give a shout-out to an ancient, still-open question of mine: "Pythagorean theorem for right-corner hyperbolic simplices? ", which specifically seeks the $4$-dimensional hyperbolic counterpart of $(\star)$. (Since spherical and hyperbolic relations are likely comparable, this also counts as an open question in spherical geometry.)

Blue
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