Talk:Central angle

Real world applications

The problem with most of these geometry articles is that they do not include any real-world applications. This would be much more helpful for the general reader. TheEvilPanda (talk) 22:24, 2 April 2009 (UTC)

Simplicity

This sentence seems to be placed too early in the article:

The "point", ${\widehat {\sigma }}\,\!$, is actually——relative to the great circle it is being measured on——the transverse colatitude ("TvL"), and the central angle/angular distance is the difference between two TvLs, $\Delta {\widehat {\sigma }}\,\!$.

Neither the symbol $\Delta {\widehat {\sigma }}\,\!$ nor the notion of transverse colatitude appear before this sentence. How the transverse colatitude, ${\widehat {\sigma }}\,\!$, may be thought of as a "point" is not explained. The location of the pole for the "the great circle it is being measured on" is not clear.

The technique for finding the central angle by Cartesian subtraction seems to be placed too far into the article. This concept may be more familiar to the general reader.

Is not the symbol for the central angle simply $\sigma \,\!$? As I read the article, it sounds like the symbol for transverse colatitude is ${\widehat {\sigma }};\,\!$ and one way to calculate the central angle is by subtracting transverse colatitudes: $\sigma =\Delta {\widehat {\sigma }}\,\!$. -Ac44ck (talk) 05:05, 10 October 2010 (UTC)

Most geodetic math papers/articles sloppily denote the central angle as just $\sigma \,\!$ (not $\Delta \sigma \,\!$), its standpoint as $\sigma _{1}\,\!$ and the forepoint just as $\sigma _{1}+\sigma \,\!$ (they similarly denote the longitude difference just as $\lambda \,\!$, not $\Delta \lambda \,\!$). The central angle is the TvL difference, as stated in the next sentence. The "hat" is added to distinguish spherical/circular from elliptical/auxiliary (capped with "tilde"): $\phi ,\lambda ({\mbox{or}}L),{\widehat {\alpha }},{\widehat {\sigma }},{\widehat {\mathrm {A} }}\,\!$ (spherical) vs. $\beta ,\omega ({\mbox{or}}\lambda ),{\tilde {\alpha }},{\tilde {\sigma }},{\tilde {\mathrm {A} }}\,\!$ (elliptical)——there is a term for this (modifying a variable to denote different "flavors", here spherical vs. elliptical), that I saw somewheres a while back, but I can't remember the word ("s______"???).

As for the TvL being a "point" and the "location of the pole for the 'the great circle it is being measured on'", another definition of the TvL is "the angular distance from the equator to a point along a great circle", thus the "pole" is on the equator and the TvL is on the point:

$\phi ,\lambda \,\!$                   ${\widehat {\sigma }},{\widehat {\mathrm {A} }}\,\!$

 ~Kaimbridge~ (talk) 02:55, 11 October 2010 (UTC)

Thanks for the clarification of the meanings of the symbols over the letters. This was new to me.

It seems to me that the central angle needs its own symbol. Given that "the central angle is the TvL difference," it doesn't seem to be the only way to compute a central angle. One can find the central angle from chord length with no awareness of transverse colatitudes.

One may note that the value 2 results from the subtraction 5-3. We might write it as $\Delta {(5,3)}\,\!$, but the notion 'two' is important enough to have its own symbol. It seems to me that the notion of 'central angle' is also important enough to have its own symbol.

The article says "A central angle is an angle whose vertex is the center of a circle." Thus, we seem to be restricted to dealing with circles (or spherical sections, which are circles). The "hat" would seem to be unnecessary due to context, as elliptical objects seem to be excluded by definition. -Ac44ck (talk) 03:23, 12 October 2010 (UTC)

Back to the original purpose

I've greatly shortened this page by removing the material related to solving for great circles. This duplicates the coverage of the Great-circle distance page and doesn't belong in an article about elementary geometry. (The coverage is now approximately the same as the central angle pages in other languages.) cffk (talk) 13:14, 25 June 2013 (UTC)

To 91.216.197.186: I removed the irrelevant material again. The subject matter doesn't match this article! If there's something of value here, consider adding it to the great circle distance or navigation pages. However I should warn you that the equations that you added back are far from being in a useable form (missing definitions and explanations, non-standard notation, multiletter symbols, etc.). This would need a lot of cleaning up before adding it to another article. cffk (talk) 01:29, 26 June 2013 (UTC)

To 162.213.209.231: I removed the irrelevant material yet again (see above). The material that I've deleted is truly awful. It uses non-standard terminology (conjugate latitude, conjugate longitude, TvL), non-standard notation (sgn with an arrow on top, hats on various quantities), and overly complicated formulas. There's no explanation of that this stuff is doing on this page. I've updated the Great-circle navigation page so that the same results are given there in a clearer manner (and this is a appropriate page for this material). cffk (talk) 15:05, 19 July 2013 (UTC)

This article now provides no help in calculating a central angle from known information. The fact that help is available in the great-circle distance article is easily missed if one's application does not involve a great-circle distance. - Ac44ck (talk) 22:16, 20 July 2013 (UTC)

I've added links to the great-circle navigation page. That page now contains complete information on determining the great circle between two points. cffk (talk) 22:40, 20 July 2013 (UTC)

Nothing in the lede of the linked article suggests it is a source of information on calculating a central angle. A ship's navigator might read further, but what about a land surveyor? Can we assume either will find the relevant information and make the connection between a great circle and an ordinary circle? What about a high school student looking for confirmation that they have a correct method to find a central angle?

I don't look for a return of transverse colatitudes and suchlike, but I think the article has become too simple.

Everything must be made as simple as possible. But not simpler. -- Albert Einstein

- Ac44ck (talk) 23:20, 20 July 2013 (UTC)

I suggest you go ahead and makes the edits you think will make the article more helpful. I'm not sure how a land surveyor would end up on this page unless a page he was reading referred to it (and then I would expect that page to provide any additional necessary information). The original purpose of this page and still the purpose with all other languages is to define the term in elementary geometry, used for instance in proving that the central angle is twice the inscribed angle. The term "central angle" is not usual in spherical trigonometry, surely. There you would normally be talking about the sides of a triangle. cffk (talk) 01:33, 21 July 2013 (UTC)

Changes by Lfahlberg

I think I stayed true to the content I found. I added formulas, images, references and links. Lfahlberg (talk) 20:49, 7 December 2013 (UTC)

I think additional formulas are fine. Whether great circles or central angles are of primary importance is in the eye of the beholder. I don't think it is necessarily true that a formula that may find application in calculating great distance circles doesn't belong here. - Ac44ck (talk) 01:04, 8 December 2013 (UTC)

Common notation of angle measure

The notation

• $m{\overset {\frown }{ABC}}$ for an arc and
• $m\angle AMC$ resp. $\mu (\angle AMC)$ for a corresponding angle

for describing the central angle seems to be quite common so it seems to be worth a mention. They are used in these documents, but I couldn’t find a source that actually describes the notation:

• Glassmeyer, David; Barton, Daniel (2020), "Applying Quantitative Reasoning to Clarify Arc Measurements", Georgia Educational Researcher, vol. 17, no. 1, Georgia Southern University, p. 4, doi:10.20429/ger.2020.170101
• The μ notation: Williams, Steven R. (2006), "Circles, Arcs, Inscribed Angles, and Power of a Point", Math 362 Notes, Brigham Young University, retrieved 2021-03-14

I could even find a rudimentary description of the notation itself here, but doubt about this source being a good source: "Central Angles and Arcs". CliffsNotes. Houghton Mifflin Harcourt. 2020. Retrieved 2021-03-14. --Yuwash (talk) 01:12, 14 March 2021 (UTC)

Odd notation

Is there any reason why capital $\Theta$ is used throughout the article instead of the more usual $\theta$ which readers will be used to seeing? Does anyone object to me changing all the instances of it to lowercase? I find it distracting when reading, and suspect others will too.

The $\theta$ in the diagram could be interpreted as either, but looks to me more like oversized lowercase than uppercase, so I don't think the change will introduce any conflict with that. Musiconeologist (talk) 00:16, 10 March 2024 (UTC)

Feel free to switch it to ⁠$\theta$⁠ or ⁠$\phi$⁠ or whatever. There's no reason to pick a particular symbol, and uppercase ⁠$\Theta$⁠ is not too common. This article is basically a stub, so don't read too much into any of the choices made. –jacobolus (t) 01:02, 10 March 2024 (UTC)

@Jacobolus Thanks. Those were my thoughts as well. Musiconeologist (talk) 01:57, 10 March 2024 (UTC)