This put up is a aspect quest within the collection on navigating by the celebs. It expands on a footnote within the earlier put up.
There are six items of data related to a spherical triangle: three sides and three angles. I mentioned within the earlier put up that given three out of those six portions you possibly can clear up for the opposite three. Then I dropped a footnote saying typically the lacking portions are uniquely decided however typically there are two options and also you want extra knowledge to uniquely decide an answer.
Todhunter’s textbook on spherical trig offers a radical account of how you can clear up spherical triangles underneath all doable instances. The primary version of the guide got here out in 1859. A gaggle of volunteers typeset the guide in TeX. Mission Gutenberg hosts the PDF model of the guide and the TeX supply.
I don’t wish to duplicate Todhunter’s work right here. As an alternative, I wish to summarize when options are or are usually not distinctive, and make comparisons with airplane triangles alongside the best way.
SSS and AAA
The best instances to explain are all sides or all angles. Given three sides of a spherical triangle (SSS), you possibly can clear up for the angles, as with a airplane triangle. Additionally, given three angles (AAA) you possibly can clear up for the remaining sides of a spherical triangle, not like a airplane triangle.
SAS and SSA
While you’re given two sides and an angle, there’s a distinctive answer if the angle is between the 2 sides (SAS), however there could also be two options if the angle is reverse one of many sides (SSA). This is similar for spherical and airplane triangles.
There could possibly be much more than two options within the spherical case. Take into account a triangle with one vertex on the North Pole and two vertices on the equator. Two sides are specified, working from the pole to the equator, and the angles on the equator are specified—each are proper angles—however the aspect of the triangle on the equator could possibly be any size.
ASA and AAS
While you’re given two angles and a aspect, there’s a distinctive answer if the aspect is frequent to the 2 angles (ASA).
If the aspect is reverse one of many angles (AAS), there could also be two options to a spherical triangle, however just one answer to a airplane triangle. It is because two angles uniquely decide the third angle in a airplane triangle, however not in a spherical triangle.
The instance above of a triangle with one vertex on the pole and two on the equator additionally reveals that an AAS drawback might have a continuum of options.
Abstract
Observe that spherical triangles have a symmetry that airplane triangles don’t: the spherical column above stays unchanged should you swap S’s and A’s. That is an instance of duality in spherical geometry.
