Spherical Trigonometry is used as a gateway to abstract mathematics because it can be taught as the study of the geometry of an intrinsically curved surface, without reference to its embedding in a “flat” 3-d space, as we experience it. So this is the preferred and usual method of teaching.
Still, I found myself a bit unsatisfied when I recently looked up, for study, the Spherical Law of Sines, which stands in direct analogy to the Law of Sines of plane trigonometry, with one important variation. That is, the sides, a and b, as referenced in the latter, become sin a and sin b in the former … very mysterious!
I found that the proofs tended to be algebraic in nature, depending on various “trigonometric identities” and I wondered if I couldn’t come up with something a little more “concrete” using constructive methods of traditional spatial ( Euclidian ) geometry.
So, doodling around, I found that I had great difficulty visualising 3-d constructions by reliance on 2-d diagrams, i.e. drawn on paper.
So I decided, well, I’ll make a model out of wood.
Of course, that requires a little doing. The obvious method is to cut out a sector of a sphere along central planes, but that isn’t exactly easy, and it wastes most of the sphere ( even if you can make several this way from one sphere. )
Also, wooden craft or carving balls are expensive, and usually rather small. So I decided to cut one from a block. I could buy a bag of 8 or 10 suitable blocks for a few dollars, and I could use two of the flat sides for the plane surfaces of the defining central planes. This would give me a nice RIGHT spherical triangle, while minimizing the carving effort.
So, here is the completed result, along with another block showing the preliminary mark-up:
BTW, forming the curved surface wasn’t as laborious as I thought it would be. I could get reasonably close by cutting easily defined tangent planes to the surface, and then sandpaper made short work. Of course, it’s not perfect!
Napier’s Rules and the Spherical Law of Sines
Well, I had already noticed that the Spherical Law of Sines could be reduced to one of Napier’s Rules, in this case (R2) as named in the Wikipedia article on Spherical Trigonometry.
(R2) sin a = sin A sin c
In the image below, angle A is at the base of the spherical triangle at the top of the photo, and measures the dihedral angle at bottom right. So it’s a simple application of “opposite over hypotenuse” to see that sin A = sin a / sin c , which is (R2) above.
Note also that sin a and sin c are defined by the central angles, a and c, which measure the sides, a and c, of the spherical triangle.
To get the Spherical Law of Sines all that is required is to place next to it another right spherical triangle sharing side a, along with the perpendicular plane face subtending it, to form a “general” triangle.
The shared altitude, sin a, then plays the same role in the proof as the shared altitude used in the proof of the Plane Law of Sines.
… and that’s all I have to say about that.
The Hot-wire Foam cutter and the Spherical jig
Now for the really fun part:
This is a hot-wire foam cutter equipped with a “spherical jig” of my own devising. As you can see it holds a 3″ foam sphere to be cut in half ( as adjusted . ) By cutting the same sphere in half “three ways”, I can form a spherical triangle along with its subtending solid central angle, just like the carved wooden model above.
Of course, I automatically get 8 such triangles. In the general case these will be 4 distinct triangles, along with their mirror inverses.
A Spherical Zoo
Here’s a “special” case of a such a production, contrived to produce a “large” equilateral triangle ( of course two of them ) and whatever else “falls out” :
The “large” equilateral triangles, inverses of each other, but in this case identical, are in the center.
Then six identical isosceles triangles are arranged on the sides. These form an equatorial ring between the two polar equilateral caps.
I first thought that this accounted for all possible isoceles triangles, but these are just a subset. They are the isoceles triangles which have the length of each ( equal ) side as the supplement of the base. That is, the base plus one side equals a 180 degree arc.
Just part of the arrangement that I think of as the Spherical Zoo.
Some “Nuts and Bolts”
I had been pondering how to organize this Zoo, and I was thinking in terms of the edges of the triangles of a partitioned sphere. This seems natural, since a “small” triangle can have arbitrarily short edges, just like plane triangles.
However, the triangles of a partition can also be specified by their ( corner ) angles, and this has the advantage that we can use the “spherical excess” rule, which gives the area of a spherical triangle in terms of the “excess” above 180 degrees, or “pi”. In fact, the area, in units of R squared, regarded as “Unity” for a given sphere, is exactly this value, expressed in radians.
So, using this scheme, I came up with a simple “map” of all the possible divisions of the sphere into triangles, in terms of A, B, and C, the angles of any one of the 8 triangles in such a division.
Well, with regard to units, we can have our cake and eat it too by specifying these angles in terms of pi/180 radians, which corresponds to “degrees”, but reminds us that the area is measured in “steradians”, of which there are 4 pi to be accounted for among the 8 triangles of a division, using the spherical excess rule. Note that this is nothing to do with “square degrees” … which we won’t touch.
In fact, since our “degree” is pi/180 radians, the 4 pi steradians of the sphere are measured by 720 degrees, or 720 X pi/180. I explain all this so that we can do examples with angles, and areas, expressed in small integers. That this is possible is remarkable in itself!
… so then my “map” :
… The angles A, B, and C , with values 0 thru 180, as indicated, will specify a valid spherical triangle whenever they are the coordinates of a point lying inside the tetrahedron drawn in red.
… to be continued