How many spherical quadrangles exist with a given ordered sequence of inner angles.

Question

Well, I think the title already explains my question. Given a sphere and an ordered sequence of inner angles ($\alpha$, $\beta$, $\gamma$, $\delta$) how many spherical quadrangles do there exist that have that sequence as angles and the added property that three of the edges need to have the same size and the fourth edge needs to have a different size.?

I was told that this on the sphere there might be several quadrangles with quite different appearances, but I can't find any reference explaining this in more detail or bounding the number of possible quadrangles.

You can assume that the angles satisfy the conditions necessary to be the angles of a spherical quadrangle.

Answer 1

The answer for the plane is wrong. You can move any vertex and the two sides incident at it while keeping the opposite vertex and the two sides incident at it fixed, then extend or clip all sides to meet in two more vertices. The result isn't a scaled version of the original quadrangle. This is easiest to imagine for a rectangle or a parallelogram, where you can simply extend two parallel sides.

The same is true for at least some sequences of angles on the sphere. For instance, take two meridians and the equator, which form two right angles, and connect two points at equal latitude on the meridians by a great circle. Now if you move the meridians apart symmetrically, they still form right angles with the equator. The angle they form with the great circle changes, but you can compensate for that by connecting them at a different latitude with a different great circle.

Answer 2

There is no bound on the sphere either. Fix an angle $\alpha > 0.$ For any $\beta > 0,$ Construct the triangle with angles $$ \frac{\pi}{2} + \alpha, \; \; \frac{\pi}{2} - \beta, \; \; \alpha + \beta. $$ The angle sum is $\pi + 2 \alpha,$ so the area is $2 \alpha.$ Now, find the midpoint of the edge opposite the angle $\frac{\pi}{2} + \alpha,$ and place a copy of the same triangle there rotated 180 degrees ($\pi$) around that point, so that the result is a quadrangle. Because $$ \frac{\pi}{2} - \beta + \alpha + \beta = \frac{\pi}{2} + \alpha, $$ the resulting quadrangle has all four angles equal to $ \frac{\pi}{2} + \alpha.$

However, by the Law of Sines on the sphere, the ratio of (the sines of) the edge $x$ opposite the angle $\alpha + \beta$ and the edge $y$ opposite the angle $ \frac{\pi}{2} - \beta$ is $$ \frac{\sin x}{\sin y} \; \; = \; \; \frac{\sin(\alpha + \beta)}{\sin{(\frac{\pi}{2} - \beta})} = \frac{\sin(\alpha + \beta)}{\cos \beta} = \sin \alpha + \cos \alpha \tan \beta. $$ That is, as we vary $\beta,$ we get uncountably many quadrangles with the same vertex angles that are not congruent.