Kendrick Hampton

2022-06-27

Let the three points A,B,C be the vertices of a moving spherical triangle on the surface of a sphere. The triangle moves so that while the vertices A,B remain fixed, the angle BCA at the vertex C stays constant. What is the locus of the moving vertex C? Is there a special name for the curve traced out by C? If A,B,C were the vertices of a plane triangle, the corresponding locus would be the arc of a circle. I have made some calculations, which-if they do not contain errors-lead me to believe that, in the spherical case, the locus I am seeking is not the arc of a small circle (on the sphere). But, if so, I do not know what type of curve it is.

feaguelaBapzo

Let O be the centre of the sphere. Then the spherical BCA angle $\gamma$ is the angle between the planes through O,A,C and O,B,C. So we can find our locus in the following way: we take a plane $\pi$ through O,A and find a plane $\tau$ through O,B such that the angle between $\pi$ and $\tau$ equals $\gamma$. By intersecting $\pi \cap \tau$ with the sphere we find every point of our locus. As an alternative, we may exploit the spherical cosine rule:
$\begin{array}{}\text{(1)}& \mathrm{cos}C=\frac{\mathrm{cos}c-\mathrm{cos}a\mathrm{cos}b}{\mathrm{sin}a\mathrm{sin}b}\end{array}$
where $c=\stackrel{^}{AOB}$ and so on. Notice that both $c$ and $C$ are fixed, so everything boils down to a not-so-trivial constraint between $a$ and $b$.

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