Let the three points A,B,C be the vertices of a moving spherical triangle on the surface of a sphere

Kendrick Hampton 2022-06-27 Answered
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.
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Answers (1)

feaguelaBapzo
Answered 2022-06-28 Author has 9 answers
Let O be the centre of the sphere. Then the spherical BCA angle γ 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 π through O,A and find a plane τ through O,B such that the angle between π and τ equals γ. By intersecting π τ with the sphere we find every point of our locus. As an alternative, we may exploit the spherical cosine rule:
(1) cos C = cos c cos a cos b sin a sin b
where c = A O B ^ 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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