Figure 3. Isosceles Spherical Triangle ∆abc

Point SC is the intersection of the z-axis with sphere S such that ∠SNSOCO = $\upsilon$ = ∠SNSOSC. There is a spherical triangle ∆abc that, when solved, will express the relationship between $\alpha$, $\phi$, $\lambda$, and $\upsilon$ such that:

Side a is great circle arc of length $\lambda$
Side b is great circle arc of length $\upsilon$, or ∠SCSOSN
Side c is great circle arc of length $\upsilon$, or ∠SCSOP $\alpha$ = ∠ $\mathbb{BG}$ $\phi$ = ∠SNCOP = dihedral ∠SN-SOCO-P $\lambda$ = ∠PSOSN $\upsilon$ = ∠SNSOCO

Since both point P and point SN are on the unit sphere with center SO, the triangle that these 3 points form is an isosceles triangle with a pair of sides, PSO and SNSO, each having length = 1.  The angle between the two equal sides is $\lambda$.  If we take $\lambda$ as given, we can find the length h of the base of this isosceles triangle, which is the distance between SN and P.  Then, using length h we can find $\phi$ by regarding the solution as a 2-dimensional problem that is set in the xy-plane where circle C lies.

For the first step, h is the chord of the great circle arc $\lambda$ of sphere S, or h = ${2}\sin\frac{\lambda}{2}$.  For the second step, with a circle of radius r and center (0,0), h will be the chord length of the arc segment from (r,0) to P, or h = ${2}{r}\sin\frac{\phi}{2}$.  These equations give a parametric representation of $\lambda$ as a function of $\phi$.