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.