**Figure 3. Isosceles Spherical Triangle ∆abc**

Point S_{C} is the intersection of the z-axis with sphere S such that ∠S_{N}S_{O}C_{O} = = ∠S_{N}S_{O}S_{C}. There is a spherical triangle ∆abc that, when solved, will express the relationship between , , , and such that:

Side a is great circle arc of length

Side b is great circle arc of length , or ∠S_{C}S_{O}S_{N}

Side c is great circle arc of length , or ∠S_{C}S_{O}P

= ∠

= ∠S_{N}C_{O}P = dihedral ∠S_{N}-S_{O}C_{O}-P

= ∠PS_{O}S_{N}

= ∠S_{N}S_{O}C_{O}

Since both point P and point S_{N} are on the unit sphere with center S_{O}, the triangle that these 3 points form is an isosceles triangle with a pair of sides, PS_{O} and S_{N}S_{O}, each having length = 1. The angle between the two equal sides is . If we take as given, we can find the length h of the base of this isosceles triangle, which is the distance between S_{N} and P. Then, using length h we can find 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 of sphere S, or h = . 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 = . These equations give a parametric representation of as a function of .