The objective is to define a family of functions which are based on different values of \upsilon, and which express α as a function of \lambda . The approach will be to find \alpha and \lambda, each as a function of \phi.  One of the methods used here will be to solve a spherical isosceles triangle using spherical trigonometry.

At the top right corner of the animation there is a set of numbers representing \lambda that change between 0° and 90°, while below them, in the rectangular box, there is another set of numbers representing \alpha that also change between 0° and 90°. The mathematical model used for the animation defines a specific member of a family of functions. The z-coordinate for the center of sphere S defines which member of the family we are analyzing. The model shows \alpha as a function of \lambda with angle \upsilon equal to 45°, or z = \frac{1}{\sqrt{2}}.


Also, because we are using a unit sphere,  r = sin \upsilon = \sqrt{1-{s_z}^2}

(Unit sphere center SO can move up or down along the z-axis and the north pole will remain the north pole and circle C will still pass though the pole SN because angle \upsilon changes accordingly in order to keep this true.)

It should be mentioned that, although a sphere is used for construction of the model that is presented here, there actually is no sphere involved in the function itself. In other words, there is no two-dimensional surface involving spherical excess (parallel transport) or anything like that. The sphere is simply used as an aid in visualizing how the object is constructed and spherical trigonometry used in solving for some of the unknowns which can be found by analysis of the sphere’s gross occupation of Euclidean 3-space.