C = circle made by intersection of the xy-plane and unit sphere S CO = (0,0,0) = origin = center of circle C r = = radius of circle C (r,0,0) = intersection of positive x-axis with unit sphere S = SN (north pole) (-r,0,0) = intersection of negative x-axis with unit sphere S […]
Formula
Use for Deriving The Formula chapters.
Alpha as a Function of Phi
Figure 4. Dihedral Angle α Referring to Fig. 4, since b and c both have endpoints on circle C, Δabc is an isosceles triangle with dihedral angle between the two equal sides. If we construct a great circle arc from point SC to midpoint D of side a, we will bisect Δabc into two congruent right […]
Lambda as a Function of Phi
Figure 3. Isosceles Spherical Triangle ∆abc Point SC is the intersection of the z-axis with sphere S such that ∠SNSOCO = = ∠SNSOSC. 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 […]
The Problem
The objective is to define a family of functions which are based on different values of , and which express α as a function of . The approach will be to find and , each as a function of . One of the methods used here will be to solve a spherical isosceles triangle using spherical […]
The Angles
Plane can be considered the longitude plane because the great circle made by its intersection with sphere S is always a line of longitude. Plane can be considered the tangent plane since it always contains a line tangent to circle C at point P. The dihedral angle between planes and is angle , the angle […]
The Planes
The two planes that remain stationary in the animation each contain SO and a line tangent to circle C. For tan-colored plane this tangent is at (-r,0,0) and for magenta-colored plane it is at SN which is also (r,0,0). Because these two planes contain the center of the sphere, which need not be in the […]
The Coordinate System
Figure 2. Coordinate System In the model as illustrated in Fig. 2, small circle C has a circumference that is 45º of latitude. We will be using the conventional terminology where a circle on the surface of a sphere that is made by the intersection of the sphere with a plane passing through the sphere center […]
The Model
Figure 3.1 Mathematica Model The animation in Fig. 3.1 above is an illustration of a mathematical model that was produced by Hans Milton in Mathematica. The model captures two smooth functions that are interdependent in normal, orthogonal, Euclidean 3-space. The illustration shows an animation of the visual components of this relationship. Also in the animation are […]
