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 […]

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# Lambda as a Function of Phi

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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

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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

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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

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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

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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

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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 […]

# Circles and Spheres, Angles and Cones

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Length in two-dimensional space (in other words, length in a plane) defines distance. If the distance is referenced from a point, then the circle is the set of all points that are this particular distance away from that point, or center. The radius is the length or measurement of the distance from the center to […]

# Volume as a Component of Spacetime

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It’s generally understood that, when the time dimension is subtracted from spacetime, we are left with Euclidean 3-space. This is usually interpreted to mean that, once we look at spacetime without the time component, we’re left with space, or what in SI units is called volume. The SI unit for volume is which is consistent […]

# What is a Base Quantity?

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Figure 2.1 – NIST Subway Diagram So, what is direction, really? Currently, direction is an orphan when it comes to understanding physical quantities and how they relate to one another. The basic unit that is used to define an angle, the radian, is the disconnected unit at the lower right in the NIST diagram. […]