There exists a deeper understanding of what constitutes -space. Each orthogonal two-dimensional plane is most commonly defined as two lengths (or axes) perpendicular to one another, sure, but what does that really mean? The concept of orthogonality brings the concept of relative direction (or orientation) into the picture. This structure of relative orientation, when combined […]

# Background

# How We Got Here

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200 years ago, Napier and others wrote about the Law of Sines for spherical angles, expanding on work begun by the Arabs. Approximately 150 years ago, a treatise on this branch of mathematics (Todhunter’s textbook “Spherical Trigonometry”) was published. From then until now, this was considered to be the complete story regarding spherical trigonometry. Nothing […]

# After Normalization

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Figure 4.3 Normalized Values for The curves representing have had the axis “normalized” and are rotated 90º to correspond to the orientation of the partial sine curve.

# Before Normalization

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Figure 4.2 Different Values of This is another graph by Strange and this one shows four different values: The curves have different extents along the x axis, which is the angle. In order to represent as a quantity that expresses direction, the curves must be “normalized” by making the extent of the curves […]

# Graphing the Function

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Figure 4.1 This graph was generated by Strange, using data from the Mathematica model written by Hans Milton.