Geometry

   

Learning Geometric Algebra by Modeling Motions of the Earth and Shadows of Gnomons to Predict Solar Azimuths and Altitudes

Authors: James A. Smith

Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to calculate Solar azimuths and altitudes as a function of time via the heliocentric model. We begin by representing the Earth's motions in GA terms. Our representation incorporates an estimate of the time at which the Earth would have reached perihelion in 2017 if not affected by the Moon's gravity. Using the geometry of the December 2016 solstice as a starting point, we then employ GA's capacities for handling rotations to determine the orientation of a gnomon at any given latitude and longitude during the period between the December solstices of 2016 and 2017. Subsequently, we derive equations for two angles: that between the Sun's rays and the gnomon's shaft, and that between the gnomon's shadow and the direction ``north" as traced on the ground at the gnomon's location. To validate our equations, we convert those angles to Solar azimuths and altitudes for comparison with simulations made by the program Stellarium. As further validation, we analyze our equations algebraically to predict (for example) the precise timings and locations of sunrises, sunsets, and Solar zeniths on the solstices and equinoxes. We emphasize that the accuracy of the results is only to be expected, given the high accuracy of the heliocentric model itself, and that the relevance of this work is the efficiency with which that model can be implemented via GA for teaching at the introductory level. On that point, comments and debate are encouraged and welcome.

Comments: 54 Pages.

Download: PDF

Submission history

[v1] 2018-04-24 20:34:14

Unique-IP document downloads: 27 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus