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steve upson
 Post subject: Expressing a New Equality  |  Posted: Sun May 29, 2016 2:33 pm

Joined: Tue May 24, 2016 12:25 am
Posts: 14

steve upson
 Post subject: Re: Expressing a New Equality  |  Posted: Sun May 29, 2016 2:36 pm

Joined: Tue May 24, 2016 12:25 am
Posts: 14

 Gratitude and credit go to Hans Milton for creating this model in Mathematica. The first angle that we are concerned with is the elevation angle E. It is the angle that is formed between the equatorial plane (beige) and the elevation plane (yellow). The second angle is more complicated. It is formed between a plane of longitude (green) and a "tangent" plane (blue) which lies in a conical orbit. The construction of the longitude plane is not too complicated, but its orientation is very specific. The orientation of the longitude plane is such that it intersects a point on a small circle which is constructed on the surface of the sphere at a 45 degree angle. This small circle can be defined as a circle at the 45 degree latitude which has been tilted 45 degrees such that it intersects both the pole and the equator. The "tangent" plane lies along the surface of a cone formed by the sphere center and the 45 degree small circle. As the elevation angle E is varied, the position of this tangent plane changes such that it remains coincident with the intersecting point on the circumference of the small circle. If we call the angle that is made between the longitude plane and the tangent plane the angle α, then the object is defined by the function which expresses the relationship between the two angles, E and α. This function is the one that is troublesome, for some reason or other. How is this function expressed? Thanks to a member of another forum, strange, for graphing the results from the model:blue: α = f(E)red: sineIt should be mentioned that, although a sphere is used for construction of the animation that is shown here, there actually is no sphere involved in the function itself. In other words, there is no two-dimensional surface involving spherical excess, or anything like that. The sphere is simply used as an aid in visualizing how the object is constructed.Also, the model isn’t really complete, as it should have another degree of freedom added to it. The following image shows how the model should work when it is complete. The illustration at a is the example shown in the model, where the relationship between the ordinal and cardinal directions is 45 degrees. Example b shows an angle less than 45 while c shows an angle greater than 45. Example d is provided to show that the model is not based on any particular size. In other words, there is no metric. It’s unfortunate that the model includes the variables E for elevation and alpha for the tangent angle, but we were dealing with spherical geometry at the time those variables were chosen and we weren’t even considering that those variables are reserved for specific usage in physics.
steve upson
 Post subject: Re: Expressing a New Equality  |  Posted: Sun May 29, 2016 3:25 pm

Joined: Tue May 24, 2016 12:25 am
Posts: 14

 The Mathematica model is contained in a .cdf file. I'm not sure of how to go about posting the file here. If anyone is interested in viewing the model (rather than a .gif animation) then let me know how to publish it and I will put it in this post. The actual model allows you to turn off certain objects such that the image is much less cluttered. Also, it lets you move the slider manually, again, for more clarity.
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 3 posts • Page 1 of 1

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