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August 5th, 2015, 08:12 AM   #21
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Quote:
 Originally Posted by steveupson It's the second case, where the intersection lies on the circumference. The plane of alpha rotates only 90°, though, when moving from the bottom of the small circle (equator) to the top (pole). When we say "plane of alpha" we are talking about the plane passing through the center of the sphere that the tangent of the intersection lies in. This plane rotates 90° (relative to a line of longitude) from the equator to the pole.
Okay, thanks for the clarification. I will try and come up with some updated mathematics as soon as I can.

 August 6th, 2015, 10:09 AM #22 Senior Member   Joined: Jun 2015 From: England Posts: 915 Thanks: 271 The mathematics of spherical gearing is quite complicated. Here is some theory http://www.iftomm.org/iftomm/proceed...apers/A385.pdf
August 8th, 2015, 07:58 AM   #23
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Quote:
 Originally Posted by studiot The mathematics of spherical gearing is quite complicated. Here is some theory http://www.iftomm.org/iftomm/proceed...apers/A385.pdf
There are a couple of things I don't really understand about this paper. First and foremost, I don't understand their figure 1 at all.

It seems to me that any "concave spherical gear" has to have internal teeth, but their illustration shows one having external teeth.

Is that even possible?

I think not. It's like an M.C. Escher creation that cannot exist in captivity.

In any event, I understand that the term "spherical gear" appears in my notes from 10 years ago, but we were working on something completely different than what's described in that paper. What we have been working on are basically two different devices; one which is a clutch that operates using mass rather than friction to do work, and the other is a gear which can produce a continuously variable ratio.

We have recently changed the nomenclature for these devices to "anti-friction clutch" and "universal gear" since these terms seem to be new and unique, and they more closely describe the actual machines.

In order to ensure that there is no confusion on this "anti-friction" aspect, that term is currently used to refer to a class of mechanical bearings which use rolling parts to control friction, as opposed to plain bearings. This concept differs greatly from your previously mentioned pumps and turbines. I do believe that friction is a necessary attribute in order for those things to function, although, as I said previously, I can easily be convinced that those things might still work absent friction. If there is such a proof, that friction is not a key component for these devices, then I would be very interested in looking at it. I don't see how you can get viscosity or Reynolds numbers absent friction.

 August 18th, 2015, 03:33 AM #24 Senior Member     Joined: Jul 2015 From: Florida Posts: 154 Thanks: 3 Math Focus: non-euclidean geometry Perhaps what needs to be said about this new function is that it simultaneously involves rotations about all three axes. It is a 3D function as opposed to the normal 2D functions normally associated with trigonometry.
 September 7th, 2015, 12:51 AM #25 Senior Member   Joined: Apr 2014 From: Glasgow Posts: 2,161 Thanks: 734 Math Focus: Physics, mathematical modelling, numerical and computational solutions Just a quick message to let you know that I haven't forgotten this problem yet... I've just been very busy recently and haven't had much time to reply to posts on this forum (except for the odd quick one). Hopefully I will have some time soon to devote to it. Thanks from steveupson
 January 18th, 2016, 02:12 PM #26 Senior Member     Joined: Jul 2015 From: Florida Posts: 154 Thanks: 3 Math Focus: non-euclidean geometry @Benit13, we've some animation fragments that show the relevant variables much more clearly. The first clip shows the elevation angle E, made between two planes. The second clip shows how the intersecting plane is drawn at 45 degrees, and how the longitude plane can be rotated to coincide with the intersection point of the 45 degree small circle and the elevation plane. The final clip shows how a plane that comprises both the tangent line of the small circle at the intersection point, and the center of the sphere, rotates as the elevation angle E is changed. The angle α that is made between the longitude plane and the tangent plane is the remaining variable in the new function. I want to let you know that when we were setting up the animation rates for the movements of the various planes, I saw the function that you plotted earlier. It seems as though the rate of rotation for the longitude plane is exactly the same as what you got for the blue trace in your model, here: Post #12: New Spherical Trig Function We run the motion of the elevation plane at a rate that is represented by a sine curve, and the longitude plane is run at a rate that is the same as that represented by your equation. I'm not sure what's going on internally in the animation package that we used, but the curves look identical to me. In that same package, the animation curve for the tangent plane is a straight curve. Hmmm...
January 29th, 2016, 07:46 PM   #27
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A model of the function has been created in Mathematica by Hans Milton.

He published a CDF of the model that folks can play with:

http://community.wolfram.com//c/port...g&userId=93385

New Spherical Trig Function - Online Technical Discussion Groups—Wolfram Community
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