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OfflineThe Mindful Mage
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Visualization / Thought Experiment: Path Around A Sphere
    #26470453 - 02/05/20 02:56 PM (4 years, 11 days ago)

During a meditation today, I was imagining the path of a point on the surface of a sphere where the sphere is rotating along the x and y axis simultaneously, at an equal rate. To begin, imagine looking top-down, with the point in the center.

I don't know the math to express this situation, nor is that my main interest (although I am certainly somewhat interested). I would like to know what the shape of this path is.

In my visualization, it appeared to create, from one angle, a lemniscate. Can anyone tell me if this is correct, and explain this situation from their own perspective?


Edited by The Mindful Mage (02/06/20 04:26 AM)


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OfflineVP123
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Re: Sphere Geometry Question [Re: The Mindful Mage]
    #26470543 - 02/05/20 03:48 PM (4 years, 11 days ago)

If it rotates around x-axis and y-axis simultaneously, wouldn't that be equivalent to rotation around an axis that makes 45 degrees with x and y in the xy plane? I don't see it making a lemniscate. Depending on the point on the sphere and the observation point, it could describe an ellipse, circle or straight line.


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OfflineThe Mindful Mage
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Re: Sphere Geometry Question [Re: VP123]
    #26470610 - 02/05/20 04:21 PM (4 years, 11 days ago)

I wonder...

Let's see if anyone else has a perspective on this.


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Offlinechibiabos
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Re: Sphere Geometry Question [Re: The Mindful Mage]
    #26470773 - 02/05/20 05:32 PM (4 years, 11 days ago)

edit:  Misread your question.  It depends on how the axis of rotation is precessing.


Edited by chibiabos (02/05/20 10:50 PM)


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OfflineThe Mindful Mage
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Re: Sphere Geometry Question [Re: chibiabos]
    #26471439 - 02/06/20 04:24 AM (4 years, 10 days ago)

This is challenging to explain without a visual representation, but I also find it enjoyable to try. I will do my best to clarify this situation. Please keep in mind that I am not very studied in mathematics, so I may be misusing some terms. I will use mostly non-mathematical terms, though some words may have a mathematical use I am not aware of.

Quote:

chibiabos said:
edit:  Misread your question.  It depends on how the axis of rotation is precessing.




If I understand precession correctly, I will say that there is no precession here. This sphere is purely imaginary and is not limited to material natural laws.

///

/////////  KEY  /////////
///  Sphere, white.  ///
///  Circles, red.      ///
///  Point, blue.      ///
///  Path, green.      ///
///      - - -            ///
///  C1, vertical.      ///
///  C2, horizontal.  ///
///  C3, around.      ///
/////////////////////////

>> Read all steps and then move through the exercise.

>> Imagine that the surface of this white sphere is entirely malleable, like a fine mist, and the points along the sphere's surface can move freely in any direction.

>> Imagine three red circles (C1, C2, and C3) crossing perpendicularly around the sphere, such that the image appears as a red cross through a red circle with a white fill (+).

>> Imagine a blue point at the middle of the cross, on the sphere's surface.

>> Imagine the surface of this sphere rotating simultaneously 'upward' along C1, and clockwise along the C3, at the same rate.

>> After 90º of rotation along both of these circles, where is the blue point? What is the shape of the green path?

>> After 180º?

>> After 360º of rotation, with the blue point returning to the original position, what is the shape of the green path from this perspective (+)?

>> Keep in mind, the red circles do not move. They are only paths of rotation. The cross remains throughout the rotation of the sphere surface.

If further explanation / clarification is needed, I will do my best to provide it.

///

When I run this exercise in my mind, after 90º of rotation along C1 and C3, the blue point is where it would be if the rotation was along C2, 90º to the right. The green path of this point is curved.


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Offlinechibiabos
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Re: Sphere Geometry Question [Re: The Mindful Mage]
    #26471958 - 02/06/20 12:26 PM (4 years, 10 days ago)

Quote:

The Mindful Mage said:
This is challenging to explain without a visual representation, but I also find it enjoyable to try. I will do my best to clarify this situation. Please keep in mind that I am not very studied in mathematics, so I may be misusing some terms. I will use mostly non-mathematical terms, though some words may have a mathematical use I am not aware of.

Quote:

chibiabos said:
edit:  Misread your question.  It depends on how the axis of rotation is precessing.




If I understand precession correctly, I will say that there is no precession here. This sphere is purely imaginary and is not limited to material natural laws.





Right, it's subject to the mathematical properties of a sphere.  You can't have a sphere in a three dimensional space with more than one axis of rotation.  You can decompose the rotation into components which are basically rotations about orthogonal axes, but the sphere will always have one unique axis of rotation.


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InvisiblebodhisattaMDiscordReddit
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Re: Sphere Geometry Question [Re: chibiabos]
    #26472415 - 02/06/20 05:06 PM (4 years, 10 days ago)



Euler has the math on finding the new rotational axis when you're "spinning on two or three"


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Offlinechibiabos
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Re: Sphere Geometry Question [Re: bodhisatta]
    #26474067 - 02/07/20 02:42 PM (4 years, 9 days ago)

Secondary circle instead of a great circle, but I think that this is more or less what you're looking for.  I'm too lazy to do the math right now, but I'm pretty sure that the curve for a great circle is just a continuous deformation of what you'd see here (being that a secondary circle of a sphere is a continuous deformation of a great circle).



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OfflineThe Mindful Mage
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Re: Sphere Geometry Question [Re: chibiabos]
    #26475314 - 02/08/20 11:12 AM (4 years, 8 days ago)

Thanks for the input, folks.


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