Le 03/10/2018 à 02:34, Kenneth a écrit :
> To take a break from my other POV-ray chores, I made a simple animated demo of
> what an object looks like when it rotates (in POV-ray), as if free-falling under
> gravity-- but discounting wind resistance or any other extra force. Its a
> comparison between applying the rotations in two axes vs. three. I made the
> animation my own purposes (to easily refer to later), but it might be of
> interest to others as well.
> 
> There's an obvious difference in the visual appearence of the objects. My own
> preference is for the 'two-rotation' scheme; it just looks more 'natural' (or
> more 'expected'?) Using all three rotation axes imparts an 'extra' force to the
> object-- kind of like wind resistance (which is also interesting, of course, but
> otherwise kind of strange.) The 3-axis scheme *might* also depend on the order
> of how those rotations are applied, to look 'more correct'-- not just straight
> <x,y,z>, in other words.
> 
> Long ago, I originally thought that a free-falling object needed all three
> rotations to look natural. The idea seemed logical-- but the visual result
> didn't bear that out (to my eyes, anyway.) I guess I could do a thorough
> analysis of the applied 'vector forces' that cause an object to rotate in the
> first place -- but that's a lot of work  ;-)  For now, I'm curious as to which
> scheme you prefer, from a purely visual standpoint.
> 
It's not the size which matters, it is how you use it.

From Euler's angles to describe a rotation, 2 angles are enough until
you want to adjust the rotation around the pole, then you need a third
angle. Any move made of 4 or more rotations can be simplified to just 3
Euler's angles, so 3 rotations is what is at most needed for any
positioning. Yet it is like saying that solving the Rubik's cube only
need at worst 17 (or 19 ?) moves, but some algorithms might generates a
hundred steps.
(and all the rotations can be composed in a single transformation, but
the equation is a bit complex)

composing rotations might not be what seems natural, especially when
they are done along fixed axes. Euler's angles have moving axis: the
first rotation is from the base(x, on z), but the second is from the
updated vector (on updated x), and so is the third (on updated z) which
in part compensate for the first one.

Natural moves usually try to conserve energy and have no acceleration.
In such setting, it is unlikely to see the axis of the last rotation be
moved by the change in the second angle: the gyroscopic effect (and our
expectation about it) would oppose to such change.

To summarize:
you need 3 angles (in Euler's system) to describes any orientation, but
natural fall would only change two angles during animation and such
change would be with a null second derivative for each unless you have
friction or engine in play.

On a table, with Inception totem:
The third angle is the proper rotation of the Inception totem, whereas
the second angle is the oscillation of the axis of the spinning top, the
speed of the oscillation being driven by the speed of the first angle.
The second angle is constant, for natural expectation of no friction.

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