Euler's Rotation Theorem Proof
in [Math]
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From: Tom on 20 May 2010 09:50 May I ask if someone can point me to a GEOMETRIC proof of Euler's Rotation Theorem? I do NOT mean the one using algebra - I really mean the one like in wiki. The problem with the wiki one is that, well, I do not understand it. There is a missing symbol "O" and I do not "see" the two angle symmetry planes. Every proof I find on the net that is NOT an alebraic one - I understand the proof involving rotation groups - is nearly a verbatim proof of the wiki one. So that gets me nowhere. (an animated proof of this one would be best... but AT LEAST a proof that explains it a bit better.) Thanks t
From: Henry on 20 May 2010 11:00 On 20 May, 14:50, "Tom" <j...(a)junk.com> wrote: > May I ask if someone can point me to a GEOMETRIC proof of Euler's Rotation > Theorem? > > I do NOT mean the one using algebra - I really mean the one like in wiki. > > The problem with the wiki one is that, well, I do not understand it. There > is a missing > symbol "O" and I do not "see" the two angle symmetry planes. > > Every proof I find on the net that is NOT an alebraic one - I understand the > proof involving rotation groups - is nearly a verbatim proof of the wiki > one. So > that gets me nowhere. > > (an animated proof of this one would be best... but AT LEAST a proof that > explains it a bit better.) > > Thanks > t The Wikipedia proof http://en.wikipedia.org/wiki/Euler's_rotation_theorem looks simple enough once you understand point \alpha goes to A in the rigid motion, while point A goes to a. The point O is on the surface of the sphere, and in the diagram would appear to be slightly above and to the left of A. If it helps visualise the (non-)movement of O, you might treat the motion as rotating the blue circle around the blue axis z until \alpha falls on A and then rotating the blue circle onto the red circle around their joint diameter N, and then O (and every point on OC) will also move anticlockwise around z and then be rotated back around N to its original position. The proof comes from the equality of angles, and that comes from the definition of O.
From: Tom on 20 May 2010 20:28 Henry, I AM SO SORRY. I am having an extreme mental block. I can almost understand: "symmetry plane of the angle αAa" But I have no idea what is meant by: "the symmetry plane of the arc Aa" I just cannot see. "Henry" <se16(a)btinternet.com> wrote in message news:88d96c47-0fdb-4259-94ba-4aebfeb41e2f(a)a20g2000vbc.googlegroups.com... On 20 May, 14:50, "Tom" <j...(a)junk.com> wrote: > May I ask if someone can point me to a GEOMETRIC proof of Euler's Rotation > Theorem? > > I do NOT mean the one using algebra - I really mean the one like in wiki. > > The problem with the wiki one is that, well, I do not understand it. There > is a missing > symbol "O" and I do not "see" the two angle symmetry planes. > > Every proof I find on the net that is NOT an alebraic one - I understand > the > proof involving rotation groups - is nearly a verbatim proof of the wiki > one. So > that gets me nowhere. > > (an animated proof of this one would be best... but AT LEAST a proof that > explains it a bit better.) > > Thanks > t The Wikipedia proof http://en.wikipedia.org/wiki/Euler's_rotation_theorem looks simple enough once you understand point \alpha goes to A in the rigid motion, while point A goes to a. The point O is on the surface of the sphere, and in the diagram would appear to be slightly above and to the left of A. If it helps visualise the (non-)movement of O, you might treat the motion as rotating the blue circle around the blue axis z until \alpha falls on A and then rotating the blue circle onto the red circle around their joint diameter N, and then O (and every point on OC) will also move anticlockwise around z and then be rotated back around N to its original position. The proof comes from the equality of angles, and that comes from the definition of O.
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