Rotation

[TheBigBadWolf]

Hi

On your page about rotation in the 4th dimension you assume that the axis of rotation in a plane. I don't think this is the case and may ask where this assumtion comes from?

thanks
wolf

[pat]

On your page about rotation in the 4th dimension you assume that the axis of rotation in a plane. I don't think this is the case and may ask where this assumtion comes from?

Throughout, I used 'simple' to mean a rotation that can occur in a body upon which no forces are acting. A rock, floating in a vacuous n-space, can only be rotating 'simply'.

When most people think of rotations, they think of something turning about an axis. This works just fine in 3-space. It works okay in 2-space as long as we can envision the axis running perpendicular to 2-space.

In 4-space, we start to run into trouble. In 4-space, it becomes clear that we should have been thinking about rotations as happening about a point in a plane from the very start.

In 2-space, a (non-trivial) rotation takes one vector that starts at the origin into a different vector that starts at the origin. The rotation leaves a point fixed.

In 3-space, a (simple, non-trivial) rotation takes one vector that starts at the origin into a different vector that starts at the origin. The rotation leaves a line fixed.

In 4-space, a basic (simple, non-trivial) rotation takes one vector that starts at the origin into a different vector that starts at the origin. The rotation leaves a plane fixed. But, it is possible to be rotating that fixed plane at the same time, too, by taking one vector in that plane which starts at the origin into a different vector in that plane which starts at the origin. Doing this leaves only a point fixed.

So, yes, rotation in 4-space is somewhat more complicated than just rotation in a plane. But, we can break it down into rotation in two planes.

It may be that the assumption was that object was set into motion by a single impulse of force. If that's the case, then the rotation has no effect on any vectors which start at the center-of-mass and are perpendicular to both the vector which starts at the center-of-mass and extends to the point of contact of the impulse and the vector which starts at the origin and is parallel to the direction of the force. In that way, it's easy to think of the rotation as happening in (really 'parallel to') the plane containing the center of mass, the point of contact, and the force vector.

[TheBigBadWolf]

In 2-space, a (non-trivial) rotation takes one vector that starts at the origin into a different vector that starts at the origin. The rotation leaves a point fixed.

In 3-space, a (simple, non-trivial) rotation takes one vector that starts at the origin into a different vector that starts at the origin. The rotation leaves a line fixed.

In 4-space, a basic (simple, non-trivial) rotation takes one vector that starts at the origin into a different vector that starts at the origin. The rotation leaves a plane fixed. But, it is possible to be rotating that fixed plane at the same time, too, by taking one vector in that plane which starts at the origin into a different vector in that plane which starts at the origin. Doing this leaves only a point fixed.

First I want to mention that I'm not a native speaker therefore maybe some expressions are not mathematically correct in english.

The problem is the step from 3d to 4d. Your assumption is that the dimension of the geometry that is fixed goes linar. That means : 2d - point is fixed, 3d - axis is fixed, therefore in 4d a plane is fixed. And I believe thats not the case.

A rotation is U*T(U) = 1 (T(U) means the transposed of U). The dimension of U is n.

Lets see what the number of degrees of freedom does. This is the number of elements in the upper right corner over the diagonal and is [d*(d-1)]/2

In two dimension the 2x2 matrix has 1 degree of freedom. This one degree of freedom in terms of rotation is the angle.

Code: Select all

            cos    sin
eg. U =
           -sin    cos

In three dimensions the number of degrees of freedom is 3. Two are used by defining a axis and one by defining a angle.

In four dimension the number of degrees of freedom is 6. So you can not use 6 degrees of freedom by defining a plane and a angle.

Code: Select all

     angle   axis      ?     degrees of freedom
_________________________________________
2d : 1                         = 1
3d : 1       + 2               = 3
4d : 1       + 2     + 3       = 6

Conclusion : number of degrees of freedom goes with d*(d-1)/2 not linear.

What do you think?

thanks
Wolf

[pat]

The problem is the step from 3d to 4d. Your assumption is that the dimension of the geometry that is fixed goes linar. That means : 2d - point is fixed, 3d - axis is fixed, therefore in 4d a plane is fixed. And I believe thats not the case.

I wasn't assuming this. In 2-D, the most that can be fixed is a point. In 3-D, the most that can be fixed is a line. In 4-D, the most that can be fixed is a plane. 4-D is the first time were the 'most that can be fixed' is different from 'the least that can be fixed'.

Lets see what the number of degrees of freedom does. This is the number of elements in the upper right corner over the diagonal and is [d*(d-1)]/2

This is the same as 'd Choose 2'. That's no coincidence. Each degree of freedom is one of the d orthogonal axises moving toward another. By virtue of the fact that it's choosing 2 axises, each degree of freedom represents rotation in the plane containing those two axises.

Conclusion : number of degrees of freedom goes with d*(d-1)/2 not linear.

What do you think?

Agreed....

[TheBigBadWolf]

By virtue of the fact that it's choosing 2 axises, each degree of freedom represents rotation in the plane containing those two axises.

Is it possible to write this down as a matrix?

wolf

[pat]

By virtue of the fact that it's choosing 2 axises, each degree of freedom represents rotation in the plane containing those two axises.

Is it possible to write this down as a matrix?

Sure is... I'm sure you can work it out by simply multiplying together a bunch of matrixes like:
<pre>[ cos a -sin a 0 0 ] [ cos b 0 -sin b 0 ]
[ sin a cos a 0 0 ] [ 0 1 0 0 ] ...
[ 0 0 1 0 ] [ sin b 0 cos b 0 ]
[ 0 0 0 1 ] [ 0 0 0 1 ]</pre>
covering all pairs of axises.

Another possibility is that every 4-d rotation can be written as a 'q v r' where q and r are unit quaternions[*] and v is a 'pure-imaginary' quaternion representing the point to be rotated. Pre-multiplication by q and post-multiplication by r can both be written in matrix form.

More later,
Patrick

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[*] and, every such multiplication with two fixed, unit quaternions is a 4-d rotation. This is another validation of the fact that there are 6 degrees of freedom since there are three degrees of freedom in choosing each unit quaternion.

[TheBigBadWolf]

thank you very much
I will study these things

wolf