Geometry of SO(3): Reflection, Composition & Rotation

[JSG31883]

1) Let P,Q be planes through the origin in R3. Let Rp, Rq be the corresponding reflections. Is Rp*Rq (where * denotes "composition") in SO(3) or O(3)/SO(3)? What is the axis of rotation of Rp*Rq?


2) For a fixed A in SO(3) show that there are infinitely many pairs of planes P,Q such that A=Rp*Rq.


3) For arbitrary A,B in SO(3) find the axis and angle of rotation of AB.

 

[AKG]

1. What does it mean for a matrix to be in SO(3)? How about O(3)/SO(3)? What can you say about the matrices Rp and Rq? What happens to any point on the axis of rotation of Rp*Rq under the action of Rp*Rq itself? What happens to any point in P under the action of Rp?

2. Do you know that in R², if you take a reflection about a line whose angle with the origin is x, and a line whose angle is y, then composing the reflections gives a rotation through an angle of x-y? Note that x-y = (x+a)-(y+a).

3. What have you tried?

 

[JSG31883]

1) SO(3) are all orthogonal matrices with det 1, and represents the rotation group in R3.

I know what the rotation matrices in R3 look like, but don't know what the reflection ones do, so I can't say what happens to a point on the axis of reflection.

2) I do know that the composition of reflections in R2 is the sum of the acute angles, but I can't figure it out for 3 dimensions and for planes (for R2 I did it with lines). So, how can I compose two reflections of planes in R3?

3) I know that the axis of rotation of AB is:
+- (cos(a/2)sin(b/2)Ub)+- (sin(a/2)cos(b/2)Ua)+- (sin(a/2)sin(b/2)sin(c)Uab) where Ua, Ub, Uab are the unit vectors.

And I know that the angle of rotation is:
2sin^-1(cos^2(a/2)sin^2(b/2)+sin^2(a/2)cos^2(b/2)+sin^2(a/2)sin^2(b/2)sin^2(c))^1/2

So I know the answer, but can't figure out how to arrive at it... any thoughts?

 

[AKG]

1. Do you know that det(AB) = det(A)det(B)? A reflection in the plane P fixes all points in P. Rotation about a line L fixes all points in L.

2. I don't know what you mean by "the composition of reflection in R2 is the sum of the acute angles," but did you note what I said?

Do you know that in R², if you take a reflection about a line whose angle with the origin is x, and a line whose angle is y, then composing the reflections gives a rotation through an angle of x-y? (EDITED)

If anything, it would be the difference of the acute angles. Anyways, that means that a reflection about a line with angle x, followed by a reflection about a line with angle y produces the same rotation as a reflection about a line with angle x+a, followed by a reflection about a line with angle y+a, and there are an infinite number of a's to choose from. So, if you take two lines, and rotate them together, then you still end up with the same rotation after composing the new reflections. Can you think of what you can do with two planes?

3. Given a rotation with the angle and axis specified, can you find it's matrix? Conversely, given a matrix in SO(3), can you find the angle and axis?

 

[JSG31883]

3) No I don't know how to find the associated matrix... can you help?

 

[AKG]

A rotation about the z axis through an angle x is represented by the matrix:

(cosx  -sinx  0)
(sinx   cosx  0)
(0      0     1)

So the matrix through an angle x about an another axis will be a similar matrix to the one above, all you need is a change of basis matrix that changes the z axis into your axis of rotation. Make sure to pick an orthonormal right-handed basis. You may not need such a basis, perhaps any basis with the third vector being on the rotation axis will do, but I'm not sure since this is just coming off the top of my head, so it's probably better to play it safe. Actually, I'm pretty sure you'd want an orthogonal basis otherwise a rotation about one vector won't guarantee that your rotation is in the plane spanned by the other two vectors, whereas when you have an orthogonal basis like your standard basis, a rotation about the z axis is the same as a rotation in the plane spanned by the other two vectors, x and y, and this matrix representation (I believe) only works when it's set up like this.