Show in R3: Product of 2 Reflexions is a Rotation?

Hi,

I'm looking for a way to show in 3d space, that the product of two reflexions is a rotation, but haven't come very far.

I tried starting like this:

Say n1, n2 are (normalized) vectors in Eucledian 3-space, c is the cross product n1 x n2 and theta is the angle between n1 and n2:

c = cross(n1, n2)
cos(theta) = dot(n1, n2)


Now, I can build two reflexion matrices S1, S2 for the reflexion in the planes that have the normals n1 and n2 and go through the origin.

S1 = I - 2 * n1 * n1^T
S2 = I - 2 * n2 * n2^T

Let S be the product of those two:

S = S2 * S1


Furtheron, I can make a rotation matrix for the rotation around an axis through the origin with direction c about the angle 2 * theta:

R = Rotation around c about 2 theta


For n1 != n2 ==> R == S (otherwise would be nonsense ;).
But, how can I show this, that in fact R == S?


bye,
Stefan.
.

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