Hi,
I am revising for an exam and working through exercise questions for which I have been given the solutions - however there is one solution I really don't understand!

We are given S a rotation about the origin anti-clockwise by theta and T a translation by b = (b1,b2).
Want to find the centre of rotation T(S(x)) and the angle and prove our answer.

The solution says that T(S(x))=Ax+b =A(x-u)+u where
u= (1/2(b1) - 1/2(b2) cot(theta/2) , 1/2(b1) cot(theta/2) + 1/2(b2) )
Apparently in order to find u I only need to solve T(x) = x and repeatedly use the identity (1-cos(theta))/sin(theta) = tan(theta/2).

As far as I can see, T(x)=x only occurs when b=0 as otherwise the point would be translated somewhere else and T(x) would not equal x. I really have no idea how to use this or how they got u!

Any help would be very much appreciated - this is an old assignment not current assessed work and I am merely using it to help me revise.

Thanks very much,
Sooz

The question must have meant you to solve T(S(x)) = x which should be simple enough.
It T is a translation then T(x) <> x for any x.
A rotation has one fixed point (or 2 if you count infinity as a point), so if you find the unique finite point where T(S(x)) you have found the centre of the rotation T(S)). Since the centre of a rotation is a fixed point.
The composition of a translation and a rotation is always another rotation (in either order)

The combination of rotation about two different points is also usually a rotation and again has one fixed point. In the exceptional cases where the equation has no solution you would find that it was a translation instead.