proof of the affine halfturn theorem
The affine halfturn theorem
If r is the halfturn with centre C, and t is any affine transformation,
then torot^{1} is the halfturn with centre t(C).
To reduce the algebra required, we begin with a lemma about affine
transformations and their matrices.
Lemma
If the affine transformations t and t' have matrices A and A' respectively, then
(1) t'ot has matrix A'A, and
(2) t^{1} has matrix A^{1}.
proof
(1) We have t(x) = Ax+b, and t'(x) = A'x+b'
for some vectors b and b'. Then
t'ot(x) = A'(t(x))+b' = A'(Ax+b)+b'
= A'Ax + (Ab+b'). Thus t'ot has matrix A'A.
(2) This follows from (1) since the identity map has matrix I, so t^{1} has matrix B,
where BA = I, i.e. B = A^{1}.
proof of the theorem
Since r is a halfturn, it has matrix I. Suppose that t has matrix A. By part (2)
of the Lemma, t^{1} has matrix A^{1}. By part (1) of the Lemma, applied twice,
torot^{1} has matrix A(I)a^{1} = AA^{1} = I, so is a halfturn.
Now, r fixes C, so
torot^{1}(t(C)) = tor(C) = t(C),
i.e. torot^{1} fixes t(C). Since a halfturn has a
unique fixed point, this is the centre of torot^{1}.
halfturns page 
