# Reflect a complex number about an arbitrary axis

This should be really obvious, but I can't quite get my head round it:

I have a complex number $z$. I want to reflect it about an axis specified by an angle $\theta$.

I thought, this should simply be, rotate $z$ by $\theta$, flip it (conj), then rotate by $-\theta$.

But this just gives $z^* (e^{-i\theta})^* e^{i\theta}$...

but this can't be right - as it's just $z^*$ rotated by angle $2\theta$, surely?

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Well, you need to make up your mind as to whether you want to reflect your rotated entity about either the real or imaginary axis... – J. M. Feb 8 '12 at 16:50
On that note: you're familiar with the matrix route for doing this operation to points? That is easily translatable to complex notation... – J. M. Feb 8 '12 at 16:51
If I am not mistaken, wouldn't a+ib reflected about x axis be a - ib and about y axis be -a + ib ? – Inquest Feb 8 '12 at 16:53

Indeed, it's just $z^*$ rotated by $2\theta$... And it's almost the right answer! You know that a symmetry composed with a rotation is still a symmetry, and you know that an (orthogonal) symmetry is characterized by its fixed points. So you want to get a symmetry that fixes the axis spanned by $e^{i\theta}$ (as an $\mathbb{R}$ vector space); an easy way to do that is, as you've noticed, to rotate by $-\theta$ (and not $\theta$ actually), flip over the real line (conjugation) and then rotate by $\theta$. So you get $(ze^{-i\theta})^* e^{i\theta} = z^* e^{2i\theta}$. It is easily checked that this fixes $e^{i\theta}$, and it's an orthogonal symmetry, therefore it's the one you're looking for.