Reflection of a curve in another

Question

What is the reflection of a line about a circle? and a circle about a circle?

Answer 1

Let’s start with a circle $C$ with center $\Bbb O$, radius $r$. If you have a point $P\ne\Bbb O$, then its reflection in $C$ is the point $P'$on the ray $\Bbb OP$ (starting at $\Bbb O$ and extending all the way out beyond $P$) such that the product of the lengths $\Bbb OP$ and $\Bbb OP'$ is $r^2$. If $P$ is inside $C$, then $P'$ is outside. If $P$ is on the circle, then $P'=P$.

In the special interesting case that the circle is the unit circle in the complex plane of all $\zeta$ with $|\zeta|=1$, then the reflection of the point $z$ is $1/\,\bar z$, the reciprocal of the conjugate of $z$.