Strip coordinates for a punctured disk: a centre from which finitely many interior points lie on distinct rays

Let $Z$ be a finite set of points of the open unit disk $\mathbb D\subset\mathbb R^2$ and write $\partial(s)=(\cos2\pi s,\sin2\pi s)$ (circlePoint). Then there is a centre $c\in\mathbb D$ such that, in the "strip coordinates" $\Psi(\mu,s)=c+\mu(\partial(s)-c)$, every $z\in Z$ has unique coordinates: there are $\lambda_z\in(0,1)$ and $s_z\in\mathbb R$ with $z=\Psi(\lambda_z,s_z)$, and $\Psi(\mu,s)=z$ with $\mu\ge0$ forces $\mu=\lambda_z$ and $s\in s_z+\mathbb Z$. Moreover distinct points of $Z$ have $s_z\not\equiv s_w \pmod 1$, i.e. they lie on distinct rays from $c$.

Proof. Choose $c\in\mathbb D$ off the finitely many lines through two points of $Z$ and off $Z$ (a finite union of lines is Lebesgue-null and $\mathbb D$ has positive measure). For $z\in Z$ the ray from $c$ through $z$ leaves the disk at a unique point $\partial(s_z)$, at distance $t_z>|z-c|$ from $c$ (the positive root of $|c+td|^2=1$); put $\lambda_z=|z-c|/t_z$. If $\Psi(\mu,s)=z$ then $\mu>0$ (as $z\ne c$) and $\partial(s)-c$ is a positive multiple of $z-c$, so $\partial(s)=\partial(s_z)$ and $\mu=\lambda_z$. If $s_z\equiv s_w$ then $c,z,w$ are collinear, which was excluded.