How can we change a non vertical line to an hyperbola while constructing a Voronoi diagram?

Question

I've been reading this article about the Voronoi diagram that uses mapping to transform a line to a hyperbola and somehow couldn't understand how the mapping works since I tried to prove it.

Here's the screenshot

Answer 1

I'll suppose in what follows that $d_P(z)=\sqrt{z_x^2+z_y^2}$.

Take a generic non-vertical line $l$ with equation $y=mx+q$. A point $P$ on $l$ has coordinates $P=(x,mx+q)$ and is mapped to $$ P'=\big(x,mx+q+\sqrt{x^2+(mx+q)^2}\big). $$ Therefore the coordinates of $P'$ obey the equation $y=mx+q+\sqrt{x^2+(mx+q)^2}$, that is: $$ y-(mx+q)=\sqrt{x^2+(mx+q)^2}. $$ Squaring and simplifying gives $$ x^2-y^2+2mxy+2qy=0, $$ which is indeed the equation of a hyperbola.