Knight's metric: ellipse and parabola. Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this new metric?
I apologize if the question is too vague.

Using Noam D. Elkies's characterization of the knight's distance, here's an animation of $d(x,y)+d(x-a,y)$ as $a$ goes from $0$ to $30$. All cells of the same colour are on the same "ellipse" (except the darkest red ones, which have distance $\ge20$).

