Knight's metric: ellipse and parabola.

Question

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.

Answer 1

huge comment/hint:

$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline 0&5&4&5&4&5&4&5&4&5&4&5&4&5&0\\ \hline 5&4&5&4&\color{orange}3&4&\color{orange}3&4&\color{orange}3&4&\color{orange}3&4&5&4&5\\ \hline 4&5&4&\color{orange}3&4&\color{orange}3&4&\color{orange}3&4&\color{orange}3&4&\color{orange}3&4&5&4\\ \hline 5&4&\color{orange}3&4&\color{orange}3&\color{blue}2&\color{orange}3&\color{blue}2&\color{orange}3&\color{blue}2&\color{orange}3&4&\color{orange}3&4&5\\ \hline 4&\color{orange}3&4&\color{orange}3&\color{blue}2&\color{orange}3&\color{blue}2&\color{orange}3&\color{blue}2&\color{orange}3&\color{blue}2&\color{orange}3&4&\color{orange}3&4\\ \hline 5&4&\color{orange}3&\color{blue}2&\color{orange}3&4&\color{red}1&\color{blue}2&\color{red}1&4&\color{orange}3&\color{blue}2&\color{orange}3&4&5\\ \hline 4&\color{orange}3&4&\color{orange}3&\color{blue}2&\color{red}1&\color{blue}2&\color{orange}3&\color{blue}2&\color{red}1&\color{blue}2&\color{orange}3&4&\color{orange}3&4\\ \hline 5&4&\color{orange}3&2&\color{orange}3&\color{blue}2&\color{orange}3&0&\color{orange}3&\color{blue}2&\color{orange}3&\color{blue}2&\color{orange}3&4&5\\ \hline 4&\color{orange}3&4&\color{orange}3&\color{blue}2&\color{red}1&\color{blue}2&\color{orange}3&\color{blue}2&\color{red}1&\color{blue}2&\color{orange}3&4&\color{orange}3&4\\ \hline 5&4&\color{orange}3&\color{blue}2&\color{orange}3&4&\color{red}1&\color{blue}2&\color{red}1&4&\color{orange}3&2&\color{orange}3&4&5\\ \hline 4&\color{orange}3&4&\color{orange}3&\color{blue}2&\color{orange}3&\color{blue}2&\color{orange}3&\color{blue}2&\color{orange}3&\color{blue}2&\color{orange}3&4&\color{orange}3&4\\ \hline 5&4&\color{orange}3&4&\color{orange}3&\color{blue}2&\color{orange}3&\color{blue}2&\color{orange}3&\color{blue}2&\color{orange}3&4&\color{orange}3&4&5\\ \hline 4&5&4&\color{orange}3&4&\color{orange}3&4&\color{orange}3&4&\color{orange}3&4&\color{orange}3&4&5&4\\ \hline 5&4&5&4&\color{orange}3&4&\color{orange}3&4&\color{orange}3&4&\color{orange}3&4&5&4&5\\ \hline 0&5&4&5&4&5&4&5&4&5&4&5&4&5&0\\ \hline \end{array}$

is the closest I could get it to work in PARI/GP right now for labeling it's too large an example for a true comment sadly. mods can size it down if they want/need.

Answer 2

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 $\ge20$).