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.

• The question is indeed vague. An ellipse with foci $a$ and $b$ could be defined as the set of of points $x$ such that $d(a,x) + d(x,b) = cst$, but how do you define a parabola? Mar 25, 2014 at 14:31
• You're right, I don't know how to define a parabola. Mar 25, 2014 at 14:33
• Find a reasonable notion of line, and consider squares equidistant from a given focus and the line. Mar 25, 2014 at 14:36
• The ellipse will be a polygon, and its shape depends on the angle between two points. This is because the knight metric is a taxicab-like metric. See my previous answer, which shows that spheres are octagons mathoverflow.net/questions/76670/… Mar 25, 2014 at 15:26
• I guess a line could be defined as the locus of points for which one has equality in the triangle inequality with respect to two given points. Then, Masked Avenger's comment gives the parabola. I would be interested to see this parabola for line $\langle (-1,2) \rangle$ and focus $(2,1)$. Mar 25, 2014 at 22:06