What shape is an ellipse?

Extract

An ellipse is ellipse-shaped, of course! Every mathematician knows what an ellipse looks like – at least, this is generally true in the Cartesian plane, with the standard distance function or metric. However, there are other metrics which can be applied to the plane ℝ² and it might be useful to know what an ellipse looks like when distances are defined by these metrics. The set of points under consideration will be ℝ², the real plane, and the metrics under consideration will be the standard metric, the Manhattan (or taxi-cab) metric and the maximum modulus metric.