A unit circle (blue curve) satisfies the equation \[ x^2 + y^2 = 1\] but if instead of the Euclidean norm you use the p-norm, you have the equation \[ {|x|}^p + {|y|}^p = 1.\]

When p=1 you get what's called the "Manhattan" metric, and the "circles" are diamond-shaped (red curve); as p goes to infinity the "circles" become more and more nearly square, and are called "squircles" (green and purple curves).

Today I learned that if one generalizes trigonometry to squircles, it's called "squigonometry" and the x and y coordinates of points on the squircle are given by the squine and cosquine functions.

https://en.wikipedia.org/wiki/Squigonometry