Triangular analog of the squircle

TimF left a comment on my guitar pick post saying the image was a “squircle-ish analog for an isosceles triangle.” That made me wonder what a more direct analog of the squircle might be for a triangle. A squircle is not exactly a square with rounded corners. The sides are continuously curved, but curved most at the corners. See, for example, this post. Suppose the sides of our triangle are given by L1(x, y) = 1 for i = 1, 2, 3. For example, We design a function f(x, y) as a soft penalty for points not being on one of the sides and look at the set of points f(x, y) = 1. You might recognize this as a Lebesgue norm, analogous to the one used to define a squircle. The larger p is, the heavier the penalty for being far from a side and the closer the level set f(x, y) = 1 comes to being a triangle. The post Triangular analog of the squircle first appeared on John D. Cook.