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.

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