I’ve written several times about the “squircle,” a sort of compromise between a square and a circle.
It looks something like a square with rounded corners, but it’s not.
Instead of having flat sizes (zero curvature) and circular corners (constant positive curvature), the curvature varies continuously.
A natural question is just what kind of circle approximates the corners.
This post answers that question, finding the radius of curvature of the osculating circle.
The squircle has a parameter p which determines how close the curve is to a circle or a square.
The case p = 2 corresponds to a circle, and in the limit as p goes to infinity you get a square.
We’ll work in the first quadrant so we can ignore absolute values.
The curvature at each point is complicated [1] but simplifies in the corner to and the radius of curvature is the reciprocal of this.
So for moderately large p, the radius of curvature is approximately √2/(p-1).
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