Ref: https://en.wikipedia.org/wiki/Cardioid
Earlier posts with similar questions: Smallest 3-ellipses that contain triangles and Curves of constant width that contain triangles
Questions: Given any triangle,
- how does one find and characterize the smallest cardioid that contains it?
- how does one find and characterize the largest cardioid contained in it?
Note: Optimal containing cardioids with "smallest" defined in terms of area and perimeter should be the same - and likewise for the embedded cardioids. The answers to above could give a couple of special centers and special directions for each triangle - for a cardioid is a curve with an orientation given by say, the line from its center to its cusp.
Note added on October 18th, 2023: We can also ask inside-out versions of these questions; for example, given a cardioid, is the triangle for which it is the smallest containing/ largest embedded cardioid unique? How to find it/them?
Note added on May 9th, 2024: The question can naturally be generalized from cardioids to limacons.
