I'm trying to smoke out the parameters for a family of curves showing up in a particular optimization problem.
I have convinced myself that the solutions always lie on a quartic curve, which is symmetric in $x=y$ and $x+y=1$. The family includes the circle as well as the pictures below. In some cases (the last picture) the curve appears to be two hyperbolas meeting at (0,0) and (1,1).
I have tried all the variants on Wikipedias entry for the Quartic Curve, but none of them appears to match the above criteria.
Do the pictures below remind you of any particular families, sections or otherwise?