A curve also known as Gutschoven's curve which was first studied by G. van Gutschoven around 1662 (MacTutor Archive). It was also studied by Newton and, some years later, by Johann Bernoulli. It is given by the Cartesian equation

by the polar equation

and the parametric equations

The curvature and tangential angle are given by

More things to try:

Weisstein, Eric W. "Kappa Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KappaCurve.html