Does a plane curve with polar equation $r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$ have a name? Does a plane curve with polar equation $$r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$$ where both $\lambda_i>0$ have a name?
It's very similar to hippopede, also known as lemniscate of Booth, but not quite the same: hippopede equation can be written in the same way but with $r^2$ instead of $r$.
Here is how it looks like for $\lambda_2=1$ and $\lambda_1 \in \{\frac{1}{2}, 1, 2, 3, 4, 5\}$:

Update: I realized that in my application it is more natural to put $r^2$ instead of $r$ in the equation above; hence what I appear to be dealing with is a hippopede after all. The family of curves displayed above is likely not to have any name (hard as I tried, I failed to find any).
 A: Do not know its name, may be called Euler normal curvature relation,  a part of the Mohr's circle in differential geometry context:

Mentioning this due to its possible interest to STEM students. If curvature of normal plane while rotating  through angle $\psi$  at a point on a surface in $\mathbb R^3$ is plotted in the tangent plane of rotating plane  in bundle then it is a representation of Euler's relation
$$ \kappa_n= \kappa_1 \cos^2\psi +\kappa_2  \sin^2 \psi$$
where $(\kappa_1, \kappa_2 )$ are principal curvatures.
Read with Enneper geodesic torsion of all rotated lines of the bundle
$$ \tau_g= (\kappa_1-\kappa_2 ) \cos \psi \sin \psi$$
Please note $\lambda_1 $ can be also negative, for asymptotic line locus on negative Gauss curvature doubly curved surfaces.
The relation forms a part of parametrization of the Mohr's circle curvature tensor.
Accordingly the curvature can be replaced by stress, strain, moment of inertia among other tensors. They are often plotted on $(x,y)$ axes respectively.
Mohr's Circle of tensors Figs 6,8 
