Name for plane curve $r=\sqrt{a^2\sin^2\theta+b^2\cos^2\theta}$

Question

I would like to know how is the following figure called (or the solid of revolution made by this figure around the horizontal axis).

$r=\sqrt{a^2\sin^2\theta+b^2\cos^2\theta}$

Plot of the figure

Answer 1

If you pass back to Cartesian coordinates using $x=r\cos\theta,r=\sin\theta$, this becomes $$r^2=a(y/r)^2+b(x/r)^2\implies r^4=(x^2+y^2)^2=ay^2+bx^2$$ which is the form of the hippopede. As noted in the linked Wikipedia article, it has a few special cases and properties; in particular, it's the pedal curve of an ellipse.