Smooth Pac-Man Curve? Idle curiosity and a basic understanding of the last example here led me to this polar curve: $$r(\theta) = \exp\left(10\frac{|2\theta|-1-||2\theta|-1|}{|2\theta|}\right)\qquad\theta\in(-\pi,\pi]$$ which Wolfram Alpha shows to look like this:

The curve is not defined at $\theta=0$, but we can augment with $r(0)=0$. If we do, then despite appearances, the curve is smooth at $\theta=0$. It is also smooth at the back where two arcs meet. However it is not differentiable at the mouth corners.
Again out of idle curiosity, can someone propose a polar equation that produces a smooth-everywhere Pac-Man on $(-\pi.\pi]$? No piece-wise definitions please, but absolute value is OK.

Not a very good one: $r(\theta) = e^{-\dfrac{1}{20 \theta^2}}$

