Geometric interpretation of convolution?

Question

The Internet has lots of nice visualizations for the convolution of two real-valued functions. But say I have two functions $f, g : \mathbb R \to \mathbb C$ with period $1$, and say that their convolution is another function with period $1$ from $\mathbb R$ to $\mathbb C$, $$ (f \ast g)(t) = \int_0^1 f(\tau)g(t-\tau) \, d\tau. $$ Is there any way to interpret the loop traced out by $f\ast g$ in the plane in terms of those traced by $f$ and $g$?