Deforming a circle into closed smooth curve

Question

If we throw a rubber band on to a smooth table it wiggles and takes variegated shapes before coming to rest. However, the deformed band while taking different shapes remains closed and smooth. I am curious to know if there is mathematically some way, say through some complex transformation, to achieve it. In other words, how can we change the boundary of circle into an arbitrary closed smooth curve.
My vague idea Take a variable point $(x,y) $ on the circle and define two functions $u(x,y)$ and $v(x,y)$ so that the point $(u,v)$ traces a requisite curve as $(x,y)$ moves on the circle. I tried a few functions but ended up obtaining some nondifferentiable points on the transformed curve. I will be highly obliged for any responses/hints/suggestions.