How to define distance between two functions in a non-linear space (example of non-linear space: shape space)?

Question

Suppose I have two parametric circle $f_1=(acost,asint)$ and $f_2=(bcos t,bsint)$, $t\in(0,2\pi),a>0,b>0$, which lies in some non-linear space. Are there any way, how to define the distance between these two functions?
I am thinking to define a function $f=f_1-f_2$, (component wise difference), and will take norm = $(\int_0^{2\pi}|f_1(t)-f_2(t)|^2 dt)^\frac{1}{2}$, as the distance between above two function. Is there any fault? Please suggest.

Answer 1

This is a perfectly valid distance definition, between two vector functions defined on the same domain.

If you want to measure the distance between two curves (shapes), this may not be a very good choice, as it will depend on the parameterization of the curves. You can consider the Hausdorff distance instead.

NB: one could try to make the first approach parameterization-independent by using the intrinsic equation based on the arc length, but for closed curves that still makes it dependent on the starting points.

Answer 2

The $L^p$ ($1\le p<\infty$) distance: $$\left(\int_0^{2\pi}|f_1(t)-f_2(t)|^p dt\right)^\frac{1}{p}$$ The $\sup$ or $L^\infty$ distance: $$\sup_{t\in[0,2\pi]}|f_1(t)-f_2(t)|$$ The "weird" $L^p$ ($0<p<1$) distance: $$\int_0^{2\pi}|f_1(t)-f_2(t)|^p dt$$ Depending of the space (set of functions) considered, this distances will have good or bad properties. The $\sup$ distange is good if you want continuous functions.