I aim to formalize the average minimum distance between any point on a circle with a radius $r$ and an infinitely long curve.
I know the location of the curve and circle, but there is no formula for the curve as its application is geographic data, e.g. borders, rivers, etc.
I want to formalize the average distance. So far, my approach, as visualized here, is as follows:
- define $d$ as the minimum distance between point A on the circle and the curve. B is the closest point on the curve.
- define $\phi$ as a random variable uniformly distributed in $[0,2]$
My results so far:
As far as I understand, there is no way to explicitly formalize $d$, i.e., I would have to compute it.
I can however define $d$ as a function of $\phi$. The average minimum distance between any point on the circle and the curve is then given by: $\frac{1}{2\pi}\int_0^{2\pi}{d(\phi) d\phi}$.
This can be computationally approximated by $\frac{1}{N}\sum_{i=1}^{N}{d(\frac{i}{N}\phi)}$
I feel like there should be a better way to formalize the relationship, but I don't know how.