I am trying to find the distance function of a point and a logarithmic spiral.
All I could find about that is this link, but it is based on the assumption that the closest point on the spiral must be on a line going through the point and the origin, which is from what I've seen not true.
Where should I start to derive this ?
If the spiral makes angle $\alpha$ to radial line,(draw the differential triangle representing infinitely small right triangle)
$$ dr = \cot \alpha . r. d\theta $$
Integrate to get logs with initial condition $ \theta=0, r= r_i $
$$ \log \frac{r}{r_i} = \cot \alpha . \theta \, \rightarrow r= r_i.e^{\cot \alpha . \theta} $$
which is log spiral with radial distances growing exponentially with $\theta.$