N particles are at the corner of a N sided polygon and move towards each other with velocity v. Determine nature of path followed by them?

Question

I can't derrive it mathematically but I have an argument :
Suppose N= infinity.

i. e. Circle.

Then path followed by them is circle.

On decreasing N you will find that they follow some sort of trajectory.
Curvature increases obviously and seems to be spiral.

As it is originated from circle how can it be logarithmic spiral(as my friend answered).