Say a set of points is convex position if no point is contained in the convex hull of the others. I am trying to grasp which Mobius Transforms will modify this property.
For simplicity I considered transforms of the form: $f(z) = \frac{1}{z-d}$
So I wrote a program that would take a set of points that were not convex and marked in green the values of $d$ which changed this:
And I wrote a program that would take a set of points that were convex in and mark in green the values of $d$ that changed this.
It seems like the values of $d$ which are contained in an odd number of cicumcircles changes the orientation. Is there an easy way of analytically verifying this? Is there a more general phenomena of which this is an instance?