This problem is in my book:
Show that as $z$ traverses a small circle in the complex plane in the positive (counterclockwise) direction, the corresponding point $P$ on the sphere traverses a small circle in the negative (clockwise) direction with respect to someone standing at the center of the circle and with body outside the sphere ( the stereographic projection is orientation reversing, as a map from the sphere with orientation determined by the unit outer normal vector to the complex plane with the usual orientation.)).
I face some problems trying to solve this exercise: The first problem is I barely remember anything from analytic geometry so I tried to do thing on my own. Second problem: How to define and express rotation mathematically? It is easy in complex plane i.e $a+bi + re^{i \theta} $ detriment a rotation for a point moving in complex plane with center at $a+bi$ and radius $r$ if $\theta$ increases from $0$ to $2\pi - \varepsilon $ this rotation is positive otherwise the rotation is negative, But how to define and express the rotation in the circle on the sphere i.e n 3D? After some thought all the ideas that I came with seems to be complicated and require a lot of calculations (I think this question is simple but I just forgot analytic geometry) one idea is to make a projection of the circle on the sphere and this shape will be an ellipse, now I need to define what does it mean for a direction of rotation on ellipse mathematically and use that to express rotation but this seem to be very complected to say the least. The third problem is How do I prove this statement?.
Pretty sure I am overthinking this problem and it has a very simple and short proof without all my complicated and dumb methods.
It is easy to see that this statement is true after visualising the problem and following the points with your finger, but how to write a Rigorous mathematical proof?