P.S.:
If the hair dome is for illustrations only, as in still photography, please have a look here
Screen capture for the adjustment options and a scene file
https://www.amazon.com/clouddrive/share/O67bUzcNbwRerJ02mfQmLosXUl5chMgffNmPaJCF9Uw
For a better viewing experience, please download the clip #471.
Perhaps that is all you need, but from there, we can harvest the splines, just a fraction of it, and go from there, which allows to use any point tool that works with Spline points.
Have a look here:
https://www.amazon.com/clouddrive/share/CZTXbdGqQFuqO7kJJou6Ts3QGCvhlKgoeW7iKGSCklg
Besides that, here is an XPresso idea:
https://www.amazon.com/clouddrive/share/4uMducU1Zf9oU70TwjI1suIZ0LSpwwil4Ko7V3xvob8
Move the midpoint!
It misses a few options of the Hair Object, but it is easy to control, and you get the “brush” you are looking for in a short time.
Edit: Sunday evening, and playing with some ideas. Here is one, it uses the minimal Python amount possible, and all there is needed to know for this process is in the screen capture. (Yes, Python can write segments and adding points, but that is not my focus here nor wanted. I hope it helps to set up any shape for a brush needed.
Scene and screen capture
https://www.amazon.com/clouddrive/share/WIn7XxCV613bOdpDOIlTJFLn0hnLdrSlKcMcVFLnFJ4
/edit.
Cheers