I was thinking something like finding the slope using differentiation. Like if you wanted to get the shape of a skateboarding pool, you would just find the slope of a upside-down semicircle. So...
x^2 + y^2 = r^2 (Formula of circle)
y = -SQRT(r^2 - x^2) (Solve for y. Take the negative root to get the bottom portion.)
dy/dx = (-1/2)*(r^2 - x^2)^(-1/2)*(-2x) (Differentiate and use chain rule)
dy/dx = -x*(r^2 - x^2)^(-1/2)
Then you just plug in your values for r and x. To get the degrees, you can just go:
arctan(-x*(r^2 - x^2)^(-1/2)) = degrees
Then since Alice is measured in revolutions:
degrees / 360 = revolutions
Now, implementing this in Alice is going to be a lot harder. Btw, let me know if my math was wrong!