You have visibly searched a lot, so I fear I will not bring any new information.
However, this post seems interesting
Hope it will at least hint in a good direction.
Here is how to compute directly what you need.
Has Apple documented exactly the algorithm they use for doing this?
It is not Apple, it is Mr Bezier, an automotive engineer in the 80's.
There is a simple parametric form of a Bezier curve with n control points (Pi).
if t is a parameter between 0 and 1 (from start to end),
The formula can be expressed explicitly as follows (cannot print the forumla with nice typo on the forum, so I did my best to edit):
B(t) = Sigma [ C(n,i) (1-t)^(n-i) * (t ^ i) ] Pi
where C(n,i) are the binomial coefficients and Pi the control points ((x,y) coordinates).
For example, for n = 5:
B(t) = (1-t)^5 P0 + 5 t (1-t)^4 P1 + 10 t^2 (1-t)^3 P2 + 10 t^3 (1-t)^2 P3 + 5 t^4 (1-t) P4 + t^5 P5 0 ≤ t ≤1
With this formula, you can compute the exact coordinates of any point on the curve and find also the orientation of the tangent (so that letter "sits" on the curve by tilting with that tangent angle).
If P is (x,y), coordinates of the tangent vector at t are given by derivative (I split in 3 lines to make it more readable):
- C(n,0) n (1-t)^(n-i-1) P0
Sigma C(n,i) [ - (n-i) (1-t)^(n- i -1) * (t ^ i) + (1-t)^(n-i) * i (t ^ (i-1) ) ] Pi
C(n,n) t^(n-1) Pn
For t = 0
-C(n,0) n P0 + C(n,1) P1 = n (P1- P0): the direction is from P0 to P1 as well known.
Note: How many control points have you for your Bezier Path ?
There is sample code from Apple called textLayoutDemo that shows how to layout text in a circle. I'm not sure if this will help but it might be worth exploring. It dates back to about 2007 (I think).
A circle can be handled by direct computation. But it is not a Bezier curve.
Have you a link to the Apple's reference ?
Did you try the proposed solution posted May 23, 2020 7:10 AM ?
Does it work as expected ?