A particle travels along an arc (in green) from A = $(x_A, y_A)$ to B= $(x_B, y_B)$. The arc is on a circle defined by its center C = $(x_C, y_C)$ and its radius r. The vector u points from C to A and the vector v points from C to B.
The goal is to find the direction vectors at the beginning (point A) and at the end (point B) of the trajectory. It is easy to find the gradient m of the tangent line at point A from the gradient n of the radius from C to A, using the fact that the radius and the tangent line are perpendicular and hence that the product of m and n is equal to -1.
For instance, the equation of the tangent line at point A is $y = m(x - x_A) + y_A$. However, I do not need a line, but rather a vector pointing toward the direction of motion at the beginning of the trajectory. Knowing the equation of the tangent line is not enough to determine the direction of this vector.
The data is from real measurements and hence all possible scenarios are present. The angle $\alpha$ between u and v can be large, as shown in the figure, or on the opposite very small, and in addition, it can run in the clockwise (as in the figure) or in the anti-clockwise direction.
The only information I have is the equation of the circle and the coordinates of the many trajectory points that form the arc along the circle.