How to get angle two curved line, which is pinned at a point?

Suppose there is a flexible line is moving and then it meets with some points and pinned to that points and then bow-out with some radius say R ,, now I want to calculate the angle formed.I have attached a figure.Please have a look.

Answer 1

[Image Analyst]

Do you have all the points coordinates along the curve? Can't you just find the vertex and then use polyfit() to get the line to the right and left and then get the angle? Here's a start

% Fit left line between x indexes k1 and k2.
coeffsLeft = polyfit(x(k1:k2), y(k1:k2), 1);
% Fit right line between x indexes k3 and k4.
coeffsRight = polyfit(x(k3:k4), y(k3:k4), 1);

yLeft = polyval(coeffsLeft, x(k1));
yRight = polyval(coeffsRight, x(k4));
% Then calculate angle

Answer 2

[Roger Stafford]

If d is the distance between the two points on the left of your picture, then

 phi = 2*acos(d/2/r);

for the phi that you show.