Coordinates of the center of a circle, radius, initial and final angles of a sector are given. How to find coordinates of the endpoints of a segment (A and B) connecting borders of sector in its widest part? The segment is parallel to axis X. On image the endpoints of a segment are noted by green points. Whether there is any known formula? I will be grateful if you give the reference to the corresponding theory as I didn't find the answer in the Web.
Thanks in advance.
Compute the endpoints of the two radii delimiting your sector. Call them $P_1=(x_1,y_1)$ and $P_2=(x_2,y_2)$. Now you need to do a bunch of case distinctions. Consider the three possible $y$ values $y_0,y_1,y_2$ of the center and the two endpoints. Sort them in increasing order. Then investigate the three cases where your line is at one of these positions, and also the two cases where the line is between two of these positions. In each of these cases, you have to check whether a line at that position would intersect the sector in one or two segments, and if it intersects in one segment, which of the endpoints lies on a radius and which lies on the circle. For points on the circle, you can compute the corresponding $x$ value as a function of $y$. That's a quadratic equation, yielding two points, one on the left half of the circle and one on the right. For cases with fixed $y$ value, simply compute the length. For ranges, express the length as a function of $y$ and describe the maximum as the position where the derivative becomes zero. This will lead you to one length for each of the cases. Pick the maximum.
All these case distinctions are far from pretty, but I can see no more elegant solution. If you have any questions how to perform one of the steps I sketched above, feel free to ask. I won't write down all the case distinctions for you, though.