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There is a segment defined by start point x1, y1 and end point x2, y2. There is also a circle with origin at x3, y3 and radius R. If there is an intersection (or intersections) between segment and circle, What will be the equation to calculate Polar coordinate (basically Angle between intersection point, circle center, and horizontal plane) of intersection points in regards to that circle.

I looked for some intersections on google, and mainly all of them answer the yes/no question. I need to not only get the exact point of intersection, but also the polar coordinate of that point (which is easy part) but is there a known equation?

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Hint

let $D$ be the distance between the center and the segment.

then

$$\cos(\frac{\theta}{2})=\frac{D}{R}$$

where $R$ is the radius of yhe circle.

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  • $\begingroup$ Just to clarify, when you say distance from segment, you mean distance to the line that carries the segment? ( as in, the perpendicular from center of circle to the line?) $\endgroup$ Nov 28 '16 at 14:57
  • $\begingroup$ Also, what if my segment is exactly intersecting circle in one point, in that case D = R which means θ = 0 which does not seem right... As I am looking for the polar coordinate of that point in circles point of reference. $\endgroup$ Nov 28 '16 at 15:33

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