# Compute angle from vertical at which a sphere strikes the lip of a cup

I'm working on a problem, wherein a sphere of known radius is dropped vertically and strikes the edge of a cup. I need to figure out the angle of deflection, which will be a function of where along the sphere contact occurs. I.e., if the contact point is dead-center (the pole), the angle of deflection is 0. The further from the pole that contact is made, the larger the deflection angle.

In the formulation of the entire problem (this is just part of it), I'm strictly using the projection onto the x-y plane, so the only information I have is about the projected circles. I know the location of the center of the ball-circle and cup-circle, their radii, and the size of the (downward projected) overlap.

I can compute the angle of deflection, if I only have the length of the chord from the pole to the contact point. How can I get the length of this chord, working from the x-y projection givens?

Because math.stackexchange ignores my rep. on stack overflow, I'm unable to attach an image which summarizes the problem (kind of). It may be found here at the initial posting on stack overflow.

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Thank you very much to my friend Dr. Andrew McHugh for helping me see the solution. I overlooked that I can compute the angle theta by knowing the horizontal distance ($X$) from the lip of the cup to the ball's vertical axis. $\theta$ is then the arcsine of the ratio of $X$ to the radius ($r$).
In the x-y plane projection, $X$ is the radius - the line connecting the cusps of the overlapping lenticular area ($d$). Hence, $\theta = sin^{-1}\left(\frac{r-d}{r}\right)$.