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I have a circle and a point in 2D. The point lies outside of the circle.

Given the distance between the point and circle center and the radius of the circle, what is the angle of the circle that the point can 'see'?

Image describing the situation

When the point is infinitely far away, it will see 180 degrees or 1 pi radians.

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  • $\begingroup$ Assuming you accept the possibility of it being "infinitely far away", yes, your two blue lines would be parallel and the visible angle would be $\pi$ radians. This is probably better formalized as a limit of sorts. $\endgroup$
    – JMoravitz
    Sep 27 '19 at 13:32
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    $\begingroup$ If you draw the line between red point and center, you have a right triangle with a known hypotenuse and a known leg. $\endgroup$
    – Michael
    Sep 27 '19 at 13:33
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We have that indicating with:

  • radius $R$
  • distance form the centre $ d$

$$\alpha = 2\arccos \left(\frac R d\right)$$

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  • $\begingroup$ More like 2*asin(R/d) $\endgroup$ Oct 27 '19 at 11:51
  • $\begingroup$ @DesmondHume I think it is correct in this way. Suppose $d \to \infty$ then we have $\alpha=2 \arccos (0)=\pi$. $\endgroup$
    – user
    Oct 27 '19 at 11:55

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