From a point outside a circle, I know the distance to two (or more if needed) points along the circle, as well as the distance to the center of the circle. Angles are unknown. How can I calculate the radius (or distance between the two points)?

enter image description here

From the image, a, b, and x are known. I also have measurements to more points along the circle if needed, say c and d.

  • $\begingroup$ Along the distances $a$ and $b$, do you have any angle? Or distance between the two points on the circle? Without further information, it's impossible to know the radius: the two points may be anywhere on the circle, that only provides a lower bound and an upper bound for the radius. $\endgroup$ – StayHomeSaveLives Jan 3 '18 at 11:06
  • $\begingroup$ @Jean-ClaudeArbaut no, unfortunately not. I had feared it was not possible to calculate without more information. Dang! $\endgroup$ – Daniel A. Burke Jan 3 '18 at 11:08
  • $\begingroup$ There's not enough information, no matter how many distances you provide. Consider: If there are points with those distances on a particular circle, then there are just as many points with those distances on a slightly-larger circle, or even on a line between your point and the "center". $\endgroup$ – Blue Jan 3 '18 at 11:09
  • 1
    $\begingroup$ Out of curiosity, where does the question come from? $\endgroup$ – StayHomeSaveLives Jan 3 '18 at 11:10
  • $\begingroup$ @Jean-ClaudeArbaut a video game, actually. The context is it's a space game with large celestial bodies with POIs scattered across them. I'm writing a "routing" program of sorts, to determine shortest possible routes, etc. To do so, I basically need to form a Cartesian coordinate system identifying the POIs' locations along the surface of a sphere (planet/moon). We aren't given coordinates, so the only I thing I have to work with are reference distances. Fortunately, I was able to retrieve the radius, but now I face the problem of calculating all the coordinates from reference distances. $\endgroup$ – Daniel A. Burke Jan 3 '18 at 14:39

If you have measurements to more points like point $Y$ then the radius is: $R=Y-X$ enter image description here

  • $\begingroup$ I do have measurements to more points, but I cannot be certain that any of those points lie on a line intersecting the center of the circle and my external reference point. $\endgroup$ – Daniel A. Burke Jan 3 '18 at 21:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.