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    <h2><a href="http://paulbourke.net/geometry/circlesphere/" rel="noreferrer">Intersection of two circles</a></h2> <p>Written by Paul Bourke</p> <blockquote> <p>The following note describes how to find the intersection point(s) between two circles on a plane, the following notation is used. The aim is to find the two points P<sub>3</sub> = (x<sub>3</sub>, y<sub>3</sub>) if they exist. </p></p> <p><img src="https://i.stack.imgur.com/aUXMY.gif" alt="Intersection of 2 circles"></p> <p> First calculate the distance d between the center of the circles. d = ||P<sub>1</sub> - P<sub>0</sub>||. </p> <ul> <li>If d > r<sub>0</sub> + r<sub>1</sub> then there are no solutions, the circles are separate. <p> <li>If d &lt; |r<sub>0</sub> - r<sub>1</sub>| then there are no solutions because one circle is contained within the other.<p> <li>If d = 0 and r<sub>0</sub> = r<sub>1</sub> then the circles are coincident and there are an infinite number of solutions.<p> </ul></p> <p>Considering the two triangles P<sub>0</sub>P<sub>2</sub>P<sub>3</sub> and P<sub>1</sub>P<sub>2</sub>P<sub>3</sub> we can write <p> a<sup>2</sup> + h<sup>2</sup> = r<sub>0</sub><sup>2</sup> and b<sup>2</sup> + h<sup>2</sup> = r<sub>1</sub><sup>2</sup> <p> Using d = a + b we can solve for a,<p> a = (r<sub>0</sub><sup>2</sup> - r<sub>1</sub><sup>2</sup> + d<sup>2</sup> ) / (2 d) <p> <p></p> It can be readily shown that this reduces to r<sub>0</sub> when the two circles touch at one point, ie: d = r<sub>0</sub> + r<sub>1</sub> </p> Solve for h by substituting a into the first equation, h<sup>2</sup> = r<sub>0</sub><sup>2</sup> - a<sup>2</sup> </p> <p>So <p> P<sub>2</sub> = P<sub>0</sub> + a ( P<sub>1</sub> - P<sub>0</sub> ) / d <p> And finally, P<sub>3</sub> = (x<sub>3</sub>,y<sub>3</sub>) in terms of P<sub>0</sub> = (x<sub>0</sub>,y<sub>0</sub>), P<sub>1</sub> = (x<sub>1</sub>,y<sub>1</sub>) and P<sub>2</sub> = (x<sub>2</sub>,y<sub>2</sub>), is <p> x<sub>3</sub> = x<sub>2</sub> +- h ( y<sub>1</sub> - y<sub>0</sub> ) / d <p> y<sub>3</sub> = y<sub>2</sub> -+ h ( x<sub>1</sub> - x<sub>0</sub> ) / d</p> </blockquote> <p>Source: <a href="http://paulbourke.net/geometry/circlesphere/" rel="noreferrer">http://paulbourke.net/geometry/circlesphere/</a></p>
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