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How can I write a program that solves x3 and y3 for the following equations, assuming all other values are known? I'm going to write this in Javascript, but any kind of pseudo-code is fine. There ought to be either 1, 2 or 0 solutions. (As you can see, I'm trying to figure out the code to find the intersection of two circles.)

(x3 - x1)2 + (y3 - y1)2 = r12

and

(x3 - x2)2 + (y3 - y2)2 = r22

Richard Ellwood
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  • This looks like CompSci homework. What have you tried? A quick Google search yielded an identical question/answer - http://stackoverflow.com/questions/12219802/a-javascript-function-that-returns-the-x-y-points-of-intersection-between-two-ci – Dave Lasley Apr 13 '15 at 00:43

1 Answers1

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I found the solution to my own question:

/* circle_circle_intersection() *
 * Determine the points where 2 circles in a common plane intersect.
 *
 * int circle_circle_intersection(
 *                                // center and radius of 1st circle
 *                                double x0, double y0, double r0,
 *                                // center and radius of 2nd circle
 *                                double x1, double y1, double r1,
 *                                // 1st intersection point
 *                                double *xi, double *yi,              
 *                                // 2nd intersection point
 *                                double *xi_prime, double *yi_prime)
 *
 * This is a public domain work. 3/26/2005 Tim Voght
 *
 */
#include <math.h>

int circle_circle_intersection(double x0, double y0, double r0,
                               double x1, double y1, double r1,
                               double *xi, double *yi,
                               double *xi_prime, double *yi_prime)
{
    double a, dx, dy, d, h, rx, ry;
    double x2, y2;

    /* dx and dy are the vertical and horizontal distances between
     * the circle centers.
     */
    dx = x1 - x0;
    dy = y1 - y0;

    /* Determine the straight-line distance between the centers. */
    //d = sqrt((dy*dy) + (dx*dx));
    d = hypot(dx,dy); // Suggested by Keith Briggs

    /* Check for solvability. */
    if (d > (r0 + r1))
    {
        /* no solution. circles do not intersect. */
        return 0;
    }
    if (d < fabs(r0 - r1))
    {
        /* no solution. one circle is contained in the other */
        return 0;
    }

    /* 'point 2' is the point where the line through the circle
     * intersection points crosses the line between the circle
     * centers.  
     */

    /* Determine the distance from point 0 to point 2. */
    a = ((r0*r0) - (r1*r1) + (d*d)) / (2.0 * d) ;

    /* Determine the coordinates of point 2. */
    x2 = x0 + (dx * a/d);
    y2 = y0 + (dy * a/d);

    /* Determine the distance from point 2 to either of the
     * intersection points.
     */
    h = sqrt((r0*r0) - (a*a));

    /* Now determine the offsets of the intersection points from
     * point 2.
     */
    rx = -dy * (h/d);
    ry = dx * (h/d);

    /* Determine the absolute intersection points. */
    *xi = x2 + rx;
    *xi_prime = x2 - rx;
    *yi = y2 + ry;
    *yi_prime = y2 - ry;

    return 1;
}
Richard Ellwood
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