Bilateration with iBeacons - ios

Question

I am trying to use multiple iBeacons to track the user's location in iOS. I know this can be done (somewhat) using 3 beacons and trilateration, but I would like to do it with two (bilateration). I know that I will likely end up with two answers. Does anyone know of a simple way to accomplish this, given the (x,y) location of the beacons (relative to the room), and an averaged RSSI from each beacon?
I have this code for trilateration that I altered to objective-c from javascript:
- (CGPoint)getTrilaterationWithBeacon1:(BBBeacon *)beacon1 Beacon2:(BBBeacon *)beacon2 Beacon3:(BBBeacon *)beacon3 {
float xa = beacon1.x;
float ya = beacon1.y;
float xb = beacon2.x;
float yb = beacon2.y;
float xc = beacon3.x;
float yc = beacon3.y;
float ra = beacon1.distance;
float rb = beacon2.distance;
float rc = beacon3.distance;
float S = (pow(xc, 2.) - pow(xb, 2.) + pow(yc, 2.) - pow(yb, 2.) + pow(rb, 2.) - pow(rc, 2.)) / 2.0;
float T = (pow(xa, 2.) - pow(xb, 2.) + pow(ya, 2.) - pow(yb, 2.) + pow(rb, 2.) - pow(ra, 2.)) / 2.0;
float y = ((T * (xb - xc)) - (S * (xb - xa))) / (((ya - yb) * (xb - xc)) - ((yc - yb) * (xb - xa)));
float x = ((y * (ya - yb)) - T) / (xb - xa);
CGPoint point = CGPointMake(x, y);
return point;
}

Answer 1

So this is the code I ended up using, thanks to ChuckCottrill's suggestion that I look for a formula to calculate intersection of two circles. It is modified from a C version I found online here: http://paulbourke.net/geometry/circlesphere/tvoght.c
The results are somewhat inconsistent due to the inconsistency of the RSSI values returned from the iBeacons.
I will still need to add code to select the correct point somehow (it gives two results).
- (CGPoint)getBilaterationWithBeacon1:(BBBeacon *)beacon1 Beacon2:(BBBeacon *)beacon2 {
float x0 = beacon1.locationX;
float y0 = beacon1.locationY;
float r0 = beacon1.filteredDistance;
float x1 = beacon2.locationX;
float y1 = beacon2.locationY;
float r1 = beacon2.filteredDistance;
float a, dx, dy, d, h, rx, ry;
float 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));
/* Check for solvability. */
if (d > (r0 + r1)) {
/* no solution. circles do not intersect. */
return CGPointMake(-1, -1);
}
if (d < abs(r0 - r1)) {
/* no solution. one circle is contained in the other */
return CGPointMake(-1, -1);
}
/* '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. */
float xi = x2 + rx;
float xi_prime = x2 - rx;
float yi = y2 + ry;
float yi_prime = y2 - ry;
CGPoint point1 = CGPointMake(xi, yi);
CGPoint point2 = CGPointMake(xi_prime, yi_prime);
//pick one
return point2;
}