Intersection of circles

This sketch contains a function "triangulate" which, given two points and the distances between these points and a third unknown point, will calculate the location of the third unknown point.

This is a common problem when working with sensors.

The main sketch simply chooses three random points and passes two of them and the distances to the triangulate function, displaying the results on Serial.

When using this function in your sketch, you would typically replace the two points and the distances between them with constants, and remove code calculating values that you know would be constant.

Likewise, if you know that your sensed point will always be "in front" of your sensors, then the correct solution will always be one of the two possible solutions (C1 or C2) and you can remove the code that produces the solution that you don't need.

You will probably also want to replace the C1x, C1y references with sketch-scoped variables rather than passing references into this function. The function is written to be general.

void triangulate(
  double Ax, double Ay,
  double Bx, double By,
  double AB, double AC, double BC,
  double &Cx1, double &Cy1,
  double &Cx2, double &Cy2)
  double cos1 = (AC * AC + AB * AB - BC * BC) / (2 * AC * AB);
  double sin1 = sqrt(1 - cos1 * cos1);

  double B1x, B1y;

  // get the location of B relative to A
  B1x = Bx - Ax;
  B1y = By - Ay;

  // scale by AC/AB,

  B1x *= AC / AB;
  B1y *= AC / AB;

  // rotate B1 by the calculated angle - plus or minus, and add A

  Cx1 = B1x * cos1 + B1y * sin1 + Ax;
  Cy1 = B1y * cos1 - B1x * sin1 + Ay;

  Cx2 = B1x * cos1 - B1y * sin1 + Ax;
  Cy2 = B1y * cos1 + B1x * sin1 + Ay;

double dist(double x1, double y1, double x2, double y2 ) {
  return sqrt((x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1));

void printxy(double x, double y) {
  Serial.print(x) ;

void setup() {
  while (!Serial);

  // seed random with A0, which is floating


void loop() {

  // create some sample data
  double Ax, Ay, Bx, By, Cx, Cy, AB, AC, BC;

  // position of sensor A
  Ax = random(-100, 100);
  Ay = random(-100, 100);

  // position of sensor B
  Bx = random(-100, 100);
  By = random(-100, 100);

  // the object that we are trying to calculate the position of
  Cx = random(-100, 100);
  Cy = random(-100, 100);

  AB = dist(Ax, Ay, Bx, By); // we know this from where we have positioned the sensors
  AC = dist(Ax, Ay, Cx, Cy); // this is given by the ultrasonic
  BC = dist(Bx, By, Cx, Cy); // this is given by the ultrasonic

  double Solution1x, Solution1y, Solution2x, Solution2y;

  triangulate(Ax, Ay, Bx, By, AB, AC, BC, Solution1x, Solution1y, Solution2x, Solution2y);

  Serial.print("Sensor A is at ");
  printxy(Ax, Ay);
  Serial.print(", Sensor B is at ");
  printxy(Bx, By);
  Serial.print("Distance AB is ");

  Serial.print("Sensed object is at ");
  printxy(Cx, Cy);
  Serial.print("Distance from sensor A to object is ");
  Serial.print("Distance from sensor B to object is ");

  Serial.print("Solution 1 is ");
  printxy(Solution1x, Solution1y);
  Serial.print(", which is accurate to ");
  Serial.print(dist(Cx, Cy, Solution1x, Solution1y));

  Serial.print("Solution 2 is ");
  printxy(Solution2x, Solution2y);
  Serial.print(", which is accurate to ");
  Serial.print(dist(Cx, Cy, Solution2x, Solution2y));