Find the intersections (two points) of two circles, if they intersect at all

IntersectTwoCircles.js

// based on the math here:
// http://math.stackexchange.com/a/1367732

// x1,y1 is the center of the first circle, with radius r1
// x2,y2 is the center of the second ricle, with radius r2
function intersectTwoCircles(x1,y1,r1, x2,y2,r2) {
  var centerdx = x1 - x2;
  var centerdy = y1 - y2;
  var R = Math.sqrt(centerdx * centerdx + centerdy * centerdy);
  if (!(Math.abs(r1 - r2) <= R && R <= r1 + r2)) { // no intersection
    return []; // empty list of results
  }
  // intersection(s) should exist

  var R2 = R*R;
  var R4 = R2*R2;
  var a = (r1*r1 - r2*r2) / (2 * R2);
  var r2r2 = (r1*r1 - r2*r2);
  var c = Math.sqrt(2 * (r1*r1 + r2*r2) / R2 - (r2r2 * r2r2) / R4 - 1);

  var fx = (x1+x2) / 2 + a * (x2 - x1);
  var gx = c * (y2 - y1) / 2;
  var ix1 = fx + gx;
  var ix2 = fx - gx;

  var fy = (y1+y2) / 2 + a * (y2 - y1);
  var gy = c * (x1 - x2) / 2;
  var iy1 = fy + gy;
  var iy2 = fy - gy;

  // note if gy == 0 and gx == 0 then the circles are tangent and there is only one solution
  // but that one solution will just be duplicated as the code is currently written
  return [[ix1, iy1], [ix2, iy2]];
}

Thanks Juppy :-)

I had to convert this to VB.Net but worked sweet with only 1 minor tweak. for calculation of var c
`

    Private Function IntersectTwoCircles(x1%, y1%, r1%, x2%, y2%, r2%) As Integer()
    'based on the math here:
    'http//math.stackexchange.com/a/1367732
    ' x1,y1 Is the center of the first circle, with radius r1
    ' x2,y2 Is the center of the second ricle, with radius r2
    Dim centerdx% = x1 - x2
    Dim centerdy% = y1 - y2
    Dim R = Math.Sqrt(centerdx * centerdx + centerdy * centerdy)


    If (Not (Math.Abs(r1 - r2) <= R) And (R <= r1 + r2)) Then
        ' no intersection
        Return Nothing ' empty list of results
    End If

    'intersection(s) should exist

    Dim R_2 = R * R
    Dim R_4 = R_2 * R_2
    Dim a = (r1 * r1 - r2 * r2) / (2 * R_2)
    Dim r2r2 = (r1 * r1 - r2 * r2)

    Dim c1 As Double = 2 * (r1 * r1 + r2 * r2) / R_2 ' As Double because default is Integer
    Dim c2 As Double = r2r2   'seperate c2 calculation avoids Integer overflow 
    c2 = (c2 * c2) / R_4        'recycle c2 
    Dim c = Math.Sqrt(c1 - c2 - 1)

    Dim fx = (x1 + x2) / 2 + a * (x2 - x1)
    Dim gx = c * (y2 - y1) / 2
    Dim ix1% = Int(fx + gx + 0.05)
    Dim ix2% = Int(fx - gx + 0.05)

    Dim fy = (y1 + y2) / 2 + a * (y2 - y1)
    Dim gy = c * (x1 - x2) / 2
    Dim iy1% = Int(fy + gy + 0.05)
    Dim iy2% = Int(fy - gy + 0.05)

    'note if gy == 0 And gx == 0 then the circles are tangent And there Is only one solution
    'but that one solution will just be duplicated as the code Is currently written
    Return {ix1, iy1, ix2, iy2}


End Function

`

Thanks!

Here's a Unity3D friendly version.

using UnityEngine;

namespace C2Intersection {

    // based on the math here:
    // http://math.stackexchange.com/a/1367732
    // based on the code from:
    // https://gist.github.com/jupdike/bfe5eb23d1c395d8a0a1a4ddd94882ac
    public static class CircleCircleIntersection {

        // circleA is the center of the first circle, with radius radiusA
        // circleB is the center of the second circle, with radius radiusB
        public static int Intersect(Vector2       circleA,
                                    float         radiusA,
                                    Vector2       circleB,
                                    float         radiusB,
                                    out Vector2[] intersections) {

            float centerDx = circleA.x - circleB.x;
            float centerDy = circleB.y - circleB.y;
            float r = Mathf.Sqrt(centerDx * centerDx + centerDy * centerDy);

            // no intersection
            if (!(Mathf.Abs(radiusA - radiusB) <= r && r <= radiusA + radiusB)) {
                intersections = new Vector2[0];
                return 0;
            }

            float r2d = r * r;
            float r4d = r2d * r2d;
            float rASquared = radiusA * radiusA;
            float rBSquared = radiusB * radiusB;
            float a = (rASquared - rBSquared) / (2 * r2d);
            float r2r2 = (rASquared - rBSquared);
            float c = Mathf.Sqrt(2 * (rASquared + rBSquared) / r2d - (r2r2 * r2r2) / r4d - 1);

            float fx = (circleA.x + circleB.x) / 2 + a * (circleB.x - circleA.x);
            float gx = c * (circleB.y - circleA.y) / 2;
            float ix1 = fx + gx;
            float ix2 = fx - gx;

            float fy = (circleA.y + circleB.y) / 2 + a * (circleB.y - circleA.y);
            float gy = c * (circleA.x - circleB.x) / 2;
            float iy1 = fy + gy;
            float iy2 = fy - gy;

            // if gy == 0 and gx == 0 then the circles are tangent and there is only one solution
            if (Mathf.Abs(gx) < float.Epsilon && Mathf.Abs(gy) < float.Epsilon) {
                intersections = new [] {
                    new Vector2(ix1, iy1)
                };
                return 1;
            }

            intersections = new [] {
                new Vector2(ix1, iy1),
                new Vector2(ix2, iy2),
            };
            return 2;
        }
    }
}

A usage example:

using mlib;
using UnityEngine;

namespace C2Intersection {

    public class IntersectionViz : MonoBehaviour {

        [SerializeField]
        private Transform circleA;
        [SerializeField]
        private float     radiusA;
        [SerializeField]
        private Color     colorA;
        [SerializeField]
        private Transform circleB;
        [SerializeField]
        private float     radiusB;
        [SerializeField]
        private Color     colorB;

        private void OnDrawGizmos() {
            GizmoUtils.DrawCircle(circleA.position, radiusA, colorA);
            GizmoUtils.DrawCircle(circleB.position, radiusB, colorB);
            Vector2[] intersections;
            int num = CircleCircleIntersection.Intersect(circleA.position,
                                                         radiusA,
                                                         circleB.position,
                                                         radiusB,
                                                         out intersections);

            Gizmos.color = Color.cyan;
            for (int i = 0; i < num; ++i) {
                Gizmos.DrawSphere(intersections[i], 0.25f);
            }
        }
    }
}

GimoUtils.DrawCircle is a utility method for drawing circles as gizmos. It looks like this:

using UnityEngine;

namespace mlib {

    public static class GizmoUtils {

        public static void DrawCircle(Vector3 position, float radius, Color color) {
            float err = 1f;
            Gizmos.color = color;
            float theta = 0f;
            float x = radius * Mathf.Cos(theta);
            float y = radius * Mathf.Sin(theta);
            Vector3 pos = position + new Vector3(x, y, 0f);
            Vector3 lastPos = pos;
            const float TWOPI = Mathf.PI * 2;
            const float inc = 0.4f;
            for (theta = inc; theta < TWOPI; theta += inc) {
                x = radius * Mathf.Cos(theta);
                y = radius * Mathf.Sin(theta);
                Vector3 newPos = position + new Vector3(x, y, 0f);
                Gizmos.DrawLine(pos, newPos);
                pos = newPos;
            }

            Gizmos.DrawLine(pos, lastPos);
        }
    }
}

Here is a FORTRAN 95 version which I had to write. A subroutine based on your code, except that it returns each coordinate separately. For this, I had to include another parameter, the 'flag' which checks if the intersection exists at all. I needed each of the intersection points as separate variables so ....

subroutine intersect2circles(x1,y1,r1,x2,y2,r2,ix1,iy1,ix2,iy2,flag)
	implicit none
	real(8),intent(in) :: x1,y1,r1,x2,y2,r2
	real(8),intent(out) :: ix1,iy1,ix2,iy2
	integer,intent(out) :: flag
	real(8) centerdx,centerdy,R,Rsq,Rquad,a,r2r2,c,fx,gx,fy,gy
	flag=0 ! This checks whether there is an intersection or not, since we cannot return empty variables in fortran as far as I know
	centerdx=x1-x2
	centerdy=y1-y2
	R=sqrt(centerdx**2+centerdy**2)
	if (.not.(abs(r1-r2)<=R.and.R<=r1+r2)) then
		print*, "No intersection, setting intersection points to 0,0, and setting flag to 1"
		ix1=0
		iy1=0
		ix2=0
		iy2=0
		flag=1
		return
	endif
	Rsq=R*R
	Rquad=Rsq*Rsq
	a=(r1**2-r2**2)/(2*Rsq)
	r2r2=(r1**2-r2**2)
	c=sqrt(2*(r1**2+r2**2)/Rsq - (r2r2**2)/Rquad -1)

	fx=(x1+x2)/2+a*(x2-x1)
	gx=c*(y2-y1)/2
	ix1=fx+gx
	ix2=fx-gx

	fy=(y1+y2)/2+a*(y2-y1)
	gy=c*(x1-x2)/2
	iy1=fy+gy
	iy2=fy-gy

	return
end subroutine intersect2circles

Hi,

Is someone able to comment the purposes of each variable? I've commented what I understand, but I would like to undestand the rest. For example, why do we have R2 and R4, is it just to neaten up the code or is there another reason. Why are fx and gx calculated in the ways they are?

Many thanks

  var centerdx = x1 - x2; // distance between x points of both circles
  var centerdy = y1 - y2; // distance between y points of both circles
  var R = Math.sqrt(centerdx * centerdx + centerdy * centerdy); // distance between centres of circles

  var R2 = R*R;
  var R4 = R2*R2;
  var a = (r1*r1 - r2*r2) / (2 * R2);
  var r2r2 = (r1*r1 - r2*r2);
  var c = Math.sqrt(2 * (r1*r1 + r2*r2) / R2 - (r2r2 * r2r2) / R4 - 1);

  var fx = (x1+x2) / 2 + a * (x2 - x1);
  var gx = c * (y2 - y1) / 2;
  var ix1 = fx + gx; // 1st intersect x-coordinate
  var ix2 = fx - gx; // 2nd intersect x-coordinate

  var fy = (y1+y2) / 2 + a * (y2 - y1);
  var gy = c * (x1 - x2) / 2;
  var iy1 = fy + gy; // 1st intersect y-coordinate
  var iy2 = fy - gy; // 2nd intersect y-coordinate

Hi,

Is someone able to comment the purposes of each variable? I've commented what I understand, but I would like to undestand the rest. For example, why do we have R2 and R4, is it just to neaten up the code or is there another reason. Why are fx and gx calculated in the ways they are?

Many thanks

  var centerdx = x1 - x2; // distance between x points of both circles
  var centerdy = y1 - y2; // distance between y points of both circles
  var R = Math.sqrt(centerdx * centerdx + centerdy * centerdy); // distance between centres of circles

  var R2 = R*R;
  var R4 = R2*R2;
  var a = (r1*r1 - r2*r2) / (2 * R2);
  var r2r2 = (r1*r1 - r2*r2);
  var c = Math.sqrt(2 * (r1*r1 + r2*r2) / R2 - (r2r2 * r2r2) / R4 - 1);

  var fx = (x1+x2) / 2 + a * (x2 - x1);
  var gx = c * (y2 - y1) / 2;
  var ix1 = fx + gx; // 1st intersect x-coordinate
  var ix2 = fx - gx; // 2nd intersect x-coordinate

  var fy = (y1+y2) / 2 + a * (y2 - y1);
  var gy = c * (x1 - x2) / 2;
  var iy1 = fy + gy; // 1st intersect y-coordinate
  var iy2 = fy - gy; // 2nd intersect y-coordinate

Hi man - Im not too great on maths either but the R2 and R4 are meant to represent R squared and R cubed.

No clue what the rest means I'm afraid!

r2 is r squared and r4 is r2 squared, or fourth power (not cubed). This saves computation. Same with fx and gx and fy and gy. They are just places to put values to be reused, for performance. 'c' is a big constant that takes a lot of computation, so is only computed once. It has been so long since I put this code here I don't really remember what the meaning of each line is, but if you care about it, you can check the Math StackExchange link for the original math and the actual formula which I translated into code: http://math.stackexchange.com/a/1367732

Thanks, very helpful, I translated it to Kotlin:

// x1,y1 is the center of the first circle, with radius r1
// x2,y2 is the center of the second ricle, with radius r2
fun intersectTwoCircles( x1: Double,y1: Double,r1: Double, x2: Double,y2: Double,r2: Double):List<Pair<Double, Double>> {
    val centerdx = x1 - x2;
    val centerdy = y1 - y2;
    val R = Math.sqrt(centerdx * centerdx + centerdy * centerdy);
    if (!(Math.abs(r1 - r2) <= R && R <= r1 + r2)) { // no intersection
        return emptyList(); // empty list of results
    }
    // intersection(s) should exist

    val R2 = R*R;
    val R4 = R2*R2;
    val a = (r1*r1 - r2*r2) / (2 * R2);
    val r2r2 = (r1*r1 - r2*r2);
    val c = Math.sqrt(2 * (r1*r1 + r2*r2) / R2 - (r2r2 * r2r2) / R4 - 1);

    val fx = (x1+x2) / 2 + a * (x2 - x1);
    val gx = c * (y2 - y1) / 2;
    val ix1 = fx + gx;
    val ix2 = fx - gx;

    val fy = (y1+y2) / 2 + a * (y2 - y1);
    val gy = c * (x1 - x2) / 2;
    val iy1 = fy + gy;
    val iy2 = fy - gy;

    // note if gy == 0 and gx == 0 then the circles are tangent and there is only one solution
    // but that one solution will just be duplicated as the code is currently written
    return listOf(ix1 to iy1, ix2 to iy2)
}

r2 is r squared and r4 is r2 squared, or fourth power (not cubed). This saves computation. Same with fx and gx and fy and gy. They are just places to put values to be reused, for performance. 'c' is a big constant that takes a lot of computation, so is only computed once. It has been so long since I put this code here I don't really remember what the meaning of each line is, but if you care about it, you can check the Math StackExchange link for the original math and the actual formula which I translated into code: http://math.stackexchange.com/a/1367732

Hi man - late reply but thanks for the correction. Cubed would have been to the power of 8, of course! And thank you for the rest of the explanation

Cubed is to the power of 3, for anyone who sees this thread. I'm glad the explanation was at all helpful. Code I wrote a long time ago!

If someone need some dirty legal python expression (might also be legal in other languages) of all the intersection points, here is this (sorry):

ix1 = (-r1**2*x1 + r1**2*x2 + r2**2*x1 - r2**2*x2 + x1**3 - x1**2*x2 - x1*x2**2 + x1*y1**2 - 2*x1*y1*y2 + x1*y2**2 + x2**3 + x2*y1**2 - 2*x2*y1*y2 + x2*y2**2 + sqrt((-r1**2 + 2*r1*r2 - r2**2 + x1**2 - 2*x1*x2 + x2**2 + y1**2 - 2*y1*y2 + y2**2)*(r1**2 + 2*r1*r2 + r2**2 - x1**2 + 2*x1*x2 - x2**2 - y1**2 + 2*y1*y2 - y2**2))*(-y1 + y2))/(2*(x1**2 - 2*x1*x2 + x2**2 + y1**2 - 2*y1*y2 + y2**2))

ix2 = (-r1**2*x1 + r1**2*x2 + r2**2*x1 - r2**2*x2 + x1**3 - x1**2*x2 - x1*x2**2 + x1*y1**2 - 2*x1*y1*y2 + x1*y2**2 + x2**3 + x2*y1**2 - 2*x2*y1*y2 + x2*y2**2 + sqrt((-r1**2 + 2*r1*r2 - r2**2 + x1**2 - 2*x1*x2 + x2**2 + y1**2 - 2*y1*y2 + y2**2)*(r1**2 + 2*r1*r2 + r2**2 - x1**2 + 2*x1*x2 - x2**2 - y1**2 + 2*y1*y2 - y2**2))*(y1 - y2))/(2*(x1**2 - 2*x1*x2 + x2**2 + y1**2 - 2*y1*y2 + y2**2))

iy1 = (-r1**2 + r2**2 + y1**2 - y2**2 + (-x1 + (-r1**2*x1 + r1**2*x2 + r2**2*x1 - r2**2*x2 + x1**3 - x1**2*x2 - x1*x2**2 + x1*y1**2 - 2*x1*y1*y2 + x1*y2**2 + x2**3 + x2*y1**2 - 2*x2*y1*y2 + x2*y2**2 + sqrt((-r1**2 + 2*r1*r2 - r2**2 + x1**2 - 2*x1*x2 + x2**2 + y1**2 - 2*y1*y2 + y2**2)*(r1**2 + 2*r1*r2 + r2**2 - x1**2 + 2*x1*x2 - x2**2 - y1**2 + 2*y1*y2 - y2**2))*(-y1 + y2))/(2*(x1**2 - 2*x1*x2 + x2**2 + y1**2 - 2*y1*y2 + y2**2)))**2 - (-x2 + (-r1**2*x1 + r1**2*x2 + r2**2*x1 - r2**2*x2 + x1**3 - x1**2*x2 - x1*x2**2 + x1*y1**2 - 2*x1*y1*y2 + x1*y2**2 + x2**3 + x2*y1**2 - 2*x2*y1*y2 + x2*y2**2 + sqrt((-r1**2 + 2*r1*r2 - r2**2 + x1**2 - 2*x1*x2 + x2**2 + y1**2 - 2*y1*y2 + y2**2)*(r1**2 + 2*r1*r2 + r2**2 - x1**2 + 2*x1*x2 - x2**2 - y1**2 + 2*y1*y2 - y2**2))*(-y1 + y2))/(2*(x1**2 - 2*x1*x2 + x2**2 + y1**2 - 2*y1*y2 + y2**2)))**2)/(2*(y1 - y2))

iy2 = (-r1**2 + r2**2 + y1**2 - y2**2 + (-x1 + (-r1**2*x1 + r1**2*x2 + r2**2*x1 - r2**2*x2 + x1**3 - x1**2*x2 - x1*x2**2 + x1*y1**2 - 2*x1*y1*y2 + x1*y2**2 + x2**3 + x2*y1**2 - 2*x2*y1*y2 + x2*y2**2 + sqrt((-r1**2 + 2*r1*r2 - r2**2 + x1**2 - 2*x1*x2 + x2**2 + y1**2 - 2*y1*y2 + y2**2)*(r1**2 + 2*r1*r2 + r2**2 - x1**2 + 2*x1*x2 - x2**2 - y1**2 + 2*y1*y2 - y2**2))*(y1 - y2))/(2*(x1**2 - 2*x1*x2 + x2**2 + y1**2 - 2*y1*y2 + y2**2)))**2 - (-x2 + (-r1**2*x1 + r1**2*x2 + r2**2*x1 - r2**2*x2 + x1**3 - x1**2*x2 - x1*x2**2 + x1*y1**2 - 2*x1*y1*y2 + x1*y2**2 + x2**3 + x2*y1**2 - 2*x2*y1*y2 + x2*y2**2 + sqrt((-r1**2 + 2*r1*r2 - r2**2 + x1**2 - 2*x1*x2 + x2**2 + y1**2 - 2*y1*y2 + y2**2)*(r1**2 + 2*r1*r2 + r2**2 - x1**2 + 2*x1*x2 - x2**2 - y1**2 + 2*y1*y2 - y2**2))*(y1 - y2))/(2*(x1**2 - 2*x1*x2 + x2**2 + y1**2 - 2*y1*y2 + y2**2)))**2)/(2*(y1 - y2))

Python

def intersect(x1, y1, r1, x2, y2, r2):
    centerdx = x1 - x2
    centerdy = y1 - y2
    R = sqrt(centerdx * centerdx + centerdy * centerdy)
    R2 = R*R
    R4 = R2*R2
    a = (r1*r1 - r2*r2) / (2 * R2)
    r2r2 = (r1*r1 - r2*r2)
    c = sqrt(2 * (r1*r1 + r2*r2) / R2 - (r2r2 * r2r2) / R4 - 1)
    fx = (x1+x2) / 2 + a * (x2 - x1)
    gx = c * (y2 - y1) / 2
    ix1 = fx + gx
    ix2 = fx - gx
    fy = (y1+y2) / 2 + a * (y2 - y1)
    gy = c * (x1 - x2) / 2
    iy1 = fy + gy
    iy2 = fy - gy`

Here is the code in Swift 5:
`
import Cocoa

struct circle{
let x: Double
let y: Double
let radius: Double
}
struct point{
let x: Double
let y: Double
}

func getCooridantes(circle1 : circle, circle2: circle) -> [point]{

let centerdx : Double = circle1.x - circle2.x
let centerdy : Double = circle1.y - circle1.y
let R = sqrt(centerdx * centerdx + centerdy * centerdy)

if (!(abs(circle1.radius - circle2.radius) <= R && R <= circle1.radius + circle2.radius)){
   
    return [point(x: 0, y: 0)]
}

let R2 = R*R;
let R4 = R2*R2;
let a = (circle1.radius * circle1.radius - circle2.radius * circle2.radius) / (2 * R2);
let r2r2 = (circle1.radius * circle1.radius - circle2.radius * circle2.radius);
let c = sqrt(2 * (circle1.radius * circle1.radius + circle2.radius * circle2.radius) / R2 - (r2r2 * r2r2) / R4 - 1);

let fx = (circle1.x+circle2.x) / 2 + a * (circle2.x - circle1.x);
let gx = c * (circle2.y - circle1.y) / 2;
let ix1 = fx + gx;
let ix2 = fx - gx;

let fy = (circle1.y + circle2.y) / 2 + a * (circle2.y - circle1.y);
let gy = c * (circle1.x - circle2.x) / 2;
let iy1 = fy + gy;
let iy2 = fy - gy;
print(ix1 ,iy1)
print(ix2, iy2)

// note if gy == 0 and gx == 0 then the circles are tangent and there is only one solution
// but that one solution will just be duplicated as the code is currently written
return [point(x: ix1, y: iy1), point(x: ix2, y: iy2)]

}

let circle1test = circle(x: 0, y: 0, radius: 10)
let circle2test = circle(x: 10, y: 0, radius: 3)
getCooridantes(circle1: circle1test, circle2: circle2test)
`

Thanks!

Here's a Unity3D friendly version.

Thanks a lot for this transcription! However I think there might be a mistakate on the line float centerDy = circleB.y - circleB.y;
I imagine it should be float centerDy = circleA.y - circleB.y;, no? Emphasis on circleA.y

Here's an implementation in no_std rust using nalgebra (can be easily substituted with a tuple instead) and num-traits.

#![no_std]
use nalgebra::Point2;
use num_traits::{Float, Pow};

type Point = Point2<f32>

pub fn intersection(
    point1: Point,
    radius1: f32,
    point2: Point,
    radius2: f32,
) -> Option<(Point, Point)> {
    let r1 = radius1.abs();
    let r2 = radius2.abs();

    let x1: f32 = point1.x;
    let y1: f32 = point1.y;

    let x2: f32 = point2.x;
    let y2: f32 = point2.y;

    let cdx: f32 = point1.x - point2.x;
    let cdy: f32 = point1.y - point2.y;

    let dist = (cdx * cdx + cdy * cdy).sqrt();

    if (r1 - r2).abs() > dist || dist > r1 + r2 {
        // no intersection
        return None;
    }

    let dist2 = dist * dist;
    let dist4 = dist2 * dist2;

    let a = (r1 * r1 - r2 * r2) / (2f32 * dist2);
    let r1r2 = r1 * r1 - r2 * r2;
    let c = (2f32 * (r1 * r1 + r2 * r2) / dist2 - (r1r2 * r1r2) / dist4 - 1f32).sqrt();

    let fx = (x1 + x2) / 2f32 + a * (x2 - x1);
    let gx = c * (y2 - y1) / 2f32;

    let fy = (y1 + y2) / 2f32 + a * (y2 - y1);
    let gy = c * (x1 - x2) / 2f32;

    Some((Point::new(fx + gx, fy + gy), Point::new(fx - gx, fy - gy)))
}

Swift:

import Foundation

/**
 Finds the points that intersect two circles.

 If the circles only intersect at one point, then the second point will be
 `nil`.

  Sources:
 
  [gist](https://gist.github.com/jupdike/bfe5eb23d1c395d8a0a1a4ddd94882ac)
 
  [stackexchange](https://math.stackexchange.com/a/1367732/825630)

 - Parameters:
   - center1: The center of the first circle.
   - radius1: The radius of the first circle. Must be positive.
   - center2: The center of the second circle.
   - radius2: The radius of the second circle. Must be positive.
 */
func intersectingPointsOfCircles(
    center1: CGPoint,
    radius1: CGFloat,
    center2: CGPoint,
    radius2: CGFloat
) -> (CGPoint, CGPoint?)? {
    
    if center1 == center2 {
        // If the centers are the same and the radii are the same, then the
        // circles intersect at an infinite number of points, so return `nil`.
        //
        // If the centers are the same, but the radii are different, then there
        // can't be any intersecting points, so also return `nil`.
        return nil
    }

    let centerDx = center1.x - center2.x
    let centerDy = center1.y - center2.y
    
    /// The distance between the centers of the circles
    let d = sqrt(pow(centerDx, 2) + pow(centerDy, 2))

    if abs(radius1 - radius2) > d || d > radius1 + radius2 {
        return nil
    }
    
    let d2 = d * d
    let d4 = d2 * d2
    let a = (radius1 * radius1 - radius2 * radius2) / (2 * d2)
    let r2r2 = (radius1 * radius1 - radius2 * radius2)
    let c = sqrt(
        2 * (radius1 * radius1 + radius2 * radius2) /
        d2 - (r2r2 * r2r2) / d4 - 1
    )
    
    let fx = (center1.x + center2.x) / 2 + a * (center2.x - center1.x)
    let gx = c * (center2.y - center1.y) / 2
    let ix1 = fx + gx
    let ix2 = fx - gx
    
    let fy = (center1.y + center2.y) / 2 + a * (center2.y - center1.y)
    let gy = c * (center1.x - center2.x) / 2
    let iy1 = fy + gy
    let iy2 = fy - gy

    // if gy == 0 and gx == 0, then the circles are tangent and there
    // is only one solution

    let intersectingPoint1 = CGPoint(x: ix1, y: iy1)
    let intersectingPoint2 = CGPoint(x: ix2, y: iy2)
    
    if intersectingPoint1 == intersectingPoint2 {
        return (intersectingPoint1, nil)
    }
    return (intersectingPoint1, intersectingPoint2)
    
}

PHP version

public function intersectTwoCircles($x1, $y1, $r1, $x2, $y2, $r2)
    {
        $centerdx = $x1 - $x2;
        $centerdy = $y1 - $y2;

        $R = sqrt($centerdx * $centerdx + $centerdy * $centerdy);

        if (!(
            abs($r1 - $r2) <= $R && 
            $R <= $r1 + $r2
        )) { // no intersection
            return []; // empty list of results
        }
        // intersection(s) should exist

        $R2 = $R * $R;
        $R4 = $R2 * $R2;
        $a = ($r1 * $r1 - $r2 * $r2) / (2 * $R2);
        $r2r2 = ($r1 * $r1 - $r2 * $r2);
        $c = sqrt(2 * ($r1 * $r1 + $r2 * $r2) / $R2 - ($r2r2 * $r2r2) / $R4 - 1);

        $fx = ($x1 + $x2) / 2 + $a * ($x2 - $x1);
        $gx = $c * ($y2 - $y1) / 2;
        $ix1 = $fx + $gx;
        $ix2 = $fx - $gx;

        $fy = ($y1 + $y2) / 2 + $a * ($y2 - $y1);
        $gy = $c * ($x1 - $x2) / 2;
        $iy1 = $fy + $gy;
        $iy2 = $fy - $gy;

        // note if gy == 0 and gx == 0 then the circles are tangent and there is only one solution
        // but that one solution will just be duplicated as the code is currently written
        return [[$ix1, $iy1], [$ix2, $iy2]];
    }

Python 3.11.2

import math
"""
x1,y1 is the center of the first circle, with radius r1
x2,y2 is the center of the second ricle, with radius r2
"""
def intersectTwoCircles(x1, y1, r1, x2, y2, r2):
    centerdx = x1 - x2
    centerdy = y1 - y2
    R = math.sqrt(centerdx**2 + centerdy**2)
    if not (abs(r1 - r2) <= R and R <= r1 + r2):
        """ No intersections """
        return []

    """ intersection(s) should exist """
    R2 = R**2
    R4 = R2**2
    a = (r1**2 - r2**2) / (2 * R2)
    r2r2 = r1**2 - r2**2
    c = math.sqrt(2 * (r1**2 + r2**2) / R2 - (r2r2**2) / R4 -1)

    fx = (x1 + x2) / 2 + a * (x2 - x1)
    gx = c * (y2 - y1) / 2
    ix1 = fx + gx
    ix2 = fx - gx

    fy = (y1 + y2) / 2 + a * (y2 - y1)
    gy = c * (x1 - x2) / 2
    iy1 = fy + gy
    iy2 = fy - gy

    return [[ix1, iy1], [ix2, iy2]]

Here is an example using turtletoy which is based on Java Script
see: https://turtletoy.net/turtle/c60ea8510d

// Locate the intersection(s) of 2 circles
// thanks to jupdike/IntersectTwoCircles.js
// https://gist.github.com/jupdike/bfe5eb23d1c395d8a0a1a4ddd94882ac

// You can find the Turtle API reference here: https://turtletoy.net/syntax
Canvas.setpenopacity(1);

const radius = 40; // min=5 max=100 step=1
const X1 = -14; // min=-100 max=100 step=1
const Y1 = -12; // min=-100 max=100 step=1
const X2 = 28; // min=-100 max=100 step=1
const Y2 = 23; // min=-100 max=100 step=1

// Global code will be evaluated once.
const turtle = new Turtle();

centeredCircle(X1, Y1, radius, 360);
centeredCircle(X2, Y2, radius, 360);

array_name = intersectTwoCircles(X1, Y1,radius, X2, Y2 ,radius)

// thanks to jupdike/IntersectTwoCircles.js
// https://gist.github.com/jupdike/bfe5eb23d1c395d8a0a1a4ddd94882ac
// based on the math here:
// http://math.stackexchange.com/a/1367732

// x1,y1 is the center of the first circle, with radius r1
// x2,y2 is the center of the second ricle, with radius r2
function intersectTwoCircles(x1,y1,r1, x2,y2,r2) {

var centerdx = x1 - x2;
var centerdy = y1 - y2;
var R = Math.sqrt(centerdx * centerdx + centerdy * centerdy);
if (!(Math.abs(r1 - r2) <= R && R <= r1 + r2)) { // no intersection
return []; // empty list of results
}
// intersection(s) should exist

var R2 = RR;
var R4 = R2R2;
var a = (r1r1 - r2r2) / (2 * R2);
var r2r2 = (r1r1 - r2r2);
var c = Math.sqrt(2 * (r1r1 + r2r2) / R2 - (r2r2 * r2r2) / R4 - 1);

var fx = (x1+x2) / 2 + a * (x2 - x1);
var gx = c * (y2 - y1) / 2;
var ix1 = fx + gx;
var ix2 = fx - gx;

var fy = (y1+y2) / 2 + a * (y2 - y1);
var gy = c * (x1 - x2) / 2;
var iy1 = fy + gy;
var iy2 = fy - gy;

centeredCircle(ix1, iy1, 2, 360); // highlight intersection point 1
centeredCircle(ix2, iy2, 2, 360); // highlight intersection point 1

// note if gy == 0 and gx == 0 then the circles are tangent and there is only one solution
// but that one solution will just be duplicated as the code is currently written
return [ix1, iy1, ix2, iy2];
}

// thanks to Reinder for this function
// Draws a circle centered a specific x,y location
// and returns the turtle to the original angle after it completes the circle.
function centeredCircle(x,y, radius, ext) {
turtle.penup();
turtle.goto(x,y);
turtle.backward(radius);
turtle.left(90);
turtle.pendown(); turtle.circle(radius, ext);
turtle.right(90); turtle.penup(); turtle.forward(radius); turtle.pendown();
}

Comparing with the math, shouldn't the denominator in line 17 be 2 * R instead of 2 * R2?
(I know this is an old thread, but still clarifying for those who use this as reference)

Never mind, I got confused by the similar notation of the math and the code! 2 * R2 is correct for a

Thanks for posting! Here's a compatible Rust version!

struct Point2 {
    x: f64,
    y: f64,
}

struct Circle2 {
    center: Point2,
    radius: f64,
}


pub fn circle_intersection(&self, circle_a: &Circle2, circle_b: &Circle2) -> Vec<Point2> {
    let center_a = circle_a.center;
    let center_b = circle_b.center;
    let r_a = circle_a.radius;
    let r_b = circle_b.radius;

    let center_dx = center_b.x - center_a.x;
    let center_dy = center_b.y - center_a.y;
    let center_dist = center_dx.hypot(center_dy);

    if !(center_dist <= r_a + r_b && center_dist >= r_a - r_b) {
        return vec![];
    }

    let r_2 = center_dist * center_dist;
    let r_4 = r_2 * r_2;
    let a = (r_a * r_a - r_b * r_b) / (2.0 * r_2);
    let r_2_r_2 = r_a * r_a - r_b * r_b;
    let c = (2.0 * (r_a * r_a + r_b * r_b) / r_2 - r_2_r_2 * r_2_r_2 / r_4 - 1.0).sqrt();

    let fx = (center_a.x + center_b.x) / 2.0 + a * (center_b.x - center_a.x);
    let gx = c * (center_b.y - center_a.y) / 2.0;
    let ix1 = fx + gx;
    let ix2 = fx - gx;

    let fy = (center_a.y + center_b.y) / 2.0 + a * (center_b.y - center_a.y);
    let gy = c * (center_a.x - center_b.x) / 2.0;
    let iy1 = fy + gy;
    let iy2 = fy - gy;

    vec![Point2 { x: ix1, y: iy1 }, Point2 { x: ix2, y: iy2}]
}