I am attempting to calculate the intersection of two circle on the Earth with a given latitude, longitude and radius. I started with this Calculating intersection of two Circles?

These circles will never be more than 100 km in diameter and will normally be "close" to each other, meaning two intersecting points for the pair of circles.

I added a comment in the code below at the place where I am having the issue.

I originally asked this question on StackOverflow, but with this being so geospatial, I did not get any answers.

var circle1 = {lat:37.673442, lon:-90.234036, r:107.5};
var circle2 = {lat:36.109997, lon:-90.953669, r:145};

function findInterscetionOfTwoCircles(circle1, circle2){
    var c1 = latLonToGeocentricCoords(circle1);
    var c2 = latLonToGeocentricCoords(circle2);
    var r1 = convertRadius(circle1.r);
    var r2 = convertRadius(circle2.r);
    var c1dotc2 = calculateDotProduct(c1, c2);

    // This is where the problem is occurring.
    // The value of 'a' should be 0.9735030 but I am getting 46.770178816522844.
    // I am certain that r1 and r2 are correct (match the values in the original post). Additionally, I believe the calculateDotProduct function to be
    // correct, therefore making c1dotc2 correct. I think I have ruled out
    // any Order of Operations error. Maybe it is a degree/radian conversion
    // issue or something like it? I tried different variations and I  and now out
    // of ideas.
    var a = (Math.cos(r1) - (Math.cos(r2) * c1dotc2)) / (1 - (c1dotc2 * c1dotc2)); 
    var b = (Math.cos(r2) - (Math.cos(r1) * c1dotc2)) / (1 - (c1dotc2 * c1dotc2)); 
    // -45.83805010474024, should be  0.0260194
    var n = calculateCrossProduct(c1, c2); 

    var x0 = calculateLinearCombination(a, b, c1, c2);

    var ndotn = calculateDotProduct(n, n);

    // var t = Math.sqrt((1 - calculateDotProduct(x0,x0))/calculateDotProduct(n,n));

    // Finish the algorithm

function latLonToGeocentricCoords(circle){
    var x = Math.cos(degToRad(circle.lon)) * Math.cos(degToRad(circle.lat));
    var y = Math.sin(degToRad(circle.lon)) * Math.cos(degToRad(circle.lat));
    var z = Math.sin(degToRad(circle.lat));

    return {x:x, y:y, z:z};

function geocentricCoordsToLatLon(x, y, z){
    var lon = Math.atan2(x,y)
    var lat = Math.atan2(Math.sqrt((x*x)+(y*y)), z)

    return { lat: lat, lon: lon};

function convertRadius(radius){
    // converts NM right now
    return degToRad(radius) / 60; 

function calculateDotProduct(input1, input2){
    if(arguments.length !== 2){
        throw new Error('You must supply two arguments to this method.');

    if(Array.isArray(input1) !== Array.isArray(input2) || typeof input1 !== typeof input2){
        throw new Error('Inputs must be the same types.');

    var dotProduct = 0;
        // process as parallel arrays
        if(input1.length === input2.length){
            throw new Error('Input lengths must be the same.');
        for(var i = 0, len = input1.length; i < len; i++){
            dotProduct += input1[i] * input2[i];
    } else if(typeof input1 === 'object' && !Array.isArray(input1)){
        // process as key-value object
        for(var key in input1){
            if (input1.hasOwnProperty(key)) {
                    throw new Error('Both inputs must have the same properties to be processed.');
                dotProduct += input1[key] * input2[key];


    return dotProduct; 

function calculateCrossProduct(circle1, circle2){
    var x = (circle1.y * circle2.z) - (circle1.z * circle2.y);
    var y = (circle1.z * circle2.x) - (circle1.x * circle2.z);
    var z = (circle1.x * circle2.y) - (circle1.y * circle2.x);

    return {x: x, y: y, z: z};

function calculateLinearCombination(a, b, c1, c2){
    var x = {x: a * c1.x, y: a * c1.y, z: a * c1.z};
    var y = {x: b * c2.x, y: b * c2.y, z: b * c2.z};

    return calculateDotProduct(x, y);

function degToRad(degree){
    return degree * Math.PI / 180;

function radToDeg(radian){
    return radian / Math.PI * 180;
  • you have mentioned radius as r=107.5 and r=145. Is radius in km or m? If 'r' is in km then how diameter will be less than 100 km?
    – Ghori
    Commented Jul 30, 2015 at 5:35
  • The units in nautical miles. Commented Jul 30, 2015 at 13:34
  • 2
    Math.cos etc. expect their arguments to be in radians, not degrees.. However, your function convertRadians clearly does not return radians (nor does it even return "NM", if that's intended to be nautical miles).
    – whuber
    Commented Jul 30, 2015 at 15:51
  • 1
    @whuber Thank you for the feedback. Based on what you said, I realized my mistake and was able to correct it. The name of the name was just a typo, but in the convertRadius function I was multiplying by 6 instead of 60. Anyways, your feedback (and you original post) was extremely helpful. Commented Jul 31, 2015 at 17:57

1 Answer 1


circle1 with r=107.5 naut.mile and circle2 with r=145 naut.mile intersect at points as shown in the image: enter image description here

One should adopt projected (Cartesian) coordinate system for this type of work. I have worked on UTM-Zone 15 because both circle center lie in this zone. If u try to do this manually/mathematically using polar coordinate system, it will be much complicated due to imperfect earth shape. One can perform more accurate calculation by adopting best projection/coordinate system for this area.


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