A JavaScript function that returns the x,y points of intersection between two circles?

Question

I got the (x,y) center location of two circles and their radius but I need to find their intersection points (marked with red) using JavaScript.

I think the best explanation as far as the math is concerned is found here (Intersection of two circles), but I don't really understand the math so I'm not able to implement it.

For example d = ||P1 - P0|| , what do the || stand for? Does it mean that the resulting number is always a positive?

And also P2 = P0 + a ( P1 - P0 ) / d , aren't the P's here something like (10, 50)? But doing (10,50)+13 in JavaScript gives you 63, so it just ignores the first number, so what's suppose to happen? Should the outcome be (23,63) here or? And also the P1-P0 part or (40,30)-(10,60), how do you express that in JavaScript?

Answer 1

Translated the C function on the site to JavaScript:

function intersection(x0, y0, r0, x1, y1, r1) {
        var a, dx, dy, d, h, rx, ry;
        var 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 = Math.sqrt((dy*dy) + (dx*dx));

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

        /* '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 = Math.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. */
        var xi = x2 + rx;
        var xi_prime = x2 - rx;
        var yi = y2 + ry;
        var yi_prime = y2 - ry;

        return [xi, xi_prime, yi, yi_prime];
    }

Answer 2

Based on AutoMonkey's answer I made some adjustments to give some more information in the response and also handle when the circles are the same.

/**
 * @description Get information about the intersection points of a circle.
 * Adapted from: https://stackoverflow.com/a/12221389/5553768.
 * @param {Object} c1 An object describing the first circle.
 * @param {float} c1.x The x coordinate of the circle.
 * @param {float} c1.y The y coordinate of the circle.
 * @param {float} c1.r The radius of the circle.
 * @param {Object} c2 An object describing the second circle.
 * @param {float} c2.x The x coordinate of the circle.
 * @param {float} c2.y The y coordinate of the circle.
 * @param {float} c2.r The radius of the circle.
 * @returns {Object} Data about the intersections of the circles.
 */
function intersection(c1, c2) {
    // Start constructing the response object.
    const result = {
        intersect_count: 0,
        intersect_occurs: true,
        one_is_in_other: false,
        are_equal: false,
        point_1: { x: null, y: null },
        point_2: { x: null, y: null },
    };

    // Get vertical and horizontal distances between circles.
    const dx = c2.x - c1.x;
    const dy = c2.y - c1.y;

    // Calculate the distance between the circle centers as a straight line.
    const dist = Math.hypot(dy, dx);

    // Check if circles intersect.
    if (dist > c1.r + c2.r) {
        result.intersect_occurs = false;
    }

    // Check one circle isn't inside the other.
    if (dist < Math.abs(c1.r - c2.r)) {
        result.intersect_occurs = false;
        result.one_is_in_other = true;
    }

    // Check if circles are the same.
    if (c1.x === c2.x && c1.y === c2.y && c1.r === c2.r) {
        result.are_equal = true;
        result.are_equal = true;
    }

    // Find the intersection points
    if (result.intersect_occurs) {
        // Centroid is the pt where two lines cross. A line between the circle centers
        // and a line between the intersection points.
        const centroid = (c1.r * c1.r - c2.r * c2.r + dist * dist) / (2.0 * dist);

        // Get the coordinates of centroid.
        const x2 = c1.x + (dx * centroid) / dist;
        const y2 = c1.y + (dy * centroid) / dist;

        // Get the distance from centroid to the intersection points.
        const h = Math.sqrt(c1.r * c1.r - centroid * centroid);

        // Get the x and y dist of the intersection points from centroid.
        const rx = -dy * (h / dist);
        const ry = dx * (h / dist);

        // Get the intersection points.
        result.point_1.x = Number((x2 + rx).toFixed(15));
        result.point_1.y = Number((y2 + ry).toFixed(15));

        result.point_2.x = Number((x2 - rx).toFixed(15));
        result.point_2.y = Number((y2 - ry).toFixed(15));

        // Add intersection count to results
        if (result.are_equal) {
            result.intersect_count = null;
        } else if (result.point_1.x === result.point_2.x && result.point_1.y === result.point_2.y) {
            result.intersect_count = 1;
        } else {
            result.intersect_count = 2;
        }
    }
    return result;
}

Usage:

intersection(
    {x: 1, y: 1, r: 2},
    {x: 0, y: -1, r: 1}
)

// Result
result = {
    intersect_count: 2,
    intersect_occurs: true,
    one_is_in_other: false,
    are_equal: false,
    point_1: { x: 1, y: -1 },
    point_2: { x: -0.6, y: -0.2 },
}

Confirmation:

https://www.desmos.com/calculator/cj12hc9jsk