javascript - 返回两个圆之间的 x,y 交点的 JavaScript 函数？

Question

我得到了两个圆的(x，y)中心位置及其半径，但我需要使用 JavaScript 找到它们的交点(用红色标记)。

我认为就数学而言最好的解释是 here (两个圆圈的交集)，但我不太了解数学，所以我无法实现它。

例如 d = ||P1 - P0|| , 做什么 ||代表？这是否意味着结果数字始终为正数？

还有 P2 = P0 + a ( P1 - P0 )/d ，这里的 P 不是 (10, 50) 吗？但是在 JavaScript 中执行 (10,50)+13 会得到 63，所以它只是忽略了第一个数字，那么会发生什么？结果应该是(23,63)还是？还有 P1-P0 部分或 (40,30)-(10,60)，你如何在 JavaScript 中表达？

Answer 1

将网站上的 C 函数翻译成 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];
    }