Question

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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，最好的解释是找到here（两个圆圈的交点），但我并不真正理解数学，所以我无法实现它。

例如d = || P1 - P0 ||什么做了||代表？这是否意味着结果数总是正数？

并且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会给你63，所以它只是忽略了第一个数字，那么假设会发生什么？结果应该是（23,63）这里还是？还有P1-P0部分或（40,30） - （10,60），你如何在JavaScript中表达它？

Answer 1

将网站上的C函数翻译为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

根据 AutoMonkey 的回答，我做了一些调整，以在响应中提供更多信息，并在圆圈相同时进行处理。

/**
 * @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;
}

用法：

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 },
}

确认：

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

一个JavaScript函数，它返回两个圆之间的x，y交点？

2 个答案: