计算SVG弧的中心

来自W3C SVG 1.1规范：从端点到中心参数化的转换

你可以看一下详细的解释。

这是一个javascript实现。

// svg : [A | a] (rx ry x-axis-rotation large-arc-flag sweep-flag x y)+
function  radian( ux, uy, vx, vy ) {
    var  dot = ux * vx + uy * vy;
    var  mod = Math.sqrt( ( ux * ux + uy * uy ) * ( vx * vx + vy * vy ) );
    var  rad = Math.acos( dot / mod );
    if( ux * vy - uy * vx < 0.0 ) {
        rad = -rad;
    }
    return rad;
}
//conversion_from_endpoint_to_center_parameterization
//sample :  svgArcToCenterParam(200,200,50,50,0,1,1,300,200)
// x1 y1 rx ry φ fA fS x2 y2
// phi, startAngle, deltaAngle, endAngle are radians not degrees.
// SVG use degrees, convert it to radians by:
// phi = degree * Math.PI / 180;
function svgArcToCenterParam(x1, y1, rx, ry, phi, fA, fS, x2, y2) {
    var cx, cy, startAngle, deltaAngle, endAngle;
    var PIx2 = Math.PI * 2.0;
    if (rx < 0) {
        rx = -rx;
    }
    if (ry < 0) {
        ry = -ry;
    }
    if (rx == 0.0 || ry == 0.0) { // invalid arguments
        throw Error('rx and ry can not be 0');
    }
    // SVG use degrees, if your input is degree from svg,
    // you should convert degree to radian as following line.
    // phi = phi * Math.PI / 180;
    var s_phi = Math.sin(phi);
    var c_phi = Math.cos(phi);
    var hd_x = (x1 - x2) / 2.0; // half diff of x
    var hd_y = (y1 - y2) / 2.0; // half diff of y
    var hs_x = (x1 + x2) / 2.0; // half sum of x
    var hs_y = (y1 + y2) / 2.0; // half sum of y
    // F6.5.1
    var x1_ = c_phi * hd_x + s_phi * hd_y;
    var y1_ = c_phi * hd_y - s_phi * hd_x;
    // F.6.6 Correction of out-of-range radii
    //   Step 3: Ensure radii are large enough
    var lambda = (x1_ * x1_) / (rx * rx) + (y1_ * y1_) / (ry * ry);
    if (lambda > 1) {
        rx = rx * Math.sqrt(lambda);
        ry = ry * Math.sqrt(lambda);
    }
    var rxry = rx * ry;
    var rxy1_ = rx * y1_;
    var ryx1_ = ry * x1_;
    var sum_of_sq = rxy1_ * rxy1_ + ryx1_ * ryx1_; // sum of square
    if (!sum_of_sq) {
        throw Error('start point can not be same as end point');
    }
    var coe = Math.sqrt(Math.abs((rxry * rxry - sum_of_sq) / sum_of_sq));
    if (fA == fS) { coe = -coe; }
    // F6.5.2
    var cx_ = coe * rxy1_ / ry;
    var cy_ = -coe * ryx1_ / rx;
    // F6.5.3
    cx = c_phi * cx_ - s_phi * cy_ + hs_x;
    cy = s_phi * cx_ + c_phi * cy_ + hs_y;
    var xcr1 = (x1_ - cx_) / rx;
    var xcr2 = (x1_ + cx_) / rx;
    var ycr1 = (y1_ - cy_) / ry;
    var ycr2 = (y1_ + cy_) / ry;
    // F6.5.5
    startAngle = radian(1.0, 0.0, xcr1, ycr1);
    // F6.5.6
    deltaAngle = radian(xcr1, ycr1, -xcr2, -ycr2);
    while (deltaAngle > PIx2) { deltaAngle -= PIx2; }
    while (deltaAngle < 0.0) { deltaAngle += PIx2; }
    if (fS == false || fS == 0) { deltaAngle -= PIx2; }
    endAngle = startAngle + deltaAngle;
    while (endAngle > PIx2) { endAngle -= PIx2; }
    while (endAngle < 0.0) { endAngle += PIx2; }
    var outputObj = { /* cx, cy, startAngle, deltaAngle */
        cx: cx,
        cy: cy,
        startAngle: startAngle,
        deltaAngle: deltaAngle,
        endAngle: endAngle,
        clockwise: (fS == true || fS == 1)
    }
    return outputObj;
}

用法示例：

svg

<path d="M 0 100 A 60 60 0 0 0 100 0"/>

js

var result = svgArcToCenterParam(0, 100, 60, 60, 0, 0, 0, 100, 0);
console.log(result);
/* will output:
{
    cx: 49.99999938964844,
    cy: 49.99999938964844,
    startAngle: 2.356194477985314,
    deltaAngle: -3.141592627780225,
    endAngle: 5.497787157384675,
    clockwise: false
}
*/

我正在考虑x轴旋转=0的情况。起点和终点方程式：

x1=cx+rx*cos（起始角度）

y1=cy+ry*sin（起始角度）

x2=cx+rx*cos（EndAngle）

y2=cy+ry*sin（EndAngle）

从方程组中排除角度得出：

ry^2*（x1 cx）^2+rx^2*（y1 cy）^2=rx^2*ry^2

ry^2*（x2 cx）^2+rx^2*（y2 cy）^2=rx^2*ry^2

这个方程组可以用手或借助数学包（Maple、Mathematica等）解析求解（cx，cy）。二次方程有两种解（由于大弧标志和扫频标志的组合）。