根据SVG Arc求出其开始角、摆动角和椭圆圆心

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撰写于 2022年 01月 22 日

SVG Arc

目前Svg的Arc的参数字符串如下:

a  rx  ry  x-axis-rotation  large-arc-flag  sweep-flag  x  y

除了a表示标识为Arc之外,其余参数说明如下:

求Arc的开始角和摆动角

实际上，在W3C的有关SVG Arc实现有相关文档和公式

当已知参数：

x1 y1 x2 y2 fA fS rx ry φ

求出以下参数的值：

cx cy θ1 Δθ

其中已知参数说明如下：

需要求的参数说明：

那么则有如下公式：

代码如下：

        /// 

/// 获取弧线的开始角度和摆动角度
/// 

/// 起点X
/// 起点Y
/// 终点X
/// 终点Y
/// 优劣弧:1 优弧  0劣弧
/// 顺逆时针绘制：1 顺时针  0 逆时针
/// 椭圆半长轴
/// 椭圆半短轴
/// 旋转角
/// 
private static (double startAngle, double swAngle) GetArcStartAngAndSwAng(double x1, double y1, double x2, double y2, double fA, double fs, double rx, double ry, double φ)
{

            var matrix1 = new Matrix { M11 = Math.Cos(φ), M12 = Math.Sin(φ), M21 = -Math.Sin(φ), M22 = Math.Cos(φ) };
var matrix2 = new Matrix { M11 = (x1 - x2) / 2, M21 = (y1 - y2) / 2 };
var matrixX1Y1 = Matrix.Multiply(matrix1, matrix2);

            var x1_ = matrixX1Y1.M11;
var y1_ = matrixX1Y1.M21;

            var a = Math.Pow(rx, 2) * Math.Pow(ry, 2) - Math.Pow(rx, 2) * Math.Pow(y1_, 2) - Math.Pow(ry, 2) * Math.Pow(x1_, 2);
var b = Math.Pow(ry, 2) * Math.Pow(y1_, 2) + Math.Pow(ry, 2) * Math.Pow(x1_, 2);

            double c = 0;
if (fA == fs)
{
c = -Math.Sqrt(a / b);
}
else
{
c = Math.Sqrt(a / b);
}

            var matrixCxCy = new Matrix { M11 = c * (rx * y1_ / ry), M21 = c * (-ry * x1_ / rx) };

            var cx_ = matrixCxCy.M11;
var cy_ = matrixCxCy.M21;

这时候我们通过矩阵运算得到了矩阵x1y1和矩阵cxcy，然后还有以下公式求开始角和摆动角：

那么代码如下：

             //求开始角
//cos<夹角> = 两向量之积 / 两向量模的乘积
//< 夹角 > = arcCos(两向量之积 / 两向量模的乘积)

            //向量U的坐标
double vectorUx = 1;
double vectorUy = 0;

            //向量V的坐标
double vectorVx = (x1_ - cx_) / rx;
double vectorVy = (y1_ - cy_) / ry;

            var multiVectorUVectorV = vectorUx * vectorVx + vectorUy * vectorVy; //两向量的乘积
var vectorUMod = Math.Sqrt(vectorUx * vectorUx + vectorUy * vectorUy);//向量U的模
var vectorVMod = Math.Sqrt(vectorVx * vectorVx + vectorVy * vectorVy);//向量V的模
var cosResult = multiVectorUVectorV / (vectorUMod * vectorVMod);

            var startAngle = Math.Acos(cosResult) * 180 / Math.PI;

            //求摆动角
//cos<夹角> = 两向量之积 / 两向量模的乘积
//< 夹角 > = arcCos(两向量之积 / 两向量模的乘积)

            //向量U的坐标
vectorUx = (x1_ - cx_) / rx;
vectorUy = (y1_ - cy_) / ry;

            //向量V的坐标
vectorVx = (-x1_ - cx_) / rx;
vectorVy = (-y1_ - cy_) / ry;

            multiVectorUVectorV = vectorUx * vectorVx + vectorUy * vectorVy; //两向量的乘积
vectorUMod = Math.Sqrt(vectorUx * vectorUx + vectorUy * vectorUy);//向量U的模
vectorVMod = Math.Sqrt(vectorVx * vectorVx + vectorVy * vectorVy);//向量V的模
cosResult = multiVectorUVectorV / (vectorUMod * vectorVMod);

            var swAngle = Math.Acos(cosResult) * 180 / Math.PI;

            if (fs == 0)
{
swAngle = -swAngle;
}
else
{
swAngle = Math.Abs(swAngle);
}

那么我们来测试下，我准备了一段Arc字符串：

"M0,0 A18.10005249343832,16.00031496062992,60,0,0,-21.634424410598417,-21.472913522584044"

然后测试代码如下：

private void ButtonBase_OnClick(object sender, RoutedEventArgs e)
{
var pathGeometry = PathGeometry.CreateFromGeometry(Geometry.Parse("M0,0 A18.10005249343832,16.00031496062992,60,0,0,-21.634424410598417,-21.472913522584044"));
var pathFigure = pathGeometry.Figures[0];

            if (pathFigure.Segments[0] is ArcSegment arcSegment)
{
var x1 = pathFigure.StartPoint.X;
var y1 = pathFigure.StartPoint.Y;
var rx = arcSegment.Size.Width;
var ry = arcSegment.Size.Height;
var φ = arcSegment.RotationAngle;
var fA = arcSegment.IsLargeArc ? 1 : 0;
var fs = arcSegment.SweepDirection is SweepDirection.Clockwise ? 1 : 0;
var x2 = arcSegment.Point.X;
var y2 = arcSegment.Point.Y;

                var (startAngle, swAngle) = GetArcStartAngAndSwAng(x1, y1, x2, y2, fA, fs, rx, ry, φ);
//算出来接近startAngle为179°，swAngle为-118°

                StringBuilder stringPath = new StringBuilder();
stringPath.Append($"M {x1} {y1}");
var openXmlArcToArcStrNew = SvgArcToArcStr(stringPath, rx, ry, φ, startAngle, swAngle, pathFigure.StartPoint);
this.NewPath.Data = Geometry.Parse(openXmlArcToArcStrNew);
}

        }

然后我们再通过求出来的开始角和摆动角求出之前的那段Arc:

         /// 

/// OpenXml Arc 转为SVG Arc 字符串
/// 

/// 路径字符串
/// 椭圆半长轴
/// 椭圆半短轴
/// 旋转角
/// 起始角
/// 摆动角
/// 当前坐标
/// 
private string SvgArcToArcStr(StringBuilder stringPath, double rx, double ry, double φ, double stAng, double swAng, Point currentPoint)
{
const string comma = ",";

            var θ1 = stAng / 180 * Math.PI;
var Δθ = swAng / 180 * Math.PI;
//是否是大弧
var isLargeArcFlag = Math.Abs(Δθ) > Math.PI;
//是否是顺时针
var isClockwise = Δθ > 0;

            //修复当椭圆弧线进行360°时，起始点和终点一样，会导致弧线变成点，因此-1°才进行计算
if (System.Math.Abs(Δθ) == 2 * System.Math.PI)
{
Δθ = Δθ - Δθ / 360;
}

            //获取终点坐标
var pt = GetArcArbitraryPoint(rx, ry, Δθ, θ1, φ, currentPoint);

            currentPoint = pt;

            // 格式如下
// A rx ry x-axis-rotation large-arc-flag sweep-flag x y
// 这里 large-arc-flag 是 1 和 0 表示
stringPath.Append("A")
.Append(rx) //rx
.Append(comma)
.Append(ry) //ry
.Append(comma)
.Append(φ) // x-axis-rotation
.Append(comma)
.Append(isLargeArcFlag ? "1" : "0") //large-arc-flag
.Append(comma)
.Append(isClockwise ? "1" : "0") // sweep-flag
.Append(comma)
.Append(pt.X)
.Append(comma)
.Append(pt.Y)
.Append(' ');
return stringPath.ToString();

        }

        /// 

/// 获取椭圆任意一点坐标(终点)
/// 

/// 椭圆半长轴
/// 椭圆半短轴
/// 摆动角度(起始角的摆动角度，也就是起始角+摆动角=结束角)
/// 起始角
/// 旋转角
/// 起点
/// 
private static Point GetArcArbitraryPoint(double rx, double ry, double Δθ, double θ1, double φ, Point currentPoint)
{
//开始点的椭圆任意一点的二维矩阵方程式
var matrixX1Y1 = new Matrix { M11 = currentPoint.X, M21 = currentPoint.Y };

            var matrix1 = new Matrix { M11 = Math.Cos(φ), M12 = -Math.Sin(φ), M21 = Math.Sin(φ), M22 = Math.Cos(φ) };
var matrix2 = new Matrix { M11 = rx * Math.Cos(θ1), M21 = ry * Math.Sin(θ1) };
var multiplyMatrix1Matrix2 = Matrix.Multiply(matrix1, matrix2);
var matrixCxCy = new Matrix { M11 = matrixX1Y1.M11 - multiplyMatrix1Matrix2.M11, M21 = matrixX1Y1.M21 - multiplyMatrix1Matrix2.M21 };

            //终点的椭圆任意一点的二维矩阵方程式
var matrix3 = new Matrix { M11 = rx * Math.Cos(θ1 + Δθ), M21 = ry * Math.Sin(θ1 + Δθ) };
var multiplyMatrix1Matrix3 = Matrix.Multiply(matrix1, matrix3);
var matrixX2Y2 = new Matrix { M11 = multiplyMatrix1Matrix3.M11 + matrixCxCy.M11, M21 = multiplyMatrix1Matrix3.M21 + matrixCxCy.M21 };

            return new Point(matrixX2Y2.M11, matrixX2Y2.M21);
}

效果如下：

可以看到根据算出来的开始角和摆动角，再带入计算出来的弧线(关于计算弧线的算法可以参考我之前的博客)是跟之前的弧线一致的，也间接验证了算法的准确性

求Arc的椭圆圆心

求圆心公式如下：

则代码如下：

        /// 

/// 获取弧线的椭圆圆心
/// 

/// 起点X
/// 起点Y
/// 终点X
/// 终点Y
/// 优劣弧:1 优弧  0劣弧
/// 顺逆时针绘制：1 顺时针  0 逆时针
/// 椭圆半长轴
/// 椭圆半短轴
/// 旋转角
/// 
private static Point GetArcCenterPoint(double x1, double y1, double x2, double y2, double fA, double fs, double rx, double ry, double φ)
{

            var matrix1 = new Matrix { M11 = Math.Cos(φ), M12 = Math.Sin(φ), M21 = -Math.Sin(φ), M22 = Math.Cos(φ) };
var matrix2 = new Matrix { M11 = (x1 - x2) / 2, M21 = (y1 - y2) / 2 };
var matrixX1Y1 = Matrix.Multiply(matrix1, matrix2);

            var x1_ = matrixX1Y1.M11;
var y1_ = matrixX1Y1.M21;

            var a = Math.Pow(rx, 2) * Math.Pow(ry, 2) - Math.Pow(rx, 2) * Math.Pow(y1_, 2) - Math.Pow(ry, 2) * Math.Pow(x1_, 2);
var b = Math.Pow(ry, 2) * Math.Pow(y1_, 2) + Math.Pow(ry, 2) * Math.Pow(x1_, 2);

            double c = 0;
if (fA == fs)
{
c = -Math.Sqrt(a / b);
}
else
{
c = Math.Sqrt(a / b);
}

            var matrixCx_Cy_ = new Matrix { M11 = c * (rx * y1_ / ry), M21 = c * (-ry * x1_ / rx) };

            var tempMatrix = new Matrix { M11 = Math.Cos(φ), M12 = -Math.Sin(φ), M21 = Math.Sin(φ), M22 = Math.Cos(φ) };
var multiplyMatrix = Matrix.Multiply(tempMatrix, matrixCx_Cy_);

            var matrixCxCy=new Matrix(){M11 = multiplyMatrix.M11+((x1+x2)/2),M21= multiplyMatrix.M21+((y1+y2)/2) };

            return new Point(matrixCxCy.M11, matrixCxCy.M21);

        }

最终通过上面Svg Arc字符串算出来的椭圆圆心为(-17.42169108128391,-5.368374418803782)

源码

BlogCodeSample/OpenxmlActToSvgSample at main · ZhengDaoWang/BlogCodeSample

参考

Implementation Notes — SVG 2

根据SVG Arc求出其开始角、摆动角和椭圆圆心的