/// 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(rx, 2) * Math.Pow(y1_, 2) + Math.Pow(ry, 2) * Math.Pow(x1_, 2); double c = 0; if (fA == fs) c = -Math.Sqrt(a / b); 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;
                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:
         /// <summary>
        /// OpenXml Arc 转为SVG Arc 字符串
        /// </summary>
        /// <param name="stringPath">路径字符串</param>
        /// <param name="rx">椭圆半长轴</param>
        /// <param name="ry">椭圆半短轴</param>
        /// <param name="φ">旋转角</param>
        /// <param name="stAng">起始角</param>
        /// <param name="swAng">摆动角</param>
        /// <param name="currentPoint">当前坐标</param>
        /// <returns></returns>
        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();
        /// <summary>
        /// 获取椭圆任意一点坐标(终点)
        /// </summary>
        /// <param name="rx">椭圆半长轴</param>
        /// <param name="ry">椭圆半短轴</param>
        /// <param name="Δθ">摆动角度(起始角的摆动角度，也就是起始角+摆动角=结束角)</param>
        /// <param name="θ1">起始角</param>
        /// <param name="φ">旋转角</param>
        /// <param name="currentPoint">起点</param>
        /// <returns></returns>
        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：
        /// <summary>
        /// 获取弧线的椭圆圆心
        /// </summary>
        /// <param name="x1">起点X</param>
        /// <param name="y1">起点Y</param>
        /// <param name="x2">终点X</param>
        /// <param name="y2">终点Y</param>
        /// <param name="fA">优劣弧:1 优弧  0劣弧</param>
        /// <param name="fs">顺逆时针绘制：1 顺时针  0 逆时针</param>
        /// <param name="rx">椭圆半长轴</param>
        /// <param name="ry">椭圆半短轴</param>
        /// <param name="φ">旋转角</param>
        /// <returns></returns>
        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);
                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)
BlogCodeSample/OpenxmlActToSvgSample at main · ZhengDaoWang/BlogCodeSample
Implementation Notes — SVG 2