SVG Arc
目前Svg的Arc的引數字串如下:
a rx ry x-axis-rotation large-arc-flag sweep-flag x y
除了a表示標識為Arc之外,其餘引數說明如下:
| 引數 | 說明 |
|---|---|
| rx | 橢圓半長軸 |
| ry | 橢圓半短軸 |
| x-axis-rotation | 橢圓相對於座標系的旋轉角度，角度數而非弧度數 |
| large-arc-flag | 是否優(大)弧：0否，1是 |
| sweep-flag | 繪製方向：0逆時針，1順時針 |
| x | 圓弧終點的x座標 |
| y | 圓弧終點的y座標 |
求Arc的開始角和擺動角
實際上，在W3C的有關SVG Arc實現有相關文件和公式
當已知引數：
x1 y1 x2 y2 fA fS rx ry φ
求出以下引數的值：
cx cy θ1 Δθ
其中已知引數說明如下：
| 引數 | 說明 | 備註 |
|---|---|---|
| (x1,y1) | 當前座標 | |
| (x2,y2) | 終點座標 | |
| fA | 是否優(大)弧 | Arc的引數字元：large-arc-flag |
| fS | 繪製方向 | Arc的引數字元：sweep-flag |
| rx | 橢圓半長軸 | Arc的引數字元：rx |
| ry | 橢圓半短軸 | Arc的引數字元：ry |
| φ | 橢圓相對於座標系的旋轉角度 | Arc的引數字元：x-axis-rotation |
需要求的引數說明：
| 引數 | 說明 | 備註 |
|---|---|---|
| (cx,cy) | 橢圓中心座標點 | |
| θ1 | 起始角 | |
| Δθ | 起始角到結束角的夾角(擺動角) | 結束角= 起始角θ1+擺動角Δθ |
那麼則有如下公式：
程式碼如下：
/// <summary>
/// 獲取弧線的開始角度和擺動角度
/// </summary>
/// <param name="x1">起點X</param>
/// <param name="y1">起點Y</param>
/// <param name="x2">終點X</param>
/// <param name="y2">終點Y</param>
/// <param name="fA">優劣弧:1 優弧 0劣弧</param>
/// <param name="fs">順逆時針繪製：1 順時針 0 逆時針</param>
/// <param name="rx">橢圓半長軸</param>
/// <param name="ry">橢圓半短軸</param>
/// <param name="φ">旋轉角</param>
/// <returns></returns>
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;
這時候我們通過矩陣運算得到了矩陣x1y1和矩陣cxcy，然後還有以下公式求開始角和擺動角：
那麼程式碼如下：
//求開始角
//cos<夾角> = 兩向量之積 / 兩向量模的乘積
//< 夾角 > = arcCos(兩向量之積 / 兩向量模的乘積)
//向量U的座標
double vectorUx = 1;
double vectorUy = 0;
//向量V的座標
double vectorVx = (x1_ - cx_) / rx;
double vectorVy = (y1_ - cy_) / ry;
var multiVectorUVectorV = vectorUx * vectorVx + vectorUy * vectorVy; //兩向量的乘積
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;
//求擺動角
//cos<夾角> = 兩向量之積 / 兩向量模的乘積
//< 夾角 > = arcCos(兩向量之積 / 兩向量模的乘積)
//向量U的座標
vectorUx = (x1_ - cx_) / rx;
vectorUy = (y1_ - cy_) / ry;
//向量V的座標
vectorVx = (-x1_ - cx_) / rx;
vectorVy = (-y1_ - cy_) / ry;
multiVectorUVectorV = vectorUx * vectorVx + vectorUy * vectorVy; //兩向量的乘積
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);
}
那麼我們來測試下，我準備了一段Arc字串：
"M0,0 A18.10005249343832,16.00031496062992,60,0,0,-21.634424410598417,-21.472913522584044"
然後測試程式碼如下：
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);
//算出來接近startAngle為179°，swAngle為-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);
}
}
然後我們再通過求出來的開始角和擺動角求出之前的那段Arc:
/// <summary>
/// OpenXml Arc 轉為SVG Arc 字串
/// </summary>
/// <param name="stringPath">路徑字串</param>
/// <param name="rx">橢圓半長軸</param>
/// <param name="ry">橢圓半短軸</param>
/// <param name="φ">旋轉角</param>
/// <param name="stAng">起始角</param>
/// <param name="swAng">擺動角</param>
/// <param name="currentPoint">當前座標</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;
//是否是順時針
var isClockwise = Δθ > 0;
//修復當橢圓弧線進行360°時，起始點和終點一樣，會導致弧線變成點，因此-1°才進行計算
if (System.Math.Abs(Δθ) == 2 * System.Math.PI)
{
Δθ = Δθ - Δθ / 360;
}
//獲取終點座標
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
// 這裡 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>
/// 獲取橢圓任意一點座標(終點)
/// </summary>
/// <param name="rx">橢圓半長軸</param>
/// <param name="ry">橢圓半短軸</param>
/// <param name="Δθ">擺動角度(起始角的擺動角度，也就是起始角+擺動角=結束角)</param>
/// <param name="θ1">起始角</param>
/// <param name="φ">旋轉角</param>
/// <param name="currentPoint">起點</param>
/// <returns></returns>
private static Point GetArcArbitraryPoint(double rx, double ry, double Δθ, double θ1, double φ, Point currentPoint)
{
//開始點的橢圓任意一點的二維矩陣方程式
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 };
//終點的橢圓任意一點的二維矩陣方程式
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：
可以看到根據算出來的開始角和擺動角，再帶入計算出來的弧線(關於計算弧線的演算法可以參考我之前的部落格)是跟之前的弧線一致的，也間接驗證了演算法的準確性
求Arc的橢圓圓心
求圓心公式如下：
則程式碼如下：
/// <summary>
/// 獲取弧線的橢圓圓心
/// </summary>
/// <param name="x1">起點X</param>
/// <param name="y1">起點Y</param>
/// <param name="x2">終點X</param>
/// <param name="y2">終點Y</param>
/// <param name="fA">優劣弧:1 優弧 0劣弧</param>
/// <param name="fs">順逆時針繪製：1 順時針 0 逆時針</param>
/// <param name="rx">橢圓半長軸</param>
/// <param name="ry">橢圓半短軸</param>
/// <param name="φ">旋轉角</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);
}
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);
}
最終通過上面Svg Arc字串算出來的橢圓圓心為(-17.42169108128391,-5.368374418803782)
原始碼
BlogCodeSample/OpenxmlActToSvgSample at main · ZhengDaoWang/BlogCodeSample