三点绘制圆弧及C#程序设计

1.   三点绘制圆

   已知三点为（x₁，y₁)、（x₂，y₂)，（x₃，y₃)，求三点形成的圆的坐标，可以设求得的圆的原点为（X，Y），半径为R，这样可以得到方程组：

 (x₁-X)²-(y₁-Y)²=R²      (1)式
 (x₂-X)²-(y₂-Y)²=R²     (2)式
 (x₃-X)²-(y₃-Y)²=R²     (3)式

由上述方程组可以得到：

 2(x₁-x₂)X +2(y₁-y₂)Y= x₁²-x₂²+y₁²-y₂²        (5)式

 2(x₁-x₃)X +2(y₁-y₃)Y= x₁²-x₃²+y₁²-y₃²        (6)式

此时，设：

   A= x₁-x₂

      B = y₁-y₂

   C= x₁-x₃

   D = y₁-y₃

E= (x₁²-x₂²+y₁²-y₂²)/2

   F= (x₁²-x₃²+y₁²-y₃²)/2

得到方程组：

AX+BY = E

CX+DY = F

求解，得到

X= -DE-BF）/ (BC-AD)

Y = -(AF-CE)/ (BC-AD)

经过计算，得到了圆心的坐标位置，也就得到了圆的半径计算公式

R = sqrt（（x1-X）（x1-X）+(y1-Y) (y1-Y)）;

2.   在圆上确定三点的角度

  根据四个象限的坐标，选定cos作为角度的值，并利用sin值扩展到360度全象限，利用三个点的位置关系确定弧度为顺时针还是逆时针绘制弧线，其中某点（x0， y0）：

Sin = （y0-Y）/R

Cos=  (x0-X）/R

3.   绘制弧线

在C#中，可以利用DrawArc()绘制弧线，其中最重要的要注意角度，绘制时角度要沿顺时针旋转，角度0度位于X轴的正方向上，在绘制弧度时，确定三个点的位置关系十分重要。

4．C#程序源码

void DrawArcFromThreePoint(Graphics mImgGraph,int x1, int y1, int x2, int y2, int x3, int y3)

{

double a = x1 - x2;

double b = y1 - y2;

double c = x1 - x3;

double d = y1 - y3;

double e = ((x1 * x1 - x2 * x2) + (y1 * y1 - y2 * y2)) / 2.0;

double f = ((x1 * x1 - x3 * x3) + (y1 * y1 - y3 * y3)) / 2.0;

 

double det = b * c - a * d;

if( Math.Abs(det) >0.001)

{

//x0,y0为计算得到的原点

double x0 = -(d * e - b * f) / det;

double y0 = -(a * f - c * e) / det;

 

SolidBrush OriginBrush = newSolidBrush(Color.Blue);

mImgGraph.FillEllipse(OriginBrush,(int)(x0-3),(int)(y0-3),6,6);

double radius = Math.Sqrt((x1 - x0)*(x1 - x0)+(y1 - y0)*(y1 - y0));

 

double angle1;

double angle2;

double angle3;

 

double sinValue1;

double cosValue1;

double sinValue2;

double cosValue2;

double sinValue3;

double cosValue3;

 

sinValue1 = (y1 - y0) / radius;

cosValue1 = (x1 - x0) / radius ;

if (cosValue1 >= 0.99999) cosValue1 = 0.99999;

if (cosValue1 <= -0.99999) cosValue1 = -0.99999;

angle1 = Math.Acos(cosValue1);

angle1 = angle1 / 3.14 * 180;

if (sinValue1 < -0.05) angle1 = 360 - angle1;

 

sinValue2 = (y2 - y0) / radius;

cosValue2 = (x2 - x0) / radius;

if (cosValue2 >= 0.99999) cosValue2 = 0.99999;

if (cosValue2 <= -0.99999) cosValue2 = -0.99999;

angle2 = Math.Acos(cosValue2);

angle2 = angle2 / 3.14 * 180;

if (sinValue2 < -0.05) angle2 = 360 - angle2;

 

sinValue3 = (y3 - y0) / radius;

cosValue3 = (x3 - x0) / radius;

if (cosValue3 >= 0.99999) cosValue3 = 0.99999;

if (cosValue3 <= -0.99999) cosValue3 = -0.99999;

angle3 = Math.Acos(cosValue3);

angle3 = angle3 / 3.14 * 180;

if (sinValue3 < -0.05) angle3 = 360-angle3 ;

Pen CurvePen = newPen(m_CurveColor);

Pen CurvePen1 = newPen(Color.Green);

 

bool PosDown = false;

double Delta13;

if (angle1 < angle3)

{

Delta13 = angle3 - angle1;

}

else Delta13 = angle3 - angle1 + 360;

double Delta12;

if (angle1 < angle2)

{

Delta12 = angle2 - angle1;

}

else Delta12 = angle2 - angle1 + 360;

 

if (Delta13 > Delta12) PosDown = true;

else PosDown = false;

if (PosDown)

{

if (angle3 > angle1) mImgGraph.DrawArc(CurvePen, (int)(x0 - radius), (int)(y0 - radius), (int)(2 * radius), (int)(2 * radius), (int)(angle1), (int)(angle3 - angle1));

else mImgGraph.DrawArc(CurvePen, (int)(x0 - radius), (int)(y0 - radius), (int)(2 * radius), (int)(2 * radius), (int)(angle1), (int)(angle3 - angle1 + 360));

}

else

{

if (angle1 > angle3) mImgGraph.DrawArc(CurvePen, (int)(x0 - radius), (int)(y0 - radius), (int)(2 * radius), (int)(2 * radius), (int)(angle3), (int)(angle1 - angle3));

else mImgGraph.DrawArc(CurvePen, (int)(x0 - radius), (int)(y0 - radius), (int)(2 * radius), (int)(2 * radius), (int)(angle3), (int)(angle1 - angle3 + 360));

}

}

}