Question

用于检查线是否与圆相交的好算法是什么？它沿圆圈边缘的坐标发生了什么？

Answer 1

使用this link中的数学，这是一个返回所有交叉点的函数。输入是圆的中心点和半径，高程m和y轴偏移c的线。

添加到链接的数学解决方案我添加了垂直行的代码。对于这些集合m = float.PositiveInfinity;和c = y-offset（即该行的x坐标）！

List<PointF> Intersections(PointF cc, float r, float m, float c)
{
    List<PointF> points = new List<PointF>();
    // vertical lines: 
    if (float.IsPositiveInfinity(m))
        {
            float b = (float) Math.Sqrt(r * r + (c - cc.X) * (c - cc.X));
            points.Add(new PointF(c, cc.Y + b));
            points.Add(new PointF(c, cc.Y - b));
            return points;
        }

    float A = (m * m + 1f);
    float B = 2*(m * c - m * cc.Y - cc.X);
    float C = cc.Y * cc.Y  - r * r  + cc.X * cc.X - 2f * c * cc.Y + c * c;
    float disc =  B * B - 4f * A * C;
    if (disc < 0) return points;
    float x1 = (float)(-B - Math.Sqrt(disc)) / 2f / A;
    float y1 = m * x1 + c;
    points.Add(new PointF(x1, y1));
    if (disc != 0)
    {
        float x2 = (float)(-B + Math.Sqrt(disc)) / 2f / A;
        float y2 = m * x2 + c;
        points.Add(new PointF(x2, y2));
    }
    return points;
}

以下是该行的两点形式的重载：

List<PointF> Intersections(PointF cc, float r, PointF p1, PointF p2)
{
    float m = 0; float c = 0;
    if (p1.X == p2.X) { m = float.PositiveInfinity; c = p1.X; }
    else              { m = (p2.Y - p1.Y) / (p2.X - p1.X);c = p1.Y - m * p1.X; }
    return Intersections(cc, r, m, c);
}

这是一个使用空Form的小型测试平台：

float radius = 200;           PointF center = new PointF(480,360);
PointF pt1 = PointF.Empty;    PointF pt2 = PointF.Empty;
List<PointF> pointsM = new List<PointF>();

private void Form1_MouseMove(object sender, MouseEventArgs e)
{   
    if (e.Button == MouseButtons.Left)
    {   // drag around and watch the points
        pt2 = e.Location;
        pointsM = Intersections(center, radius, pt1, pt2, false);
        Invalidate();
    }
}

private void Form1_MouseClick(object sender, MouseEventArgs e)
{   // right-click to set the first point
    if (e.Button == MouseButtons.Right) pt1 = e.Location;
    Invalidate();
}

private void Form1_Paint(object sender, PaintEventArgs e)
{
    e.Graphics.DrawEllipse(Pens.Blue, center.X - radius, center.Y - radius, 
                           radius * 2, radius * 2);
    e.Graphics.DrawLine(Pens.Green, pt1, pt2);
    e.Graphics.FillEllipse(Brushes.Blue, center.X - 4, center.Y - 4, 8, 8);
    foreach (PointF pt in pointsM)
        e.Graphics.FillEllipse(Brushes.DarkOrange, pt.X - 4, pt.Y - 4, 8, 8);
    e.Graphics.FillEllipse(Brushes.Green, pt1.X - 4, pt1.Y - 4, 8, 8);
    e.Graphics.FillEllipse(Brushes.DeepPink, pt2.X - 4, pt2.Y - 4, 8, 8);
    Text = pointsM.Count + " intersection points.";
}

Answer 2

如here所述，计算圆心和线之间的距离。

检查最近点是否在线段的端点之间。如果是在它们之间，检查距离是否小于半径;否则检查其中一个边缘点是否在圆圈内。

圆和线段碰撞

2 个答案: