logo标签列表

如何检查两个线段是否相交？

pythonmathgeometry
161

如何检查两个线段是否相交？

我有以下数据：

Segment1 [ {x1,y1}, {x2,y2} ]
Segment2 [ {x1,y1}, {x2,y2} ]

我需要用Python编写一个小算法来检测这两条线是否相交。


alt text

2
请参见http://www.topcoder.com/tc?module=Static&d1=tutorials&d2=geometry2。 - FMc
26个回答
0

如果您的数据定义了一条直线，您只需要证明它们不是平行的即可。为此，您可以计算

alpha = float(y2 - y1) / (x2 - x1).

如果Line1和Line2的这个系数相等，它意味着这两条线是平行的。如果不相等，则它们会相交。

如果它们是平行的，那么你必须证明它们不是相同的。为此，你需要计算

beta = y1 - alpha*x1

如果Line1和Line2的beta相同，则它们相交，因为它们是相等的。
如果它们是线段，则仍然必须计算每条线的alpha和beta，如上所述。然后，您必须检查(beta1-beta2)/(alpha1-alpha2)是否大于x1_line1和x2_line1中的最小值，并且小于x1_line1和x2_line1中的最大值。
0

这是我在AS3方面得到的，对于Python我不是很了解，但概念在那里。

    public function getIntersectingPointF($A:Point, $B:Point, $C:Point, $D:Point):Number {
        var A:Point = $A.clone();
        var B:Point = $B.clone();
        var C:Point = $C.clone();
        var D:Point = $D.clone();
        var f_ab:Number = (D.x - C.x) * (A.y - C.y) - (D.y - C.y) * (A.x - C.x);

        // are lines parallel
        if (f_ab == 0) { return Infinity };

        var f_cd:Number = (B.x - A.x) * (A.y - C.y) - (B.y - A.y) * (A.x - C.x);
        var f_d:Number = (D.y - C.y) * (B.x - A.x) - (D.x - C.x) * (B.y - A.y);
        var f1:Number = f_ab/f_d
        var f2:Number = f_cd / f_d
        if (f1 == Infinity || f1 <= 0 || f1 >= 1) { return Infinity };
        if (f2 == Infinity || f2 <= 0 || f2 >= 1) { return Infinity };
        return f1;
    }

    public function getIntersectingPoint($A:Point, $B:Point, $C:Point, $D:Point):Point
    {
        var f:Number = getIntersectingPointF($A, $B, $C, $D);
        if (f == Infinity || f <= 0 || f >= 1) { return null };

        var retPoint:Point = Point.interpolate($A, $B, 1 - f);
        return retPoint.clone();
    }
0

计算线段上的线的交点（基本上是解一个线性方程组），然后检查它是否在您的线段的起点和终点之间。

0
这里是一些Python代码，它能够完全应对共线性边缘情况和舍入误差问题。
epsilon = 0.000001

def is_intersecting(x1, y1, x2, y2, x3, y3, x4, y4):
    c_area = area(x1, y1, x2, y2, x3, y3)
    d_area = area(x1, y1, x2, y2, x4, y4)
    if abs(c_area) < epsilon:
        if abs(x3-x1) < epsilon:
            if min(y1,y2)-epsilon < y3 < max(y1,y2)+epsilon:
                return True
        else:
            if min(x1,x2)-epsilon < x3 < max(x1,x2)+epsilon:
                return True
        if abs(d_area) > epsilon:
            return False
    if abs(d_area) < epsilon:
        if abs(x4-x1) < epsilon:
            if min(y1,y2)-epsilon < y4 < max(y1,y2)+epsilon:
                return True
        else:
            if min(x1,x2)-epsilon < x4 < max(x1,x2)+epsilon:
                return True
        if abs(c_area) > epsilon:
            return False
        if abs(x3-x1) < epsilon:
            return (y1 < y3) != (y1 < y4)
        else:
            return (x1 < x3) != (x1 < x4)
    if (c_area > 0) == (d_area > 0):
        return False
    a_area = area(x3, y3, x4, y4, x1, y1)
    b_area = area(x3, y3, x4, y4, x2, y2)
    return (a_area > 0) != (b_area > 0)
        
def area(x1, y1, x2, y2, x3, y3):
    return (x2-x1)*(y3-y1)-(x3-x1)*(y2-y1)
-1

使用JAVA实现。然而，似乎对于共线线段（即存在于彼此内部的线段L1（0,0）（10,10）L2（1,1）（2,2））无法正常工作。

public class TestCode
{

  public class Point
  {
    public double x = 0;
    public double y = 0;
    public Point(){}
  }

  public class Line
  {
    public Point p1, p2;
    public Line( double x1, double y1, double x2, double y2) 
    {
      p1 = new Point();
      p2 = new Point();
      p1.x = x1;
      p1.y = y1;
      p2.x = x2;
      p2.y = y2;
    }
  }

  //line segments
  private static Line s1;
  private static Line s2;

  public TestCode()
  {
    s1 = new Line(0,0,0,10);
    s2 = new Line(-1,0,0,10);
  }

  public TestCode(double x1, double y1, 
    double x2, double y2,
    double x3, double y3,
    double x4, double y4)
  {
    s1 = new Line(x1,y1, x2,y2);
    s2 = new Line(x3,y3, x4,y4);
  }

  public static void main(String args[])
  {
     TestCode code  = null;
////////////////////////////
     code = new TestCode(0,0,0,10,
                         0,1,0,5);
     if( intersect(code) )
     { System.out.println( "OK COLINEAR: INTERSECTS" ); }
     else
     { System.out.println( "ERROR COLINEAR: DO NOT INTERSECT" ); }
////////////////////////////
     code = new TestCode(0,0,0,10,
                         0,1,0,10);
     if( intersect(code) )
     { System.out.println( "OK COLINEAR: INTERSECTS" ); }
     else
     { System.out.println( "ERROR COLINEAR: DO NOT INTERSECT" ); }
////////////////////////////
     code = new TestCode(0,0,10,0,
                         5,0,15,0);
     if( intersect(code) )
     { System.out.println( "OK COLINEAR: INTERSECTS" ); }
     else
     { System.out.println( "ERROR COLINEAR: DO NOT INTERSECT" ); }
////////////////////////////
     code = new TestCode(0,0,10,0,
                         0,0,15,0);
     if( intersect(code) )
     { System.out.println( "OK COLINEAR: INTERSECTS" ); }
     else
     { System.out.println( "ERROR COLINEAR: DO NOT INTERSECT" ); }

////////////////////////////
     code = new TestCode(0,0,10,10,
                         1,1,5,5);
     if( intersect(code) )
     { System.out.println( "OK COLINEAR: INTERSECTS" ); }
     else
     { System.out.println( "ERROR COLINEAR: DO NOT INTERSECT" ); }
////////////////////////////
     code = new TestCode(0,0,0,10,
                         -1,-1,0,10);
     if( intersect(code) )
     { System.out.println( "OK SLOPE END: INTERSECTS" ); }
     else
     { System.out.println( "ERROR SLOPE END: DO NOT INTERSECT" ); }
////////////////////////////
     code = new TestCode(-10,-10,10,10,
                         -10,10,10,-10);
     if( intersect(code) )
     { System.out.println( "OK SLOPE Intersect(0,0): INTERSECTS" ); }
     else
     { System.out.println( "ERROR SLOPE Intersect(0,0): DO NOT INTERSECT" ); }
////////////////////////////
     code = new TestCode(-10,-10,10,10,
                         -3,-2,50,-2);
     if( intersect(code) )
     { System.out.println( "OK SLOPE Line2 VERTIAL: INTERSECTS" ); }
     else
     { System.out.println( "ERROR SLOPE Line2 VERTICAL: DO NOT INTERSECT" ); }
////////////////////////////
     code = new TestCode(-10,-10,10,10,
                         50,-2,-3,-2);
     if( intersect(code) )
     { System.out.println( "OK SLOPE Line2 (reversed) VERTIAL: INTERSECTS" ); }
     else
     { System.out.println( "ERROR SLOPE Line2 (reversed) VERTICAL: DO NOT INTERSECT" ); }
////////////////////////////
     code = new TestCode(0,0,0,10,
                         1,0,1,10);
     if( intersect(code) )
     { System.out.println( "ERROR PARALLEL VERTICAL: INTERSECTS" ); }
     else
     { System.out.println( "OK PARALLEL VERTICAL: DO NOT INTERSECT" ); }
////////////////////////////
     code = new TestCode(0,2,10,2,
                         0,10,10,10);
     if( intersect(code) )
     { System.out.println( "ERROR PARALLEL HORIZONTAL: INTERSECTS" ); }
     else
     { System.out.println( "OK PARALLEL HORIZONTAL: DO NOT INTERSECT" ); }
////////////////////////////
     code = new TestCode(0,10,5,13.75,
                         0,18.75,10,15);
     if( intersect(code) )
     { System.out.println( "ERROR PARALLEL SLOPE=.75: INTERSECTS" ); }
     else
     { System.out.println( "OK PARALLEL SLOPE=.75: DO NOT INTERSECT" ); }
////////////////////////////
     code = new TestCode(0,0,1,1,
                         2,-1,2,10);
     if( intersect(code) )
     { System.out.println( "ERROR SEPERATE SEGMENTS: INTERSECTS" ); }
     else
     { System.out.println( "OK SEPERATE SEGMENTS: DO NOT INTERSECT" ); }
////////////////////////////
     code = new TestCode(0,0,1,1,
                         -1,-10,-5,10);
     if( intersect(code) )
     { System.out.println( "ERROR SEPERATE SEGMENTS 2: INTERSECTS" ); }
     else
     { System.out.println( "OK SEPERATE SEGMENTS 2: DO NOT INTERSECT" ); }
  }

  public static boolean intersect( TestCode code )
  {
    return intersect( code.s1, code.s2);
  }

  public static boolean intersect( Line line1, Line line2 )
  {
    double i1min = Math.min(line1.p1.x, line1.p2.x);
    double i1max = Math.max(line1.p1.x, line1.p2.x);
    double i2min = Math.min(line2.p1.x, line2.p2.x);
    double i2max = Math.max(line2.p1.x, line2.p2.x);

    double iamax = Math.max(i1min, i2min);
    double iamin = Math.min(i1max, i2max);

    if( Math.max(line1.p1.x, line1.p2.x) < Math.min(line2.p1.x, line2.p2.x) )
      return false;

    double m1 = (line1.p2.y - line1.p1.y) / (line1.p2.x - line1.p1.x );
    double m2 = (line2.p2.y - line2.p1.y) / (line2.p2.x - line2.p1.x );

    if( m1 == m2 )
        return false;

    //b1 = line1[0][1] - m1 * line1[0][0]
    //b2 = line2[0][1] - m2 * line2[0][0]
    double b1 = line1.p1.y - m1 * line1.p1.x;
    double b2 = line2.p1.y - m2 * line2.p1.x;
    double x1 = (b2 - b1) / (m1 - m2);
    if( (x1 < Math.max(i1min, i2min)) || (x1 > Math.min(i1max, i2max)) )
        return false;
    return true;
  }
}

到目前为止的输出是

ERROR COLINEAR: DO NOT INTERSECT
ERROR COLINEAR: DO NOT INTERSECT
ERROR COLINEAR: DO NOT INTERSECT
ERROR COLINEAR: DO NOT INTERSECT
ERROR COLINEAR: DO NOT INTERSECT
OK SLOPE END: INTERSECTS
OK SLOPE Intersect(0,0): INTERSECTS
OK SLOPE Line2 VERTIAL: INTERSECTS
OK SLOPE Line2 (reversed) VERTIAL: INTERSECTS
OK PARALLEL VERTICAL: DO NOT INTERSECT
OK PARALLEL HORIZONTAL: DO NOT INTERSECT
OK PARALLEL SLOPE=.75: DO NOT INTERSECT
OK SEPERATE SEGMENTS: DO NOT INTERSECT
OK SEPERATE SEGMENTS 2: DO NOT INTERSECT
它也不能处理斜率为无穷大的线条... - Felipe
-2

已解决，但为什么不用Python呢... :)

def islineintersect(line1, line2):
    i1 = [min(line1[0][0], line1[1][0]), max(line1[0][0], line1[1][0])]
    i2 = [min(line2[0][0], line2[1][0]), max(line2[0][0], line2[1][0])]
    ia = [max(i1[0], i2[0]), min(i1[1], i2[1])]
    if max(line1[0][0], line1[1][0]) < min(line2[0][0], line2[1][0]):
        return False
    m1 = (line1[1][1] - line1[0][1]) * 1. / (line1[1][0] - line1[0][0]) * 1.
    m2 = (line2[1][1] - line2[0][1]) * 1. / (line2[1][0] - line2[0][0]) * 1.
    if m1 == m2:
        return False
    b1 = line1[0][1] - m1 * line1[0][0]
    b2 = line2[0][1] - m2 * line2[0][0]
    x1 = (b2 - b1) / (m1 - m2)
    if (x1 < max(i1[0], i2[0])) or (x1 > min(i1[1], i2[1])):
        return False
    return True

这个：

print islineintersect([(15, 20), (100, 200)], [(210, 5), (23, 119)])

输出：

True

还有这个：

print islineintersect([(15, 20), (100, 200)], [(-1, -5), (-5, -5)])

输出：

False
网页内容由stack overflow 提供, 点击上面的
可以查看英文原文，
原文链接