檢測兩個重合線段

Question

的一致子集這個問題是有關：

• How do I determine the intersection point of two lines in GDI+?（代數的很好的解釋，但無碼）
• How do you detect where two line segments intersect?（接受的答案並不實際工作）

但是請注意，一個有趣的子問題在大多數解決方案中完全掩蓋了，即使存在三種子情況，這些問題也只是返回null：

• 一致，但不重疊
• 例如，我們可以設計這樣一個C＃函數感人只是點和重合
• 重疊/重合線子段

：

public static PointF[] Intersection(PointF a1, PointF a2, PointF b1, PointF b2) 

其中（a1，a2）是一條線段，（b1，b2）是另一條線段。

此功能需要涵蓋大多數實現或解釋所掩蓋的所有奇怪情況。爲了解釋的重合線的古怪，該函數可以返回的PointF的數組：

• 零結果的點（或空），如果線是平行的或不相交（無限線相交但線段是不相交，或線是平行）
• 一個結果點（包含相交位置），如果他們相交或者如果它們是一致的在一個點
• 兩個結果的點（對於重疊線段的一部分），如果兩行是重合

Answer 1

聽起來像你有你的解決方案，這是偉大的。我有一些改進它的建議。

該方法有一個主要的可用性問題，因爲它非常容易理解（1）參數意味着什麼，以及（2）結果是什麼意思。這兩個都是你想要使用該方法的小謎題。

我會更傾向於使用類型系統來更清楚地說明這種方法的作用。

我會首先定義一個類型 - 可能是一個結構，特別是如果它是不可變的 - 稱爲LineSegment。一個LineSegment由兩個表示結束點的PointF結構組成。第二，如果需要表示兩個或多個基因座聯合的基因座，我將定義一個抽象基類型「基因座」和派生類型EmptyLocus，PointLocus，LineSegmentLocus和可能的UnionLocus。一個空的軌跡只是一個單獨的點，一個點軌跡只是一個點，等等。

現在你的方法簽名變得更加清晰：

static Locus Intersect(LineSegment l1, LineSegment l2) 

這個方法有兩個線段和計算的點的軌跡就是它們的交集 - 無論是空的，單點，或線段。

請注意，您可以概括此方法。計算線段與線段的交點是棘手的，但是計算線段與點或點與點的交點或任何具有空軌跡的點是容易。將交集擴展到任意位置的聯合並不難。因此，你實際上可以寫：

static Locus Intersect(Locus l1, Locus l2) 

哎，現在它變得清晰，相交可能是在軌跡的擴展方法：

static Locus Intersect(this Locus l1, Locus l2) 

從的PointF到PointLocus和線段添加的隱式轉換到LineSegmentLocus ，你可以說像

var point = new PointF(whatever); 
var lineseg = new LineSegment(somepoint, someotherpoint); 
var intersection = lineseg.Intersect(point); 
if (intersection is EmptyLocus) ... 

使用類型系統很好可以大大提高程序的可讀性。

Answer 2

// port of this JavaScript code with some changes: 
    // http://www.kevlindev.com/gui/math/intersection/Intersection.js 
    // found here: 
    // http://stackoverflow.com/questions/563198/how-do-you-detect-where-two-line-segments-intersect/563240#563240 

public class Intersector 
{ 
    static double MyEpsilon = 0.00001; 

    private static float[] OverlapIntervals(float ub1, float ub2) 
    { 
     float l = Math.Min(ub1, ub2); 
     float r = Math.Max(ub1, ub2); 
     float A = Math.Max(0, l); 
     float B = Math.Min(1, r); 
     if (A > B) // no intersection 
      return new float[] { }; 
     else if (A == B) 
      return new float[] { A }; 
     else // if (A < B) 
      return new float[] { A, B }; 
    } 

    // IMPORTANT: a1 and a2 cannot be the same, e.g. a1--a2 is a true segment, not a point 
    // b1/b2 may be the same (b1--b2 is a point) 
    private static PointF[] OneD_Intersection(PointF a1, PointF a2, PointF b1, PointF b2) 
    { 
     //float ua1 = 0.0f; // by definition 
     //float ua2 = 1.0f; // by definition 
     float ub1, ub2; 

     float denomx = a2.X - a1.X; 
     float denomy = a2.Y - a1.Y; 

     if (Math.Abs(denomx) > Math.Abs(denomy)) 
     { 
      ub1 = (b1.X - a1.X)/denomx; 
      ub2 = (b2.X - a1.X)/denomx; 
     } 
     else 
     { 
      ub1 = (b1.Y - a1.Y)/denomy; 
      ub2 = (b2.Y - a1.Y)/denomy; 
     } 

     List<PointF> ret = new List<PointF>(); 
     float[] interval = OverlapIntervals(ub1, ub2); 
     foreach (float f in interval) 
     { 
      float x = a2.X * f + a1.X * (1.0f - f); 
      float y = a2.Y * f + a1.Y * (1.0f - f); 
      PointF p = new PointF(x, y); 
      ret.Add(p); 
     } 
     return ret.ToArray(); 
    } 

    private static bool PointOnLine(PointF p, PointF a1, PointF a2) 
    { 
     float dummyU = 0.0f; 
     double d = DistFromSeg(p, a1, a2, MyEpsilon, ref dummyU); 
     return d < MyEpsilon; 
    } 

    private static double DistFromSeg(PointF p, PointF q0, PointF q1, double radius, ref float u) 
    { 
     // formula here: 
     //http://mathworld.wolfram.com/Point-LineDistance2-Dimensional.html 
     // where x0,y0 = p 
     //  x1,y1 = q0 
     //  x2,y2 = q1 
     double dx21 = q1.X - q0.X; 
     double dy21 = q1.Y - q0.Y; 
     double dx10 = q0.X - p.X; 
     double dy10 = q0.Y - p.Y; 
     double segLength = Math.Sqrt(dx21 * dx21 + dy21 * dy21); 
     if (segLength < MyEpsilon) 
      throw new Exception("Expected line segment, not point."); 
     double num = Math.Abs(dx21 * dy10 - dx10 * dy21); 
     double d = num/segLength; 
     return d; 
    } 

    // this is the general case. Really really general 
    public static PointF[] Intersection(PointF a1, PointF a2, PointF b1, PointF b2) 
    { 
     if (a1.Equals(a2) && b1.Equals(b2)) 
     { 
      // both "segments" are points, return either point 
      if (a1.Equals(b1)) 
       return new PointF[] { a1 }; 
      else // both "segments" are different points, return empty set 
       return new PointF[] { }; 
     } 
     else if (b1.Equals(b2)) // b is a point, a is a segment 
     { 
      if (PointOnLine(b1, a1, a2)) 
       return new PointF[] { b1 }; 
      else 
       return new PointF[] { }; 
     } 
     else if (a1.Equals(a2)) // a is a point, b is a segment 
     { 
      if (PointOnLine(a1, b1, b2)) 
       return new PointF[] { a1 }; 
      else 
       return new PointF[] { }; 
     } 

     // at this point we know both a and b are actual segments 

     float ua_t = (b2.X - b1.X) * (a1.Y - b1.Y) - (b2.Y - b1.Y) * (a1.X - b1.X); 
     float ub_t = (a2.X - a1.X) * (a1.Y - b1.Y) - (a2.Y - a1.Y) * (a1.X - b1.X); 
     float u_b = (b2.Y - b1.Y) * (a2.X - a1.X) - (b2.X - b1.X) * (a2.Y - a1.Y); 

     // Infinite lines intersect somewhere 
     if (!(-MyEpsilon < u_b && u_b < MyEpsilon)) // e.g. u_b != 0.0 
     { 
      float ua = ua_t/u_b; 
      float ub = ub_t/u_b; 
      if (0.0f <= ua && ua <= 1.0f && 0.0f <= ub && ub <= 1.0f) 
      { 
       // Intersection 
       return new PointF[] { 
        new PointF(a1.X + ua * (a2.X - a1.X), 
         a1.Y + ua * (a2.Y - a1.Y)) }; 
      } 
      else 
      { 
       // No Intersection 
       return new PointF[] { }; 
      } 
     } 
     else // lines (not just segments) are parallel or the same line 
     { 
      // Coincident 
      // find the common overlapping section of the lines 
      // first find the distance (squared) from one point (a1) to each point 
      if ((-MyEpsilon < ua_t && ua_t < MyEpsilon) 
       || (-MyEpsilon < ub_t && ub_t < MyEpsilon)) 
      { 
       if (a1.Equals(a2)) // danger! 
        return OneD_Intersection(b1, b2, a1, a2); 
       else // safe 
        return OneD_Intersection(a1, a2, b1, b2); 
      } 
      else 
      { 
       // Parallel 
       return new PointF[] { }; 
      } 
     } 
    } 


} 

下面是測試代碼：

public class IntersectTest 
    { 
     public static void PrintPoints(PointF[] pf) 
     { 
      if (pf == null || pf.Length < 1) 
       System.Console.WriteLine("Doesn't intersect"); 
      else if (pf.Length == 1) 
      { 
       System.Console.WriteLine(pf[0]); 
      } 
      else if (pf.Length == 2) 
      { 
       System.Console.WriteLine(pf[0] + " -- " + pf[1]); 
      } 
     } 

     public static void TestIntersect(PointF a1, PointF a2, PointF b1, PointF b2) 
     { 
      System.Console.WriteLine("----------------------------------------------------------"); 
      System.Console.WriteLine("Does  " + a1 + " -- " + a2); 
      System.Console.WriteLine("intersect " + b1 + " -- " + b2 + " and if so, where?"); 
      System.Console.WriteLine(""); 
      PointF[] result = Intersect.Intersection(a1, a2, b1, b2); 
      PrintPoints(result); 
     } 

     public static void Main() 
     { 
      System.Console.WriteLine("----------------------------------------------------------"); 
      System.Console.WriteLine("line segments intersect"); 
      TestIntersect(new PointF(0, 0), 
          new PointF(100, 100), 
          new PointF(100, 0), 
          new PointF(0, 100)); 
      TestIntersect(new PointF(5, 17), 
          new PointF(100, 100), 
          new PointF(100, 29), 
          new PointF(8, 100)); 
      System.Console.WriteLine("----------------------------------------------------------"); 
      System.Console.WriteLine(""); 

      System.Console.WriteLine("----------------------------------------------------------"); 
      System.Console.WriteLine("just touching points and lines cross"); 
      TestIntersect(new PointF(0, 0), 
          new PointF(25, 25), 
          new PointF(25, 25), 
          new PointF(100, 75)); 
      System.Console.WriteLine("----------------------------------------------------------"); 
      System.Console.WriteLine(""); 

      System.Console.WriteLine("----------------------------------------------------------"); 
      System.Console.WriteLine("parallel"); 
      TestIntersect(new PointF(0, 0), 
          new PointF(0, 100), 
          new PointF(100, 0), 
          new PointF(100, 100)); 
      System.Console.WriteLine("----------------------------------------------------------"); 
      System.Console.WriteLine(""); 

      System.Console.WriteLine("----"); 
      System.Console.WriteLine("lines cross but segments don't intersect"); 
      TestIntersect(new PointF(50, 50), 
          new PointF(100, 100), 
          new PointF(0, 25), 
          new PointF(25, 0)); 
      System.Console.WriteLine("----------------------------------------------------------"); 
      System.Console.WriteLine(""); 

      System.Console.WriteLine("----------------------------------------------------------"); 
      System.Console.WriteLine("coincident but do not overlap!"); 
      TestIntersect(new PointF(0, 0), 
          new PointF(25, 25), 
          new PointF(75, 75), 
          new PointF(100, 100)); 
      System.Console.WriteLine("----------------------------------------------------------"); 
      System.Console.WriteLine(""); 

      System.Console.WriteLine("----------------------------------------------------------"); 
      System.Console.WriteLine("touching points and coincident!"); 
      TestIntersect(new PointF(0, 0), 
          new PointF(25, 25), 
          new PointF(25, 25), 
          new PointF(100, 100)); 
      System.Console.WriteLine("----------------------------------------------------------"); 
      System.Console.WriteLine(""); 

      System.Console.WriteLine("----------------------------------------------------------"); 
      System.Console.WriteLine("overlap/coincident"); 
      TestIntersect(new PointF(0, 0), 
          new PointF(75, 75), 
          new PointF(25, 25), 
          new PointF(100, 100)); 
      TestIntersect(new PointF(0, 0), 
          new PointF(100, 100), 
          new PointF(0, 0), 
          new PointF(100, 100)); 
      System.Console.WriteLine("----------------------------------------------------------"); 
      System.Console.WriteLine(""); 

      while (!System.Console.KeyAvailable) { } 
     } 

    } 

，這裏是輸出：

 
---------------------------------------------------------- 
line segments intersect 
---------------------------------------------------------- 
Does  {X=0, Y=0} -- {X=100, Y=100} 
intersect {X=100, Y=0} -- {X=0, Y=100} and if so, where? 

{X=50, Y=50} 
---------------------------------------------------------- 
Does  {X=5, Y=17} -- {X=100, Y=100} 
intersect {X=100, Y=29} -- {X=8, Y=100} and if so, where? 

{X=56.85001, Y=62.30054} 
---------------------------------------------------------- 

---------------------------------------------------------- 
just touching points and lines cross 
---------------------------------------------------------- 
Does  {X=0, Y=0} -- {X=25, Y=25} 
intersect {X=25, Y=25} -- {X=100, Y=75} and if so, where? 

{X=25, Y=25} 
---------------------------------------------------------- 

---------------------------------------------------------- 
parallel 
---------------------------------------------------------- 
Does  {X=0, Y=0} -- {X=0, Y=100} 
intersect {X=100, Y=0} -- {X=100, Y=100} and if so, where? 

Doesn't intersect 
---------------------------------------------------------- 

---- 
lines cross but segments don't intersect 
---------------------------------------------------------- 
Does  {X=50, Y=50} -- {X=100, Y=100} 
intersect {X=0, Y=25} -- {X=25, Y=0} and if so, where? 

Doesn't intersect 
---------------------------------------------------------- 

---------------------------------------------------------- 
coincident but do not overlap! 
---------------------------------------------------------- 
Does  {X=0, Y=0} -- {X=25, Y=25} 
intersect {X=75, Y=75} -- {X=100, Y=100} and if so, where? 

Doesn't intersect 
---------------------------------------------------------- 

---------------------------------------------------------- 
touching points and coincident! 
---------------------------------------------------------- 
Does  {X=0, Y=0} -- {X=25, Y=25} 
intersect {X=25, Y=25} -- {X=100, Y=100} and if so, where? 

{X=25, Y=25} 
---------------------------------------------------------- 

---------------------------------------------------------- 
overlap/coincident 
---------------------------------------------------------- 
Does  {X=0, Y=0} -- {X=75, Y=75} 
intersect {X=25, Y=25} -- {X=100, Y=100} and if so, where? 

{X=25, Y=25} -- {X=75, Y=75} 
---------------------------------------------------------- 
Does  {X=0, Y=0} -- {X=100, Y=100} 
intersect {X=0, Y=0} -- {X=100, Y=100} and if so, where? 

{X=0, Y=0} -- {X=100, Y=100} 
---------------------------------------------------------- 

Answer 3

這真的很簡單。如果你有兩條線，你可以找到y = mx + b形式的兩個方程。例如：

y = 2x + 5 
y = x - 3 

所以，兩線相交時，Y1 = Y2在同一x座標，所以......

2x + 5 = x - 3 
x + 5 = -3 
x = -8 

當x = -8 Y1 = Y2和你已經找到了交點。這應該是非常簡單的翻譯成代碼。如果沒有交點，則每條線的斜率將爲m，在這種情況下，您甚至不需要執行計算。

Answer 4

@Jared，很好的問題和很好的答案。

如約瑟夫奧洛克的CGA常見問題解答here中所解釋的，通過將點的位置表示爲單個參數的函數，可以簡化問題。

令R爲一個參數，以指示沿包含AB， 具有以下含義的線P的 位置：

 r=0  P = A 
     r=1  P = B 
     r<0  P is on the backward extension of AB 
     r>1  P is on the forward extension of AB 
     0<r<1 P is interior to AB 

沿着這些線路的思考，對於任何點C（CX，CY ）我們計算R如下：

double deltax = bx - ax; 
double deltay = by - ay; 
double l2 = deltax * deltax + deltay * deltay; 
double r = (ay - cy) * (ay - by) - (ax - cx) * (bx - ax)/l2; 

這應該更容易計算的重疊部分。

請注意，我們避免採取平方根，因爲只需要長度的平方。