Обнаружение совпадающего подмножества двух совпадающих линейных сегментов

    // 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[] { };
            }
        }
    }


}

Вот тестовый код:

    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) { }
        }

    }

, а вот результат:

----------------------------------------------------------
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}
----------------------------------------------------------

Это действительно очень просто. Если у вас есть две строки, вы можете найти два уравнения в виде y = mx + b. Например:

y = 2x + 5
y = x - 3

Итак, две прямые пересекаются, когда y1 = y2 в одной и той же координате x, поэтому ...

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

Когда x = -8 y1 = y2 и вы нашли точку пересечения. Это должно быть очень легко перевести в код. Если точки пересечения нет, то наклон м каждой линии будет одинаковым, и в этом случае вам даже не нужно выполнять вычисления.

@ Джаред, Отличный вопрос и отличный ответ.

Задачу можно упростить, представив положение точки на линии как функцию одного параметра, как это объясняется в FAQ Джозефа О'Рурка по CGA здесь .

Пусть r будет параметром для обозначения P местоположение по линии, содержащей AB, со следующим значением:

  r = 0 P = A
г = 1 П = В
r <0 P находится на обратном продолжении AB
r> 1 P находится на прямом продолжении AB
0 

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;