How to check if two given line segments intersect?

Given two line segments (p1, q1) and (p2, q2), find if the given line segments intersect with each other.
Before we discuss the solution, let us define notion of orientation. Orientation of an ordered triplet of points in the plane can be 
–counterclockwise 
–clockwise 
–collinear 

The following diagram shows different possible orientations of (a, b, c) 

How is Orientation useful here?
Two segments (p1,q1) and (p2,q2) intersect if and only if one of the following two conditions is verified

1. General Case:
– (p1, q1, p2) and (p1, q1, q2) have different orientations and 
– (p2, q2, p1) and (p2, q2, q1) have different orientations.

Examples:

2. Special Case 
– (p1, q1, p2), (p1, q1, q2), (p2, q2, p1), and (p2, q2, q1) are all collinear and 
– the x-projections of (p1, q1) and (p2, q2) intersect 
– the y-projections of (p1, q1) and (p2, q2) intersect

Examples:

Following is the implementation based on the above idea.  

C++

// A C++ program to check if two given line segments intersect
#include <iostream>
using namespace std;

struct Point
{
    int x;
    int y;
};

// Given three collinear points p, q, r, the function checks if
// point q lies on line segment 'pr'
bool onSegment(Point p, Point q, Point r)
{
    if (q.x <= max(p.x, r.x) && q.x >= min(p.x, r.x) &&
        q.y <= max(p.y, r.y) && q.y >= min(p.y, r.y))
       return true;

    return false;
}

// To find orientation of ordered triplet (p, q, r).
// The function returns following values
// 0 --> p, q and r are collinear
// 1 --> Clockwise
// 2 --> Counterclockwise
int orientation(Point p, Point q, Point r)
{
    // See https://www.geeksforgeeks.org/orientation-3-ordered-points/
    // for details of below formula.
    int val = (q.y - p.y) * (r.x - q.x) -
              (q.x - p.x) * (r.y - q.y);

    if (val == 0) return 0;  // collinear

    return (val > 0)? 1: 2; // clock or counterclock wise
}

// The main function that returns true if line segment 'p1q1'
// and 'p2q2' intersect.
bool doIntersect(Point p1, Point q1, Point p2, Point q2)
{
    // Find the four orientations needed for general and
    // special cases
    int o1 = orientation(p1, q1, p2);
    int o2 = orientation(p1, q1, q2);
    int o3 = orientation(p2, q2, p1);
    int o4 = orientation(p2, q2, q1);

    // General case
    if (o1 != o2 && o3 != o4)
        return true;

    // Special Cases
    // p1, q1 and p2 are collinear and p2 lies on segment p1q1
    if (o1 == 0 && onSegment(p1, p2, q1)) return true;

    // p1, q1 and q2 are collinear and q2 lies on segment p1q1
    if (o2 == 0 && onSegment(p1, q2, q1)) return true;

    // p2, q2 and p1 are collinear and p1 lies on segment p2q2
    if (o3 == 0 && onSegment(p2, p1, q2)) return true;

     // p2, q2 and q1 are collinear and q1 lies on segment p2q2
    if (o4 == 0 && onSegment(p2, q1, q2)) return true;

    return false; // Doesn't fall in any of the above cases
}

// Driver program to test above functions
int main()
{
    struct Point p1 = {1, 1}, q1 = {10, 1};
    struct Point p2 = {1, 2}, q2 = {10, 2};

    doIntersect(p1, q1, p2, q2)? cout << "Yes\n": cout << "No\n";

    p1 = {10, 0}, q1 = {0, 10};
    p2 = {0, 0}, q2 = {10, 10};
    doIntersect(p1, q1, p2, q2)? cout << "Yes\n": cout << "No\n";

    p1 = {-5, -5}, q1 = {0, 0};
    p2 = {1, 1}, q2 = {10, 10};
    doIntersect(p1, q1, p2, q2)? cout << "Yes\n": cout << "No\n";

    return 0;
}

Java

// Java program to check if two given line segments intersect
class GFG
{

static class Point
{
    int x;
    int y;

        public Point(int x, int y)
        {
            this.x = x;
            this.y = y;
        }

};

// Given three collinear points p, q, r, the function checks if
// point q lies on line segment 'pr'
static boolean onSegment(Point p, Point q, Point r)
{
    if (q.x <= Math.max(p.x, r.x) && q.x >= Math.min(p.x, r.x) &&
        q.y <= Math.max(p.y, r.y) && q.y >= Math.min(p.y, r.y))
    return true;

    return false;
}

// To find orientation of ordered triplet (p, q, r).
// The function returns following values
// 0 --> p, q and r are collinear
// 1 --> Clockwise
// 2 --> Counterclockwise
static int orientation(Point p, Point q, Point r)
{
    // See https://www.geeksforgeeks.org/orientation-3-ordered-points/
    // for details of below formula.
    int val = (q.y - p.y) * (r.x - q.x) -
            (q.x - p.x) * (r.y - q.y);

    if (val == 0) return 0; // collinear

    return (val > 0)? 1: 2; // clock or counterclock wise
}

// The main function that returns true if line segment 'p1q1'
// and 'p2q2' intersect.
static boolean doIntersect(Point p1, Point q1, Point p2, Point q2)
{
    // Find the four orientations needed for general and
    // special cases
    int o1 = orientation(p1, q1, p2);
    int o2 = orientation(p1, q1, q2);
    int o3 = orientation(p2, q2, p1);
    int o4 = orientation(p2, q2, q1);

    // General case
    if (o1 != o2 && o3 != o4)
        return true;

    // Special Cases
    // p1, q1 and p2 are collinear and p2 lies on segment p1q1
    if (o1 == 0 && onSegment(p1, p2, q1)) return true;

    // p1, q1 and q2 are collinear and q2 lies on segment p1q1
    if (o2 == 0 && onSegment(p1, q2, q1)) return true;

    // p2, q2 and p1 are collinear and p1 lies on segment p2q2
    if (o3 == 0 && onSegment(p2, p1, q2)) return true;

    // p2, q2 and q1 are collinear and q1 lies on segment p2q2
    if (o4 == 0 && onSegment(p2, q1, q2)) return true;

    return false; // Doesn't fall in any of the above cases
}

// Driver code
public static void main(String[] args)
{
    Point p1 = new Point(1, 1);
    Point q1 = new Point(10, 1);
    Point p2 = new Point(1, 2);
    Point q2 = new Point(10, 2);

    if(doIntersect(p1, q1, p2, q2))
        System.out.println("Yes");
    else
        System.out.println("No");

    p1 = new Point(10, 1); q1 = new Point(0, 10);
    p2 = new Point(0, 0); q2 = new Point(10, 10);
    if(doIntersect(p1, q1, p2, q2))
            System.out.println("Yes");
    else
        System.out.println("No");

    p1 = new Point(-5, -5); q1 = new Point(0, 0);
    p2 = new Point(1, 1); q2 = new Point(10, 10);;
    if(doIntersect(p1, q1, p2, q2))
        System.out.println("Yes");
    else
        System.out.println("No");
}
}

// This code is contributed by Princi Singh

Python3

# A Python3 program to find if 2 given line segments intersect or not

class Point:
    def __init__(self, x, y):
        self.x = x
        self.y = y

# Given three collinear points p, q, r, the function checks if
# point q lies on line segment 'pr'
def onSegment(p, q, r):
    if ( (q.x <= max(p.x, r.x)) and (q.x >= min(p.x, r.x)) and
           (q.y <= max(p.y, r.y)) and (q.y >= min(p.y, r.y))):
        return True
    return False

def orientation(p, q, r):
    # to find the orientation of an ordered triplet (p,q,r)
    # function returns the following values:
    # 0 : Collinear points
    # 1 : Clockwise points
    # 2 : Counterclockwise

    # See https://www.geeksforgeeks.org/orientation-3-ordered-points/amp/
    # for details of below formula.

    val = (float(q.y - p.y) * (r.x - q.x)) - (float(q.x - p.x) * (r.y - q.y))
    if (val > 0):

        # Clockwise orientation
        return 1
    elif (val < 0):

        # Counterclockwise orientation
        return 2
    else:

        # Collinear orientation
        return 0

# The main function that returns true if
# the line segment 'p1q1' and 'p2q2' intersect.
def doIntersect(p1,q1,p2,q2):

    # Find the 4 orientations required for
    # the general and special cases
    o1 = orientation(p1, q1, p2)
    o2 = orientation(p1, q1, q2)
    o3 = orientation(p2, q2, p1)
    o4 = orientation(p2, q2, q1)

    # General case
    if ((o1 != o2) and (o3 != o4)):
        return True

    # Special Cases

    # p1 , q1 and p2 are collinear and p2 lies on segment p1q1
    if ((o1 == 0) and onSegment(p1, p2, q1)):
        return True

    # p1 , q1 and q2 are collinear and q2 lies on segment p1q1
    if ((o2 == 0) and onSegment(p1, q2, q1)):
        return True

    # p2 , q2 and p1 are collinear and p1 lies on segment p2q2
    if ((o3 == 0) and onSegment(p2, p1, q2)):
        return True

    # p2 , q2 and q1 are collinear and q1 lies on segment p2q2
    if ((o4 == 0) and onSegment(p2, q1, q2)):
        return True

    # If none of the cases
    return False

# Driver program to test above functions:
p1 = Point(1, 1)
q1 = Point(10, 1)
p2 = Point(1, 2)
q2 = Point(10, 2)

if doIntersect(p1, q1, p2, q2):
    print("Yes")
else:
    print("No")

p1 = Point(10, 0)
q1 = Point(0, 10)
p2 = Point(0, 0)
q2 = Point(10,10)

if doIntersect(p1, q1, p2, q2):
    print("Yes")
else:
    print("No")

p1 = Point(-5,-5)
q1 = Point(0, 0)
p2 = Point(1, 1)
q2 = Point(10, 10)

if doIntersect(p1, q1, p2, q2):
    print("Yes")
else:
    print("No")

# This code is contributed by Ansh Riyal

C#

// C# program to check if two given line segments intersect
using System;
using System.Collections.Generic;

class GFG
{

public class Point
{
    public int x;
    public int y;

    public Point(int x, int y)
    {
        this.x = x;
        this.y = y;
    }

};

// Given three collinear points p, q, r, the function checks if
// point q lies on line segment 'pr'
static Boolean onSegment(Point p, Point q, Point r)
{
    if (q.x <= Math.Max(p.x, r.x) && q.x >= Math.Min(p.x, r.x) &&
        q.y <= Math.Max(p.y, r.y) && q.y >= Math.Min(p.y, r.y))
    return true;

    return false;
}

// To find orientation of ordered triplet (p, q, r).
// The function returns following values
// 0 --> p, q and r are collinear
// 1 --> Clockwise
// 2 --> Counterclockwise
static int orientation(Point p, Point q, Point r)
{
    // See https://www.geeksforgeeks.org/orientation-3-ordered-points/
    // for details of below formula.
    int val = (q.y - p.y) * (r.x - q.x) -
            (q.x - p.x) * (r.y - q.y);

    if (val == 0) return 0; // collinear

    return (val > 0)? 1: 2; // clock or counterclock wise
}

// The main function that returns true if line segment 'p1q1'
// and 'p2q2' intersect.
static Boolean doIntersect(Point p1, Point q1, Point p2, Point q2)
{
    // Find the four orientations needed for general and
    // special cases
    int o1 = orientation(p1, q1, p2);
    int o2 = orientation(p1, q1, q2);
    int o3 = orientation(p2, q2, p1);
    int o4 = orientation(p2, q2, q1);

    // General case
    if (o1 != o2 && o3 != o4)
        return true;

    // Special Cases
    // p1, q1 and p2 are collinear and p2 lies on segment p1q1
    if (o1 == 0 && onSegment(p1, p2, q1)) return true;

    // p1, q1 and q2 are collinear and q2 lies on segment p1q1
    if (o2 == 0 && onSegment(p1, q2, q1)) return true;

    // p2, q2 and p1 are collinear and p1 lies on segment p2q2
    if (o3 == 0 && onSegment(p2, p1, q2)) return true;

    // p2, q2 and q1 are collinear and q1 lies on segment p2q2
    if (o4 == 0 && onSegment(p2, q1, q2)) return true;

    return false; // Doesn't fall in any of the above cases
}

// Driver code
public static void Main(String[] args)
{
    Point p1 = new Point(1, 1);
    Point q1 = new Point(10, 1);
    Point p2 = new Point(1, 2);
    Point q2 = new Point(10, 2);

    if(doIntersect(p1, q1, p2, q2))
        Console.WriteLine("Yes");
    else
        Console.WriteLine("No");

    p1 = new Point(10, 1); q1 = new Point(0, 10);
    p2 = new Point(0, 0); q2 = new Point(10, 10);
    if(doIntersect(p1, q1, p2, q2))
            Console.WriteLine("Yes");
    else
        Console.WriteLine("No");

    p1 = new Point(-5, -5); q1 = new Point(0, 0);
    p2 = new Point(1, 1); q2 = new Point(10, 10);;
    if(doIntersect(p1, q1, p2, q2))
        Console.WriteLine("Yes");
    else
        Console.WriteLine("No");
}
}

/* This code contributed by PrinciRaj1992 */

Javascript

<script>
// Javascript program to check if two given line segments intersect

class Point
{
    constructor(x, y)
    {
        this.x = x;
            this.y = y;
    }
}

// Given three collinear points p, q, r, the function checks if
// point q lies on line segment 'pr'
function onSegment(p, q, r)
{
    if (q.x <= Math.max(p.x, r.x) && q.x >= Math.min(p.x, r.x) &&
        q.y <= Math.max(p.y, r.y) && q.y >= Math.min(p.y, r.y))
    return true;

    return false;
}

// To find orientation of ordered triplet (p, q, r).
// The function returns following values
// 0 --> p, q and r are collinear
// 1 --> Clockwise
// 2 --> Counterclockwise
function orientation(p, q, r)
{

    // See https://www.geeksforgeeks.org/orientation-3-ordered-points/
    // for details of below formula.
    let val = (q.y - p.y) * (r.x - q.x) -
            (q.x - p.x) * (r.y - q.y);

    if (val == 0) return 0; // collinear

    return (val > 0)? 1: 2; // clock or counterclock wise
}

// The main function that returns true if line segment 'p1q1'
// and 'p2q2' intersect.
function doIntersect(p1, q1, p2, q2)
{

    // Find the four orientations needed for general and
    // special cases
    let o1 = orientation(p1, q1, p2);
    let o2 = orientation(p1, q1, q2);
    let o3 = orientation(p2, q2, p1);
    let o4 = orientation(p2, q2, q1);

    // General case
    if (o1 != o2 && o3 != o4)
        return true;

    // Special Cases
    // p1, q1 and p2 are collinear and p2 lies on segment p1q1
    if (o1 == 0 && onSegment(p1, p2, q1)) return true;

    // p1, q1 and q2 are collinear and q2 lies on segment p1q1
    if (o2 == 0 && onSegment(p1, q2, q1)) return true;

    // p2, q2 and p1 are collinear and p1 lies on segment p2q2
    if (o3 == 0 && onSegment(p2, p1, q2)) return true;

    // p2, q2 and q1 are collinear and q1 lies on segment p2q2
    if (o4 == 0 && onSegment(p2, q1, q2)) return true;

    return false; // Doesn't fall in any of the above cases
}

// Driver code
let p1 = new Point(1, 1);
let q1 = new Point(10, 1);
let p2 = new Point(1, 2);
let q2 = new Point(10, 2);

if(doIntersect(p1, q1, p2, q2))
    document.write("Yes<br>");
else
    document.write("No<br>");

p1 = new Point(10, 1); q1 = new Point(0, 10);
p2 = new Point(0, 0); q2 = new Point(10, 10);
if(doIntersect(p1, q1, p2, q2))
    document.write("Yes<br>");
else
    document.write("No<br>");

p1 = new Point(-5, -5); q1 = new Point(0, 0);
p2 = new Point(1, 1); q2 = new Point(10, 10);;
if(doIntersect(p1, q1, p2, q2))
    document.write("Yes<br>");
else
    document.write("No<br>");

// This code is contributed by avanitrachhadiya2155
</script>

Output: 

No
Yes
No

Time Complexity: O(1)

Space Complexity: O(1)

Sources:
http://www.dcs.gla.ac.uk/~pat/52233/slides/Geometry1x1.pdf
Introduction to Algorithms 3rd Edition by Clifford Stein, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest
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