How to check if a given point lies inside or outside a polygon? | GeeksforGeeks

Given a polygon and a point ‘p’, find if ‘p’ lies inside the polygon or not. The points lying on the border are considered inside.

Following is a simple idea to check whether a point is inside or outside.

1) Draw a horizontal line to the right of each point and extend it to infinity

1) Count the number of times the line intersects with polygon edges.

2) A point is inside the polygon if either count of intersections is odd or
   point lies on an edge of polygon.  If none of the conditions is true, then 
   point lies outside.

// Returns true if the point p lies inside the polygon[] with n vertices

bool isInside(Point polygon[], int n, Point p)

{

    // There must be at least 3 vertices in polygon[]

    if (n < 3)  return false;

 

    // Create a point for line segment from p to infinite

    Point extreme = {INF, p.y};

 

    // Count intersections of the above line with sides of polygon

    int count = 0, i = 0;

    do

    {

        int next = (i+1)%n;

 

        // Check if the line segment from 'p' to 'extreme' intersects

        // with the line segment from 'polygon[i]' to 'polygon[next]'

        if (doIntersect(polygon[i], polygon[next], p, extreme))

        {

            // If the point 'p' is colinear with line segment 'i-next',

            // then check if it lies on segment. If it lies, return true,

            // otherwise false

            if (orientation(polygon[i], p, polygon[next]) == 0)

               return onSegment(polygon[i], p, polygon[next]);

 

            count++;

        }

        i = next;

    } while (i != 0);

 

    // Return true if count is odd, false otherwise

    return count&1;  // Same as (count%2 == 1)

}

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 colinear

// 1 --> Clockwise

// 2 --> Counterclockwise

int orientation(Point p, Point q, Point r)

{

    int val = (q.y - p.y) * (r.x - q.x) -

              (q.x - p.x) * (r.y - q.y);

 

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

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

}

 

// The 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 colinear and p2 lies on segment p1q1

    if (o1 == 0 && onSegment(p1, p2, q1)) return true;

 

    // p1, q1 and p2 are colinear and q2 lies on segment p1q1

    if (o2 == 0 && onSegment(p1, q2, q1)) return true;

 

    // p2, q2 and p1 are colinear and p1 lies on segment p2q2

    if (o3 == 0 && onSegment(p2, p1, q2)) return true;

 

     // p2, q2 and q1 are colinear 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

}

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