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

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.

Examples:

👁 check if a given point lies inside or outside a polygon 1

Approach: The idea to solve this problem is based on How to check if two given line segments intersect, and to be used as follows:

1. Draw a horizontal line to the right of each point and extend it to infinity
2. Count the number of times the line intersects with polygon edges.
3. 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.

👁 check if a given point lies inside or outside a polygon 2

How to handle point 'g' in the above figure?

Note that we should return true if the point lies on the line or the same as one of the vertices of the given polygon. To handle this, after checking if the line from 'p' to extreme intersects, we check whether 'p' is collinear with vertices of the current line of polygon. If it is collinear, then we check if the point 'p' lies on current side of polygon, if it lies, we return true, else false.

Following is the implementation of the above approach: 

Point is outside the polygon

Time Complexity: O(n) where n is the number of vertices in the given polygon.
Auxiliary Space: O(1), since no extra space has been taken.

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