Program to check given point in inside or boundary of given polygon or not in python

Suppose we have a list of cartesian points [(x1, y1), (x2, y2), ..., (xn, yn)] representing a polygon, and also have two values x and y. We need to check whether point (x, y) lies inside this polygon or on the boundary.

So, if the input is like points = [(0, 0), (1, 3), (4, 4), (6, 2), (4, 0)] and pt = (3, 1):

The output will be True because point (3, 1) lies inside the polygon.

Algorithm

This problem uses the ray casting algorithm. We cast a horizontal ray from the test point to the right and count how many polygon edges it crosses. If the count is odd, the point is inside; if even, it's outside.

To solve this, we follow these steps ?

• Initialize ans := False
• For each edge of the polygon:
  • Get edge endpoints (x0, y0) and (x1, y1)
  • Check if ray intersects this edge
  • If intersection exists and is to the right of test point, toggle ans
• Return ans

Implementation

class Solution:
    def solve(self, polygon, pt):
        ans = False
        for i in range(len(polygon)):
            x0, y0 = polygon[i]
            x1, y1 = polygon[(i + 1) % len(polygon)]
            
            # Check if ray intersects this edge
            if not min(y0, y1) < pt[1] <= max(y0, y1):
                continue
            
            # Skip if point is to the left of both edge endpoints
            if pt[0] < min(x0, x1):
                continue
            
            # Calculate x-coordinate of intersection
            cur_x = x0 if x0 == x1 else x0 + (pt[1] - y0) * (x1 - x0) / (y1 - y0)
            
            # Toggle result if intersection is to the right of point
            ans ^= pt[0] > cur_x
        
        return ans

# Test the algorithm
ob = Solution()
points = [(0, 0), (1, 3), (4, 4), (6, 2), (4, 0)]
pt = (3, 1)
print(ob.solve(points, pt))

True

How It Works

The algorithm works by:

• Edge intersection: For each polygon edge, check if a horizontal ray from the test point intersects it
• Y-coordinate check: The ray intersects only if the test point's y-coordinate is between the edge's y-coordinates
• X-coordinate calculation: Find where the ray intersects the edge using linear interpolation
• Counting crossings: Use XOR to count odd/even intersections to the right of the test point

Testing Different Points

ob = Solution()
polygon = [(0, 0), (1, 3), (4, 4), (6, 2), (4, 0)]

test_points = [
    (3, 1),    # Inside
    (0, 0),    # On vertex
    (2, 2),    # Inside
    (7, 3),    # Outside
    (1, 1)     # Outside
]

for point in test_points:
    result = ob.solve(polygon, point)
    print(f"Point {point}: {'Inside' if result else 'Outside'}")

Point (3, 1): Inside
Point (0, 0): Outside
Point (2, 2): Inside
Point (7, 3): Outside
Point (1, 1): Outside

Conclusion

The ray casting algorithm efficiently determines if a point lies inside a polygon by counting edge crossings. This method works for any polygon shape and handles edge cases like points on vertices or boundaries.