判定任意形状多边形中的点

判断一个点是否在凸多边形内，可以使用著名的射线法算法。

def point_inside_polygon(x, y, poly):
    """ Deciding if a point is inside (True, False otherwise) a polygon,
    where poly is a list of pairs (x,y) containing the polygon's vertices.
    The algorithm is called the 'Ray Casting Method' """
    n = len(poly)
    inside = False
    p1x, p1y = poly[0]
    for i in range(n):
        p2x, p2y = poly[i % n]
        if y > min(p1y, p2y):
            if y <= max(p1y, p2y):
                if x <= max(p1x, p2x):
                    if p1y != p2y:
                        xinters = (y-p1y) * (p2x-p1x) / (p2y-p1y) + p1x
                    if p1x == p2x or x <= xinters:
                        inside = not inside
        p1x, p1y = p2x, p2y
    return inside

但是，如果多边形不是完全凸的，该怎么办？已知边界点，如何确定一个点是否在任意形状的多边形内？

给定一个如下所示的边界点多边形：

       (1, 2)
       / \
     (0, 0)---(2, 0)
     |   |
     |___|
       (1, -2)

如何判断一个点是否在多边形内？

最好用 Python 实现，但任何通用的解决方案也都是受欢迎的。

解决方案

我们可以使用一种称为“扫描线法”的算法来解决这个问题。该算法通过将多边形与水平线进行交叉来工作。

1. 将多边形的边界点按 y 轴坐标从小到大排序。
2. 将一条水平扫描线从多边形的底部向上移动，直到到达多边形的顶部。
3. 在每一步，计算扫描线与多边形边界的交点。
4. 将交点按 x 轴坐标从小到大排序。
5. 对于每个交点，检查该交点是否位于多边形的内部。
6. 如果交点位于多边形的内部，则将该交点添加到多边形的内部点集合中。

如果一个点在多边形的内部点集合中，则该点在多边形内。否则，该点在多边形外。

def point_inside_polygon(x, y, poly):
    """ Deciding if a point is inside (True, False otherwise) a polygon,
    where poly is a list of pairs (x,y) containing the polygon's vertices.
    The algorithm is called the 'Scan Line Method' """
    poly.sort(key=lambda p: p[1])
    intersections = []
    for i in range(1, len(poly)):
        p1, p2 = poly[i-1], poly[i]
        if p1[1] <= y <= p2[1] or p2[1] <= y <= p1[1]:
            x_inter = (y-p1[1]) * (p2[0]-p1[0]) / (p2[1]-p1[1]) + p1[0]
            intersections.append(x_inter)
    intersections.sort()
    inside = False
    for j in range(0, len(intersections), 2):
        if intersections[j] <= x <= intersections[j+1]:
            inside = not inside
    return inside

以下是该算法的 Python 实现：

def point_inside_polygon(x, y, poly):
    """ Deciding if a point is inside (True, False otherwise) a polygon,
    where poly is a list of pairs (x,y) containing the polygon's vertices.
    The algorithm is called the 'Ray Casting Method' """
    n = len(poly)
    inside = False
    p1x, p1y = poly[0]
    for i in range(n):
        p2x, p2y = poly[i % n]
        if y > min(p1y, p2y):
            if y <= max(p1y, p2y):
                if x <= max(p1x, p2x):
                    if p1y != p2y:
                        xinters = (y-p1y) * (p2x-p1x) / (p2y-p1y) + p1x
                    if p1x == p2x or x <= xinters:
                        inside = not inside
        p1x, p1y = p2x, p2y
    return inside

def point_inside_polygon_scanline(x, y, poly):
    """ Deciding if a point is inside (True, False otherwise) a polygon,
    where poly is a list of pairs (x,y) containing the polygon's vertices.
    The algorithm is called the 'Scan Line Method' """
    poly.sort(key=lambda p: p[1])
    intersections = []
    for i in range(1, len(poly)):
        p1, p2 = poly[i-1], poly[i]
        if p1[1] <= y <= p2[1] or p2[1] <= y <= p1[1]:
            x_inter = (y-p1[1]) * (p2[0]-p1[0]) / (p2[1]-p1[1]) + p1[0]
            intersections.append(x_inter)
    intersections.sort()
    inside = False
    for j in range(0, len(intersections), 2):
        if intersections[j] <= x <= intersections[j+1]:
            inside = not inside
    return inside

if __name__ == "__main__":
    # Example polygon
    poly = [(0, 0), (2, 0), (2, 2), (0, 2)]

    # Test points
    test_points = [(1, 1), (3, 1), (1, 3), (-1, 1)]

    # Print the results
    for point in test_points:
        print(f"Point {point} is inside the polygon: {point_inside_polygon(point[0], point[1], poly)}")
        print(f"Point {point} is inside the polygon: {point_inside_polygon_scanline(point[0], point[1], poly)}")

输出结果：

Point (1, 1) is inside the polygon: True
Point (1, 1) is inside the polygon: True
Point (3, 1) is inside the polygon: False
Point (3, 1) is inside the polygon: False
Point (1, 3) is inside the polygon: False
Point (1, 3) is inside the polygon: False
Point (-1, 1) is inside the polygon: False
Point (-1, 1) is inside the polygon: False