手动实现Delaunay三角剖分算法

Delaunay三角剖分算法是一种常用的二维平面三角形剖分方法，常用于计算机图形学、地理信息系统、计算几何等领域。下面是一种手动实现的Delaunay三角剖分算法，供参考。

1. 输入点集P，初始化三角形集T为一个超级三角形
2. 将P中每个点插入到T中
3. 对于P中每个点p，查找在T中包含p的三角形t
4. 将t的三条边删除，得到三个新的三角形，如果这些三角形不满足Delaunay条件，则进行翻转操作
5. 将新的三角形加入T中
6. 删除T中所有与超级三角形相邻的三角形

以下是Python代码实现：

from typing import List, Tuple


class Triangle:
    def __init__(self, points: List[Tuple[float, float]]):
        self.points = points
        self.circumcircle = self.get_circumcircle()

    def get_circumcircle(self):
        x1, y1 = self.points[0]
        x2, y2 = self.points[1]
        x3, y3 = self.points[2]

        D = (x1 - x2) * (y2 - y3) - (x2 - x3) * (y1 - y2)
        if abs(D) < 1e-8:
            return None

        x = ((x1 ** 2 + y1 ** 2) * (y2 - y3) + (x2 ** 2 + y2 ** 2) * (y3 - y1) + (x3 ** 2 + y3 ** 2) * (y1 - y2)) / (2 * D)
        y = -((x1 ** 2 + y1 ** 2) * (x2 - x3) + (x2 ** 2 + y2 ** 2) * (x3 - x1) + (x3 ** 2 + y3 ** 2) * (x1 - x2)) / (2 * D)

        r = ((x - x1) ** 2 + (y - y1) ** 2) ** 0.5

        return (x, y, r)

    def contains(self, point: Tuple[float, float]):
        x, y = point
        x1, y1 = self.points[0]
        x2, y2 = self.points[1]
        x3, y3 = self.points[2]

        if (y2 - y1) * (x - x1) - (x2 - x1) * (y - y1) < 0:
            return False
        if (y3 - y2) * (x - x2) - (x3 - x2) * (y - y2) < 0:
            return False
        if (y1 - y3) * (x - x3) - (x1 - x3) * (y - y3) < 0:
            return False

        return True

    def flip(self):
        p1, p2, p3 = self.points

        if self.circumcircle is None:
            return []

        (xc, yc, r) = self.circumcircle

        for triangle in self.neighbours:
            if triangle is None:
                continue
            if triangle.circumcircle is None:
                continue

            (xc1, yc1, r1) = triangle.circumcircle
            if (xc - xc1) ** 2 + (yc - yc1) ** 2 < r ** 2 + r1 ** 2:
                p4 = [point for point in triangle.points if point not in self.points]
                break
        else:
            return []

        t1 = Triangle([p1, p2, p4])
        t2 = Triangle([p2, p3, p4])
        t3 = Triangle([p3, p1, p4])

        t1.add_neighbours([t2, t3, self.neighbours[2]])
        t2.add_neighbours([t3, t1, self.neighbours[0]])
        t3.add_neighbours([t1, t2, self.neighbours[1]])

        if self.neighbours[2] is not None:
            self.neighbours[2].swap_neighbour(self, t1)
        if self.neighbours[0] is not None:
            self.neighbours[0].swap_neighbour(self, t2)
        if self.neighbours[1] is not None:
            self.neighbours[1].swap_neighbour(self, t3)

        return [t1, t2, t3]

    def add_neighbours(self, neighbours: List['Triangle']):
        self.neighbours = neighbours

    def swap_neighbour(self, old: 'Triangle', new: 'Triangle'):
        if self.neighbours[0] == old:
            self.neighbours[0] = new
        elif self.neighbours[1] == old:
            self.neighbours[1] = new
        elif self.neighbours[2] == old:
            self.neighbours[2] = new

    def __eq__(self, other):
        if isinstance(other, Triangle):
            return set(self.points) == set(other.points)
        return False

    def __hash__(self):
        return hash(tuple(sorted(self.points)))


class Delaunay:
    def __init__(self, points: List[Tuple[float, float]]):
        self.points = points
        self.triangles = self.triangulate()

    def triangulate(self):
        x_min = min(self.points, key=lambda p: p[0])[0] - 1
        y_min = min(self.points, key=lambda p: p[1])[1] - 1
        x_max = max(self.points, key=lambda p: p[0])[0] + 1
        y_max = max(self.points, key=lambda p: p[1])[1] + 1

        super_triangle = Triangle([(x_min, y_min), (x_max, y_min), (x_min, y_max)])
        triangles = [super_triangle]

        for point in self.points:
            bad_triangles = []
            for triangle in triangles:
                if triangle.contains(point):
                    bad_triangles.append(triangle)

            polygon = []
            for triangle in bad_triangles:
                for p in triangle.points:
                    if all([p != tp for tp in point] for point in polygon):
                        polygon.append(p)

            for triangle in bad_triangles:
                triangles.remove(triangle)

            new_triangles = []
            for i in range(len(polygon)):
                new_triangles.append(Triangle([polygon[i], polygon[(i + 1) % len(polygon)], point]))

            for triangle in new_triangles:
                for other_triangle in new_triangles:
                    if triangle == other_triangle:
                        continue
                    if set(triangle.points).intersection(set(other_triangle.points)):
                        triangle.add_neighbours([other_triangle if p in other_triangle.points else None for p in triangle.points])

            for triangle in new_triangles:
                triangles.append(triangle)

        for triangle in triangles:
            if any([p in super_triangle.points for p in triangle.points]):
                continue
            for i in range(3):
                if triangle.neighbours[i] is None:
                    continue
                if any([p in super_triangle.points for p in triangle.neighbours[i].points]):
                    continue
                edge = {triangle.points[i], triangle.points[(i + 1) % 3]}
                for j in range(3):
                    if triangle.neighbours[i].points[j] not in edge:
                        break
                else:
                    triangle.neighbours[i] = None

        triangles = [triangle for triangle in triangles if not any([p in super_triangle.points for p in triangle.points])]

        return triangles

该实现中，Triangle类表示一个三角形，包含三个点和一个外接圆的信息。Delaunay类表示Delaunay三角剖分算法，包含一个点集和一个三角形集。在实现中，使用了超级三角形的概念，即将点集中所有点都插入到一个包含所有点的三角形中，然后再去除与超级三角形相邻的三角形。