用python写三维泰森多边形代码

泰森多边形是一种用于最近邻搜索的数据结构，它可以快速地找到某个点周围最近的点集。以下是用Python实现三维泰森多边形的代码：

import numpy as np
from scipy.spatial import Delaunay

def compute_3d_convex_hull(points):
    """
    Computes the 3D convex hull of given points.
    """
    tri = Delaunay(points)
    return tri.convex_hull

def compute_3d_tetrahedra(points):
    """
    Computes the tetrahedra of the 3D convex hull of given points.
    """
    tri = Delaunay(points)
    return tri.simplices

def circumcenter(p1, p2, p3):
    """
    Computes the circumcenter of a triangle defined by three points.
    """
    m1 = np.array([p1,p2,p3])
    m2 = np.vstack((np.ones(3), np.sum(m1**2,axis=1)))
    v = np.linalg.solve(m2, np.array([1,1,1]))
    return np.dot(m1.T,v[1:])

def in_circumsphere(p1, p2, p3, p4):
    """
    Checks if a point p4 lies inside the circumsphere of a tetrahedron defined by three points.
    """
    m1 = np.array([p1,p2,p3,p4])
    m2 = np.hstack((np.ones((4,1)), m1))
    m3 = np.array([np.sum(m1**2,axis=1)]).T
    m4 = np.vstack((m2.T,m3.T))
    return np.linalg.det(m4) > 0

def compute_3d_alpha_shapes(points, alpha):
    """
    Computes the 3D alpha shape of given points.
    """
    tetrahedra = compute_3d_tetrahedra(points)
    edge_points = set()
    tri_points = set()
    for tetra in tetrahedra:
        for i, j, k in [(0,1,2), (1,0,3), (2,1,3), (0,2,3)]:
            triangle = (tetra[i], tetra[j], tetra[k])
            circum = circumcenter(points[triangle[0]],
                                  points[triangle[1]],
                                  points[triangle[2]])
            if all(np.linalg.norm(circum - points[pt]) <= alpha for pt in tetra if pt not in triangle):
                tri_points |= set(triangle)
                for edge in [(triangle[0],triangle[1]), (triangle[1],triangle[2]), (triangle[2],triangle[0])]:
                    if edge in edge_points:
                        edge_points.remove(edge)
                    else:
                        edge_points.add(edge)
    return tri_points, edge_points

def compute_3d_tessellation(points, alpha):
    """
    Computes the 3D alpha shape tessellation of given points.
    """
    tri_points, edge_points = compute_3d_alpha_shapes(points, alpha)
    vertices = sorted(set([v for e in edge_points for v in e]).union(tri_points))
    vertex_map = dict(zip(vertices, range(len(vertices))))
    triangles = [[vertex_map[v] for v in triangle] for triangle in tri_points]
    edges = [[vertex_map[v1], vertex_map[v2]] for v1, v2 in edge_points]
    return vertices, triangles, edges

以下是一个示例，演示如何使用3D alpha形来绘制一个球体上的点云：

import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d.art3d import Poly3DCollection

def plot_3d_tessellation(vertices, triangles, edges):
    """
    Plots the 3D alpha shape tessellation.
    """
    fig = plt.figure()
    ax = fig.add_subplot(111, projection='3d')
    ax.set_box_aspect([1,1,1])
    for triangle in triangles:
        poly = Poly3DCollection([vertices[triangle]], alpha=0.1, facecolor='b')
        ax.add_collection3d(poly)
    for edge in edges:
        ax.plot(vertices[edge,0], vertices[edge,1], vertices[edge,2], 'k-', alpha=0.2)
    ax.scatter(vertices[:,0], vertices[:,1], vertices[:,2], c='r', alpha=0.5, s=10)
    plt.show()

# Generate random points on a sphere
theta = np.random.uniform(0, 2*np.pi, 1000)
phi = np.random.uniform(0, np.pi, 1000)
x = np.sin(phi) * np.cos(theta)
y = np.sin(phi) * np.sin(theta)
z = np.cos(phi)
points = np.vstack((x,y,z)).T

# Compute the 3D alpha shape tessellation
vertices, triangles, edges = compute_3d_tessellation(points, 0.25)

# Plot the tessellation
plot_3d_tessellation(vertices, triangles, edges)

这将生成一个球体上的点云，并绘制其3D alpha形。可以通过调整alpha值来改变形状。