python实现Delaunay三角网算法,完成XYMaxMin函数BuildMaxTriangle函数ContainsInCircumCircle函数完成Delaunay函数
时间: 2024-12-08 21:23:25 浏览: 17
在Python中实现Delaunay三角网(也称为Delaunay三角形)通常会涉及一些几何计算库,如`scipy.spatial`或者自定义几何操作。下面简要概述三个关键函数的实现:
1. **XYMaxMin函数**(BuildMaxTriangle): 这个函数用于从一组点中找出最大三角形。它通常涉及到对所有可能的三角形组合进行比较,找到边长最大的那个作为候选。假设我们有一个点集`points`:
```python
import numpy as np
def BuildMaxTriangle(points):
max_area = 0
max_triangle = None
for i in range(len(points)):
for j in range(i+1, len(points)):
for k in range(j+1, len(points)):
# 计算当前三角形面积
area = _triangle_area(points[i], points[j], points[k])
if area > max_area:
max_area = area
max_triangle = (i, j, k)
return max_triangle
# 自定义三角形面积计算函数
def _triangle_area(p1, p2, p3):
a = np.linalg.norm(p2 - p1)
b = np.linalg.norm(p3 - p1)
c = np.linalg.norm(p3 - p2)
s = (a + b + c) / 2
return np.sqrt(s * (s - a) * (s - b) * (s - c))
```
2. **ContainsInCircumCircle函数**:检查给定点是否位于某个三角形的外接圆内。这需要计算三角形的重心、半周长以及三点之间的距离,然后判断该点到重心的距离小于等于半周长减去两点间的距离的一半。如果成立,则包含。
```python
def ContainsInCircumCircle(triangle_points, point):
circum_center = _circum_center(triangle_points)
a, b, c = triangle_points
pa = circum_center - a
pb = circum_center - b
pc = circum_center - c
# 计算半径 r
r = _radius(pa, pb, pc)
return _is_point_in_circle(point, circum_center, r)
def _circum_center(p1, p2, p3):
s = (p1 + p2 + p3) / 3
return s
def _radius(p1, p2, p3):
a = np.linalg.norm(p1 - p2)
b = np.linalg.norm(p2 - p3)
c = np.linalg.norm(p3 - p1)
s = (a + b + c) / 2
return np.sqrt(s * (s - a) * (s - b) * (s - c)) / s
def _is_point_in_circle(point, center, radius):
distance = np.linalg.norm(center - point)
return distance <= radius
```
3. **Delaunay函数**:这是一个完整的Delaunay三角化过程,它首先生成所有的可能三角形,然后使用`ContainsInCircumCircle`函数过滤出非法的(不满足Delauany条件的)三角形。剩下的就是合法的Delaunay三角网。
```python
from scipy.spatial import Voronoi, voronoi_plot_2d
def Delaunay(points):
triangles = []
for triangle in BuildMaxTriangle(points):
if not ContainsInCircumCircle(points[triangle], points):
continue
triangles.append(triangle)
# 使用Voronoi图生成更准确的Delaunay划分
vor = Voronoi(points)
for simplex in vor.ridge_vertices:
if simplex is not None and len(set(simplex)) == 3:
simplex.sort()
t = tuple(points[v] for v in simplex)
triangles.append(t)
return triangles
```
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