The Delaunay triangulation is a popular method for dividing a given set of points into a set of non-overlapping triangles. It has a wide range of applications, including computer graphics, mesh generation, and computational geometry. In this blog post, we will explore how to use the scipy library in Python to perform Delaunay triangulation on boundary problems.

What is a Delaunay Triangulation?

A Delaunay triangulation of a set of points is a triangulation such that no point is inside the circumcircle of any triangle. In other words, it maintains the maximum minimum angle property, which makes it an excellent choice for generating well-shaped triangles.

Scipy Library

scipy is a popular scientific computing library in Python that provides numerous numerical algorithms and tools. It includes a module called scipy.spatial that provides functions for performing spatial computations, including Delaunay triangulation.

Performing Delaunay Triangulation on Boundary Problems

To perform Delaunay triangulation on boundary problems using scipy, we need to follow these steps:

Step 1: Import the Required Libraries

import numpy as np
from scipy.spatial import Delaunay

Step 2: Prepare the Boundary Points

boundary_points = np.array([[0, 0], [1, 0], [1, 1], [0, 1]])

Here, boundary_points is a 2D NumPy array containing the coordinates of the boundary points.

Step 3: Perform Delaunay Triangulation

triangulation = Delaunay(boundary_points)

The Delaunay function in scipy.spatial performs the Delaunay triangulation on the given set of points.

Step 4: Accessing Delaunay Triangles

triangles = boundary_points[triangulation.simplices]

The simplices attribute of the Delaunay object provides the indices of the vertices that form each triangle. We can use these indices to extract the triangle coordinates from the original boundary points.

Step 5: Visualization

import matplotlib.pyplot as plt
from matplotlib.tri import Triangulation

triangulation_plot = Triangulation(boundary_points[:, 0], boundary_points[:, 1], triangles=triangulation.simplices)
plt.triplot(triangulation_plot)
plt.plot(boundary_points[:, 0], boundary_points[:, 1], 'ro')
plt.show()

This code snippet visualizes the triangulation by plotting the triangles and the original boundary points using the matplotlib library.

Conclusion

In this blog post, we learned how to perform Delaunay triangulation on boundary problems using the scipy library in Python. The combination of scipy’s Delaunay function and matplotlib’s plotting capabilities allows us to generate and visualize well-shaped triangles for various applications.