Scipy 3D Delaunay. Different results when translating the set of points

Question

I thought that given a set of points, the 3D Delaunay network would be unique. But moving all the points (without changing the relative distances) gives a different result.

import numpy as np
from scipy.spatial import Delaunay

points = np.array([ [-0.6, -0.1, -1.5], [0, 0, 0], [-1,-1,-1],[1, 0, 1], [1, 1, 2], [0, 1.1, -1]])
points = points+[0,5,0] #for translation

tri = Delaunay(points)

I have compared the results and the connection matrices, and they differ.

Case 0. No translation. Connection matrix.

array([[0., 1., 1., 1., 0., 0.],
       [1., 0., 1., 1., 1., 1.],
       [1., 1., 0., 1., 0., 1.],
       [1., 1., 1., 0., 1., 1.],
       [0., 1., 0., 1., 0., 1.],
       [0., 1., 1., 1., 1., 0.]])

Image.
enter image description here

Case 1. Translating 5 units in y-direction. Connection matrix.

array([[0., 1., 0., 1., 0., 1.],
       [1., 0., 1., 1., 1., 0.],
       [0., 1., 0., 1., 0., 1.],
       [1., 1., 1., 0., 1., 1.],
       [0., 1., 0., 1., 0., 1.],
       [1., 0., 1., 1., 1., 0.]])

Image.
enter image description here

Any idea why they are different?

The connection matrix is computed this way:

matrix = np.zeros((len(points),len(points))

for triangle in tri.simplices:
    for i in range(4): 
        point_initial = triangulo[i] 
        point_final = triangulo[(i+1)%4] 

        if matrix[point_initial,point_final] == 1:
            continue
        
        matrix[point_initial,point_final] = 1
        matrix[point_final,point_initial] = 1

You can see tri.simplices are also different in both cases:

array([[1, 4, 3, 2],
       [1, 4, 5, 2],
       [1, 4, 5, 3],
       [1, 0, 3, 2],
       [1, 0, 5, 2],
       [1, 0, 5, 3]], dtype=int32)

array([[1, 4, 3, 2],
       [1, 4, 5, 2],
       [1, 4, 5, 3],
       [0, 1, 3, 2],
       [0, 1, 5, 2],
       [0, 1, 5, 3]], dtype=int32)

Asked By: geofisue

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Source

Answer 1

@geofisue Thanks for the proposition for writing the answer.

As I described in the comment, the reason for the problem is that when transitioning from a 2D plane point to a 3D solid point, the simplices in the triangulation change from a triangle to a triangular pyramid.

So the number of the connections change from 3 to 6.

Then I modified the function for calculating the connection matrix, which can be used in nD:

import itertools
matrix = np.zeros((len(points),len(points)))
    
for triangle in tri.simplices:
    connections = list(itertools.combinations(triangle, 2))
    for connection in connections:
        point_initial = connection [0] 
        point_final = connection[1]

        if matrix[point_initial,point_final] == 1:
            continue
        
        matrix[point_initial,point_final] = 1
        matrix[point_final,point_initial] = 1

The full codes:

import itertools
import numpy as np
from scipy.spatial import Delaunay
import matplotlib.pyplot as plt

points = np.array([ [-0.6, -0.1, -1.5], [0, 0, 0], [-1,-1,-1],[1, 0, 1], [1, 1, 2], [0, 1.1, -1]])
points_trans = points+[0,5,0] #for translation

tri = Delaunay(points)
tri_trans = Delaunay(points_trans)



def get_connection_matrix(points):
    matrix = np.zeros((len(points),len(points)))
        
    for triangle in tri.simplices:
        connections = list(itertools.combinations(triangle, 2))
        for connection in connections:
            point_initial = connection [0] 
            point_final = connection[1]
    
            if matrix[point_initial,point_final] == 1:
                continue
            
            matrix[point_initial,point_final] = 1
            matrix[point_final,point_initial] = 1
    return matrix

matrix =  get_connection_matrix(points)
matrix_trans =  get_connection_matrix(points_trans)

print(np.array_equal(matrix, matrix_trans))

The result :

> True

Answered By: HMH1013

Scipy 3D Delaunay. Different results when translating the set of points

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