Delaunay triangulation in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Delaunay simplices array

What is the output of the following code that computes Delaunay triangulation simplices for given points?

SciPy

import numpy as np
from scipy.spatial import Delaunay
points = np.array([[0, 0], [1, 0], [0, 1], [1, 1]])
delaunay = Delaunay(points)
print(delaunay.simplices)

A

[[0 2 3]
 [0 3 1]]

B

[[0 1 2]
 [1 3 2]]

C

[[1 3 0]
 [3 2 0]]

D

[[2 3 1]
 [0 1 3]]

💡 Hint

Remember that Delaunay triangulation divides the convex hull into triangles connecting points.

The Delaunay triangulation for these four points forms two triangles: one with points 0,1,2 and another with points 1,3,2. The simplices attribute lists these triangles as arrays of point indices.

❓ data_output

intermediate

Number of triangles in Delaunay triangulation

Given 6 points arranged in a circle, how many triangles does the Delaunay triangulation produce?

SciPy

import numpy as np
from scipy.spatial import Delaunay
angles = np.linspace(0, 2*np.pi, 6, endpoint=False)
points = np.c_[np.cos(angles), np.sin(angles)]
delaunay = Delaunay(points)
print(len(delaunay.simplices))

A 4

B 10

C 8

D 6

💡 Hint

Delaunay triangulation of points on a circle forms triangles inside the polygon formed by the points.

For 6 points on a circle, the convex polygon is a hexagon. The triangulation divides it into 6 triangles.

🔧 Debug

advanced

Error raised by invalid input to Delaunay

What error does the following code raise when trying to create a Delaunay triangulation with only one point?

SciPy

import numpy as np
from scipy.spatial import Delaunay
points = np.array([[0, 0]])
delaunay = Delaunay(points)

A ValueError: too few points for triangulation

B TypeError: unsupported operand type(s)

C QhullError: input data must contain at least 3 points

D IndexError: index out of bounds

💡 Hint

Delaunay triangulation requires at least 3 points in 2D.

The scipy.spatial.Delaunay function uses Qhull internally, which raises QhullError if the input points are insufficient for triangulation.

🧠 Conceptual

advanced

Understanding convex hull relation to Delaunay triangulation

Which statement correctly describes the relationship between the convex hull and the Delaunay triangulation of a set of points?

A The convex hull is the union of all triangles in the Delaunay triangulation.

B The convex hull is the intersection of all triangles in the Delaunay triangulation.

C The convex hull is unrelated to the Delaunay triangulation.

D The convex hull is always smaller than the area covered by the Delaunay triangulation.

💡 Hint

Think about how the Delaunay triangulation covers the points.

The Delaunay triangulation connects points to form triangles that fill the convex hull area. Thus, the convex hull is the outer boundary enclosing all these triangles.

🚀 Application

expert

Using Delaunay triangulation to find nearest neighbors

Given a set of points, which code snippet correctly finds the indices of points that are neighbors of point 0 using Delaunay triangulation?

SciPy

import numpy as np
from scipy.spatial import Delaunay
points = np.array([[0, 0], [1, 0], [0, 1], [1, 1], [0.5, 0.5]])
delaunay = Delaunay(points)

A

neighbors = set()
for simplex in delaunay.simplices:
    if 0 in simplex:
        neighbors.update(simplex)
neighbors.remove(0)
print(sorted(neighbors))

B

neighbors = [i for i in range(len(points)) if i != 0]
print(neighbors)

C

neighbors = delaunay.neighbors[0]
print(neighbors[neighbors != -1])

D

neighbors = delaunay.vertex_neighbor_vertices[1][delaunay.vertex_neighbor_vertices[0][0]:delaunay.vertex_neighbor_vertices[0][1]]
print(neighbors)

💡 Hint

Neighbors share a triangle with the point of interest.

Option A correctly collects all points that share a simplex (triangle) with point 0, excluding point 0 itself. Other options either list all points or misuse attributes.