Convex hull computation in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Convex Hull vertices indices

What is the output of this code snippet that computes the convex hull of a set of 2D points using scipy?

SciPy

import numpy as np
from scipy.spatial import ConvexHull
points = np.array([[0, 0], [1, 1], [1, 0], [0, 1], [0.5, 0.5]])
hull = ConvexHull(points)
print(list(hull.vertices))

A [0, 2, 1, 3]

B [0, 1, 2, 3, 4]

C [4, 2, 1]

D [1, 2, 3]

💡 Hint

The convex hull vertices are the outermost points forming the polygon around all points.

The points at indices 0, 2, 1, and 3 form the convex polygon enclosing all points. The point at index 4 lies inside and is not part of the hull vertices.

❓ data_output

intermediate

Number of simplices in 3D convex hull

Given a set of 3D points, how many simplices (triangular faces) does the convex hull have?

SciPy

import numpy as np
from scipy.spatial import ConvexHull
points = np.random.rand(10, 3)
hull = ConvexHull(points)
print(len(hull.simplices))

A 8

B 10

C 20

D 16

💡 Hint

The number of simplices depends on the shape formed by the points, but for 10 random points in 3D, it is usually around 16.

For 10 random points in 3D, the convex hull typically has 16 triangular faces (simplices).

🔧 Debug

advanced

Identify the error in convex hull computation

What error does this code raise when trying to compute the convex hull of a single point?

SciPy

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

A IndexError: list index out of range

B QhullError: input must contain at least 3 points

C TypeError: unsupported operand type(s)

D No error, returns empty hull

💡 Hint

Convex hull requires at least 3 points in 2D to form a polygon.

The ConvexHull function raises a QhullError because the input has fewer than 3 points, which is insufficient to form a hull in 2D.

❓ visualization

advanced

Visualizing convex hull edges in 2D

Which option correctly plots the convex hull edges for a set of 2D points using matplotlib and scipy?

SciPy

import numpy as np
import matplotlib.pyplot as plt
from scipy.spatial import ConvexHull
points = np.random.rand(15, 2)
hull = ConvexHull(points)
plt.plot(points[:,0], points[:,1], 'o')
for simplex in hull.simplices:
    plt.plot(points[simplex, 0], points[simplex, 1], 'r-')
plt.show()

A Plots points and blue lines connecting all points, not just hull edges.

B Plots points but no lines connecting hull edges.

C Plots points and red lines connecting hull edges correctly.

D Raises an error due to incorrect indexing.

💡 Hint

The hull.simplices attribute contains pairs of indices for edges of the hull.

The code plots the points and draws red lines between points that form the convex hull edges, correctly visualizing the hull.

🚀 Application

expert

Using convex hull to filter outliers in 2D data

You have 2D data points with some outliers far from the main cluster. Which approach uses the convex hull to identify and remove outliers?

A Compute the convex hull of the main cluster only, then remove points outside this hull.

B Compute the convex hull of all points, then remove points inside the hull polygon.

C Compute the convex hull of all points, then remove points that are vertices of the hull.

D Compute the convex hull of all points, then remove points outside the hull's polygon.

💡 Hint

Outliers lie outside the main cluster's convex hull.

To remove outliers, first identify the main cluster, compute its convex hull, and remove points outside this hull. Removing hull vertices or points inside the hull does not remove outliers.

Practice

1. What does the scipy.spatial.ConvexHull function compute for a set of points?

easy

A. The smallest convex shape enclosing all points

B. The average distance between points

C. The maximum value among the points

D. The sum of all point coordinates

Solution

1. Step 1: Understand the purpose of ConvexHull

   The ConvexHull function finds the smallest convex polygon that contains all given points.

2. Step 2: Compare options with definition

   Only The smallest convex shape enclosing all points describes this smallest convex shape enclosing all points.

3. Final Answer:

   The smallest convex shape enclosing all points -> Option A

4. Quick Check:

   Convex hull = smallest convex shape [OK]

Hint: Convex hull wraps points tightly in a convex boundary [OK]

Common Mistakes:

• Confusing convex hull with average or sum calculations
• Thinking it finds maximum or minimum coordinate values
• Assuming it returns all points instead of boundary points

2. Which of the following is the correct way to import the ConvexHull class from scipy?

easy

A. from scipy.spatial import ConvexHull

B. import scipy.ConvexHull

C. from scipy import ConvexHull

D. import ConvexHull from scipy.spatial

Solution

1. Step 1: Recall correct import syntax in Python

   To import a class from a submodule, use 'from module.submodule import Class'.

2. Step 2: Match with scipy.spatial.ConvexHull

   The correct import is 'from scipy.spatial import ConvexHull'. Other options have syntax errors or wrong paths.

3. Final Answer:

   from scipy.spatial import ConvexHull -> Option A

4. Quick Check:

   Correct import = from scipy.spatial import ConvexHull [OK]

Hint: Use 'from module.submodule import Class' for importing classes [OK]

Common Mistakes:

• Using 'import scipy.ConvexHull' which is invalid syntax
• Trying 'import ConvexHull from ...' which is wrong order
• Importing directly from 'scipy' without 'spatial'

3. What will be the output of the following code?

import numpy as np
from scipy.spatial import ConvexHull
points = np.array([[0, 0], [1, 0], [1, 1], [0, 1], [0.5, 0.5]])
hull = ConvexHull(points)
print(hull.vertices)

medium

A. [1 2 3]

B. [0 1 2 3 4]

C. [4]

D. [0 1 2 3]

Solution

1. Step 1: Identify points forming the convex hull

   The points at indices 0,1,2,3 form the square boundary; point 4 is inside.

2. Step 2: Understand hull.vertices output

   hull.vertices returns indices of points forming the hull boundary, excluding interior points.

3. Final Answer:

   [0 1 2 3] -> Option D

4. Quick Check:

   Vertices exclude inside points = [0 1 2 3] [OK]

Hint: Hull vertices exclude points inside the shape [OK]

Common Mistakes:

• Including interior points in hull vertices
• Expecting all points as hull vertices
• Misunderstanding output as coordinates instead of indices

4. The following code throws an error. What is the likely cause?

import numpy as np
from scipy.spatial import ConvexHull
points = np.array([[0, 0], [1, 1]])
hull = ConvexHull(points)
print(hull.vertices)

medium

A. Missing import statement

B. Points are not in ascending order

C. Not enough points to form a convex hull

D. Points array has wrong data type

Solution

1. Step 1: Check minimum points needed for convex hull

   Convex hull in 2D requires at least 3 points to form a polygon.

2. Step 2: Analyze given points

   Only 2 points are given, so ConvexHull cannot compute a polygon, causing an error.

3. Final Answer:

   Not enough points to form a convex hull -> Option C

4. Quick Check:

   Minimum 3 points needed for 2D hull [OK]

Hint: Convex hull needs at least 3 points in 2D [OK]

Common Mistakes:

• Assuming any number of points works
• Thinking order of points matters for hull computation
• Ignoring error message about input size

5. You have a set of 2D points representing city locations. You want to find the smallest fence enclosing all cities. Which steps correctly use scipy.spatial.ConvexHull to find the fence points?

hard

A. Import ConvexHull, sort points by x-coordinate, use hull.simplices to get fence points

B. Import ConvexHull, create numpy array of points, compute hull, use hull.vertices to get fence points

C. Import ConvexHull, compute hull, use hull.area to get fence points

D. Import ConvexHull, compute hull, use hull.points to get fence points

Solution

1. Step 1: Prepare data and compute hull

   Import ConvexHull, convert city coordinates to numpy array, then compute hull = ConvexHull(points).

2. Step 2: Extract fence points using hull.vertices

   hull.vertices gives indices of points forming the convex hull boundary (fence).

3. Final Answer:

   Import ConvexHull, create numpy array of points, compute hull, use hull.vertices to get fence points -> Option B

4. Quick Check:

   hull.vertices = fence points [OK]

Hint: Use hull.vertices for boundary points, not hull.area or hull.points [OK]

Common Mistakes:

• Using hull.area which is a number, not points
• Using hull.points which is all points, not boundary
• Sorting points manually instead of using hull.vertices