Convex hull computation in SciPy - Mini Project: Build & Apply

Convex Hull Computation with SciPy

📖 Scenario: Imagine you have a set of points on a map representing locations of trees in a park. You want to find the smallest fence that can enclose all these trees. This fence shape is called the convex hull.

🎯 Goal: You will create a list of points, configure a helper variable, compute the convex hull using SciPy, and finally print the vertices of the hull.

📋 What You'll Learn

Create a list of 2D points with exact coordinates

Create a variable to hold the number of points

Use SciPy's ConvexHull to compute the hull

Print the indices of the hull vertices

💡 Why This Matters

🌍 Real World

Convex hulls are used in geography to find boundaries around locations, in robotics for obstacle avoidance, and in computer graphics for shape analysis.

💼 Career

Understanding convex hulls helps in data analysis, spatial data processing, and computational geometry tasks common in data science and machine learning roles.

1

Create the list of points

Create a variable called points that is a list of these exact 2D points: [1, 1], [2, 5], [3, 3], [5, 3], [3, 2], [2, 2].

SciPy

# Create the list of points called points
# Your code here

Hint

Use a list of lists to store the points exactly as shown.

2

Create a variable for the number of points

Create a variable called num_points that stores the number of points in the points list.

SciPy

points = [[1, 1], [2, 5], [3, 3], [5, 3], [3, 2], [2, 2]]
# Create num_points as the length of points
# Your code here

Hint

Use the len() function to count the points.

3

Compute the convex hull

Import ConvexHull from scipy.spatial. Then create a variable called hull by passing points to ConvexHull.

SciPy

points = [[1, 1], [2, 5], [3, 3], [5, 3], [3, 2], [2, 2]]
num_points = len(points)
# Import ConvexHull and compute hull from points
# Your code here

Hint

Use from scipy.spatial import ConvexHull and then call ConvexHull(points).

4

Print the convex hull vertices

Print the vertices attribute of the hull object to show the indices of points forming the convex hull.

SciPy

from scipy.spatial import ConvexHull

points = [[1, 1], [2, 5], [3, 3], [5, 3], [3, 2], [2, 2]]
num_points = len(points)
hull = ConvexHull(points)
# Print the hull vertices
# Your code here

Hint

Use print(hull.vertices) to show the hull points.

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