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Polytopes in Python
  • Last Updated : 31 Dec, 2019

Python provides us with a third-party module to operate on polytopes. For one who doesn’t know what polytopes are, Polytopes are a geometrical form in an n-dimensional geometry corresponding to a polygon or polyhedron. Like other geometrical figures such as a square or a cube, a polytope isn’t limited to a single dimension. It can be 2-D, 3-D, or n-D.

polytopes

An n-dimensional polytope is generally represented as n-polytope. For example, a 2-D polytope will be represented as 2-polytope, a 3-D polytope as 3-polytope and so on. A polytope is stored inside Python using half-space representation or H-representation. This means that a polytope is represented as vector-matrix multiplication of two matrices and a vector:

Ax <= B

Here, A is an m*n matrix, x is a set of coordinates multiplied to A, and B is an m*1 column matrix. The polytope module will allow us to perform some geometrical operations on these polytopes. Let’s get started with the installation:

A polytope is a third-party module, so we would need to install it on our machine before we can use it. To install the module, just type this in the terminal:



pip install polytope

After installing, we will import the module into our IDE as:

import polytope as pc

We will now create a polytope using the polytope module:




# Python program to demonstrate
# polytopes
  
  
# Using numpy to create matrices
import numpy as np 
import polytope as pc
  
  
# Declaring numpy arrays
A = np.array([[1.0, 2.0],
              [3.0, -1.0],
              [-1.0, 4.0],
              [0.0, -2.0]])
  
b = np.array([2.0, 0.0, 3.0, 1.0])
  
p = pc.Polytope(A, b)
  
print(p)

Output:

polytopes

Here, we created matrices A and b using numpy arrays. Then we used the .Polytope() function to create a polytope using those matrices. Here, we didn’t specify any value of “x”.

Methods and Operations on Polytopes:

Now that we have created a simple polytope, we can perform some basic operations on them. For example, the polytope module gives us the power to compare two polytopes.
Consider two polytopes, p1 and p2, then:

  • p1 == p2: This means that every element of p1 is in p2 and every element of p2 is in p1. Hence, p1 and p2 are exactly identical.
  • p1 <= p2: This means that p1 is a subset of p2. Therefore, every element of p1 is in p2 but the same may not be true for p2.
  • p2 <= p1: This means that p2 is a subset of p1. Therefore, every element of p2 is in p1 but the same may not be true for p1.

We can also check if an element is in a polytope using:

[4.0, 5.0] in p1

Some basic mathematical operations available for polytopes include:



  • p1.diff(p2)
  • p1.union(p2)
  • p1.intersect(p2)

For example:




# Python program to demonstrate
# polytopes
  
  
# Using numpy to create matrices
import numpy as np 
import polytope as pc 
  
# Declaring numpy arrays
A = np.array([[1.02.0],
              [3.0, -1.0],
              [-1.0, 4.0],
              [0.0, -2.0]])
  
b = np.array([2.0, 0.0, 3.0, 1.0])
  
C = np.array([[1.00.0],
              [2.04.0],
              [3.0, -1.0],
              [1.09.0]]) 
  
d = np.array([2.0, 0.0, -1.0, 3.0])
  
p1 = pc.Polytope(A, b)
p2 = pc.Polytope(C, d)
  
# Using diff method
p1.diff(p2)

Output:

polytope

We can iterate over a region of polytopes. A region can be understood as a container containing two polytopes, where one is the beginning and the other is the end. Consider the following code.




# Python program to demonstrate
# polytopes
  
  
import numpy as np 
import polytope as pc 
  
  
# Declaring numpy arrays
A = np.array([[1.02.0],
              [3.0, -1.0],
              [-1.0, 4.0],
              [0.0, -2.0]])
  
b = np.array([2.0, 0.0, 3.0, 1.0])
  
C = np.array([[1.00.0],
              [2.04.0],
              [3.0, -1.0],
              [1.09.0]]) 
  
d = np.array([2.0, 0.0, -1.0, 3.0])
  
p1 = pc.Polytope(A, b)
p2 = pc.Polytope(C, d)
r = pc.Region([p1, p2])

The above code will create a region of two polytopes. This can be used as an iterator as:




for polytope in r:
    print(polytope)

output:

polytopes

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