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.
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:
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([[1.0, 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]) C = np.array([[1.0, 0.0], [2.0, 4.0], [3.0, -1.0], [1.0, 9.0]]) 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:
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([[1.0, 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]) C = np.array([[1.0, 0.0], [2.0, 4.0], [3.0, -1.0], [1.0, 9.0]]) 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:
