Getting Started

The simplest tasks are pypolycontain is defining polytopic objects, performing operations, and visualizing them. First, import the package alongside numpy.

import numpy as np
import pypolycontain as pp

Objects

H-polytope

We define an H-polytope \(\mathbb{P}=\{x \in \mathbb{R}^2 | Hx \le h\}\). We give the following numericals for H and h:

\[\begin{split}H=\left( \begin{array}{cc} 1 & 1 \\ -1 & 1 \\ 0 & -1 \end{array}\right), h= \left( \begin{array}{c} 1 \\ 1 \\ 0 \end{array}\right).\end{split}\]

H=np.array([[1,1],[-1,1],[0,-1]])
h=np.array([1,1,0])
A=pp.H_polytope(H,h)

This a triangle as it is defined by intersection of 3 half-spaces in \(\mathbb{R}^2\). In order to visualizate the polytope, we call the following function. Note the brackets around visualize function - it takes in a list of polytopes as its primary argument.

pp.visualize([A],title=r'$A$')

AH-polytope

We define an AH-polytope as \(t+T\mathbb{P}\) with the following numbers. The transformation represents a rotation of \(30^\circ\) and translation in \(x\) direction.

t=np.array([5,0]).reshape(2,1) # offset
theta=np.pi/6 # 30 degrees
T=np.array([[np.cos(theta),np.sin(theta)],[-np.sin(theta),np.cos(theta)]]) # Linear transformation
B=pp.AH_polytope(t,T,A)
pp.visualize([B],title=r'$B$')

Zonotope

We define a zonotope as \(\mathbb{Z}=x+G[-1,1]^{n_p}\), where \(n_p\) is the number of rows in \(p\).

x=np.array([4,0]).reshape(2,1) # offset
G=np.array([[1,0,0.5],[0,0.5,-1]]).reshape(2,3)
C=pp.zonotope(x=x,G=G)
pp.visualize([C],title=r'$C$')

Visualization

The visualize function allows for visualizing multiple polytopic objects.

pp.visualize([A,C,B],title=r'$A,B,C$')

You may have noticed the colors. Here are the default colors for various polytopic objects. Using color argument you can change the color.

Object	Default Color
H-polytope	Red
Zonotope	Green
AH-polytope	Blue

Options

visualize has a set of options. While it only supports 2D plotting, it can take high dimensional polytopes alongside the argument list_of_dimensions. Its default is [0,1], meaning the projection into the first and second axis is demonstrated.

You can also add a separate subplot environment to plot the polytopes. Take the following example:

import matplotlib.pyplot as plt
fig,ax=plt.subplots()
fig.set_size_inches(6, 3)
pp.visualize([A,B,C],ax=ax,fig=fig)
ax.set_title(r'A triangle (red), rotated by 30 degrees (blue), and a zonotope (green)',FontSize=15)
ax.set_xlabel(r'$x$',FontSize=15)
ax.set_ylabel(r'$y$',FontSize=15)
ax.axis('equal')

(-1.343301270189222,
 6.209326673973662,
 -1.6499999999999997,
 1.6499999999999997)

Operations

pypolycontain supports a broad range of polytopic operations. The complete list of operations is here.

Minkowski sum

D=pp.operations.minkowski_sum(A,B)
D.color=(0.9, 0.9, 0.1)
pp.visualize([D,A,B],title=r'$A$ (red),$B$ (blue), $D=B\oplus C$ (yellow)')

Convex Hull

Let’s take the intersection of \(A\) and \(C\).

E=pp.convex_hull(A,C)
E.color='purple'
pp.visualize([E,A,C],alpha=0.5,title=r'$A$ (red),$C$ (green), $E={ConvexHull}(A,C)$ (purple)')

Intersection

Let’s take the intersection of \(B\) and \(D\).

F=pp.intersection(D,E)
F.color='black'
pp.visualize([D,E,F],alpha=1,title=r'$D$ (yellow), $E$ (purple), $F=D \cap E$ (black)')

Bounding Box

We can compute the bounding box of a polytopic object. For instance, let’s compute the bounding box of \(D\).

G=pp.bounding_box(D)
pp.visualize([G,D],title=r'$D$ and its bounding box')

Or the following:

mylist=[A,B,C]
pp.visualize([pp.bounding_box(p) for p in mylist]+mylist,title=r'$A,B,C$ and their bounding boxes')