The Python library continuoussets provides classes for the representation of continuous sets and the evaluation of set operations on them.
The library continuoussets
requires Python 3.9 or higher.
The only direct dependencies are numpy, scipy, and matplotlib.
Installation via PyPI is recommended:
pip install continuoussets
Proficient users may choose their own installation path.
The implemented classes inherit from the abstract base class ConvexSet
.
They represent continuous sets of n-dimensional vectors.
[!TIP] Many operations, notably including the constructors, also support scalar types, such as
int
andfloat
, as well as vectors defined using the typelist
. However, these are internally converted to numpy arrays, which may slow down the computation.[!IMPORTANT] The usage of keyword arguments for constructors is mandatory.
An interval I
is defined using a lower bound lb
and an upper bound ub
:
I = { x | lb <= x <= ub }
where lb
and ub
are vectors of equal length, and the inequality holds elementwise.
The Interval
class allows to instantiate such objects:
I = Interval(lb = numpy.array([-2., 0.]), ub = numpy.array([2., 1.]))
A zonotopes Z
is defined using a center c
and a generator matrix G
:
Z = { c + sum_i G_i a_i | -1 <= a_i <= 1 }
where G_i
are the columns of G
.
In contrast to intervals, zonotopes can represent dependencies between different dimensions.
The Zonotope
class allows to instantiate such objects:
Z = Zonotope(c = numpy.array([1., 0.]), G = numpy.array([[1., 0., 2.], [-1., 1., 1.]]))
Many standard set operations are implemented:
boundary_point
: Computation of boundary pointscartesian_product
: Cartesian productcompact
: Reduction to the minimal set representation size without changing the setconvex_hull
: Convex hullmatmul
: Linear mapminkowski_sum
: Minkowski summinkowski_difference
: Minkowski/Pontryagin differenceproject
: Projection of the set onto an axis-aligned subspacereduce
: Reduction of the set representation size, possibly incurring enlargementsupport_function
: Support function evaluationvertices
: Vertex enumerationvolume
: Volume computation[!IMPORTANT] All operations return new class instances.
[!NOTE] Operands for binary operations can also be vectors, represented by 1D numpy arrays.
[!TIP] Many operations support various evaluation modes via the keyword argument
mode
. These detail whether an exact solution, an outer approximation or an inner approximation should be computed. Not all modes are supported for each operation, some operations cannot be evaluated exactly, and runtime may differ strongly between modes.
Furthermore, the following checks are supported:
contains
: Containment of one set or vector in another setintersects
: Intersection between a set and another set or vector__eq__
: Equality of a set and another set or vectorrepresents
: Equivalent representation of a set by another set representationThe Interval
class additionally supports interval arithmetic for range bounding purposes. This includes
__add__
), subtraction (__sub__
), multiplication (__mul__
), division (__truediv__
), exponentiation (__pow__
)sin
, cos
, tan
, arcsin
, arccos
, arctan
sqrt
, log
, log10
This library is heavily inspired by the MATLAB toolbox CORA. However, this implementation is based on original sources, e.g.,