https://github.com/tebartsch/rportion
Represent unions of rectangular regions in the plane and efficiently obtain all maximum empty/used rectangles.
https://github.com/tebartsch/rportion
interval polygon python rectangle
Last synced: 29 days ago
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Represent unions of rectangular regions in the plane and efficiently obtain all maximum empty/used rectangles.
- Host: GitHub
- URL: https://github.com/tebartsch/rportion
- Owner: tebartsch
- License: mit
- Created: 2022-07-25T03:48:01.000Z (over 3 years ago)
- Default Branch: master
- Last Pushed: 2024-04-02T12:55:51.000Z (almost 2 years ago)
- Last Synced: 2025-12-08T07:58:49.992Z (about 1 month ago)
- Topics: interval, polygon, python, rectangle
- Language: Python
- Homepage:
- Size: 1.14 MB
- Stars: 5
- Watchers: 1
- Forks: 1
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- Changelog: CHANGELOG.md
- License: LICENSE
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README
# rportion - data structure and operations for rectilinear polygons
[](https://pypi.org/project/rportion/)
[](https://github.com/tilmann-bartsch/portion/actions/workflows/test.yaml)
[](https://coveralls.io/github/tilmann-bartsch/rportion?branch=master)
[](https://opensource.org/licenses/MIT)
[](https://github.com/tilmann-bartsch/rportion/commits/master)
The `rportion` library provides data structure to represent
2D [rectilinear polygons](https://en.wikipedia.org/wiki/Rectilinear_polygon) (unions of 2D-intervals) in Python 3.9+.
It is built upon the library [`portion`](https://github.com/AlexandreDecan/portion) and follows its concepts.
The following features are provided:
- 2D-Intervals (rectangles) which can be open/closed and finite/infinite at every boundary
- intersection, union, complement and difference of rectilinear polygons
- iterator over all maximum rectangles inside and outside a given polygon
In the case of integers/floats it can be used to keep track of the area resulting
from the union/difference of rectangles:
Internally the library uses an [interval tree](https://en.wikipedia.org/wiki/Interval_tree) to represent a polygon.
## Table of contents
* [Installation](#installation)
* [Documentation & usage](#documentation--usage)
* [Polygon creation](#polygon-creation)
* [Polygon bounds & attributes](#polygon-bounds--attributes)
* [Polygon operations](#polygon-operations)
* [Rectangle partitioning iterator](#rectangle-partitioning-iterator)
* [Maximum rectangle iterator](#maximum-rectangle-iterator)
* [Boundary](#boundary)
* [Internal data structure](#internal-data-structure)
* [Changelog](#changelog)
* [Contributions](#contributions)
* [License](#license)
## Installation
`rportion` can be installed from [PyPi](https://pypi.org/project/rportion/) with `pip` using
```bash
pip install rportion
```
Alternatively, clone the repository and run
```bash
pip install -e ".[test]"
python -m unittest discover -s tests
```
## Documentation & usage
### Polygon creation
Atomic polygons (rectangles) can be created by one of the following:
```python
>>> import rportion as rp
>>> rp.ropen(0, 2, 0, 1)
(x=(0,2), y=(0,1))
>>> rp.rclosed(0, 2, 0, 1)
(x=[0,2], y=[0,1])
>>> rp.ropenclosed(0, 2, 0, 1)
(x=(0,2], y=(0,1])
>>> rp.rclosedopen(0, 2, 0, 1)
(x=[0,2), y=[0,1))
>>> rp.rsingleton(0, 1)
(x=[0], y=[1])
>>> rp.rempty()
(x=(), y=())
```
Polygons can also be created by using two intervals of the underlying library
[`portion`](https://github.com/AlexandreDecan/portion):
```python
>>> import portion as P
>>> import rportion as rp
>>> rp.RPolygon.from_interval_product(P.openclosed(0, 2), P.closedopen(0, 1))
(x=(0,2], y=[0,1))
```
[↑ back to top](#table-of-contents)
### Polygon bounds & attributes
An `RPolygon` defines the following properties
- `empty` is true if the polygon is empty.
```python
>>> rp.rclosed(0, 2, 1, 2).empty
False
>>> rp.rempty().empty
True
```
- `atomic` is true if the polygon can be expressed by a single rectangle.
```python
>>> rp.rempty().atomic
True
>>> rp.rclosedopen(0, 2, 1, 2).atomic
True
>>> (rp.rclosed(0, 2, 1, 2) | rp.rclosed(0, 2, 1, 3)).atomic
True
>>> (rp.rclosed(0, 2, 1, 2) | rp.rclosed(1, 2, 1, 3)).atomic
False
```
- `enclosure` is the smallest rectangle containing the polygon.
```python
>>> (rp.rclosed(0, 2, 0, 2) | rp.rclosed(1, 3, 0, 1)).enclosure
(x=[0,3], y=[0,2])
>>> (rp.rclosed(0, 1, -3, 3) | rp.rclosed(-P.inf, P.inf, -1, 1)).enclosure
(x=(-inf,+inf), y=[-3,3])
```
- `enclosure_x_interval` is the smallest rectangle containing the polygon's extension in x-dimension.
```python
>>> (rp.rclosed(0, 2, 0, 2) | rp.rclosed(1, 3, 0, 1)).x_enclosure_interval
x=[0,3]
>>> (rp.rclosed(0, 1, -3, 3) | rp.rclosed(-P.inf, P.inf, -1, 1)).x_enclosure_interval
(-inf,+inf)
```
- `enclosure_y_interval` is the smallest interval containing the polygon's extension in y-dimension.
```python
>>> (rp.rclosed(0, 2, 0, 2) | rp.rclosed(1, 3, 0, 1)).y_enclosure_interval
[0,2]
>>> (rp.rclosed(0, 1, -3, 3) | rp.rclosed(-P.inf, P.inf, -1, 1)).y_enclosure_interval
[-3,3]
```
- `x_lower`, `x_upper`, `y_lower` and `y_upper` yield the boundaries of the rectangle enclosing
the polygon.
```python
>>> p = rp.rclosedopen(0, 2, 1, 3)
>>> p.x_lower, p.x_upper, p.y_lower, p.y_upper
(0, 2, 1, 3)
```
- `x_left`, `x_right`, `y_left` and `y_right` yield the type of the boundaries of the rectangle enclosing
the polygon.
```python
>>> p = rp.rclosedopen(0, 2, 1, 3)
>>> p.x_left, p.x_right, p.y_left, p.y_right
(CLOSED, OPEN, CLOSED, OPEN)
```
[↑ back to top](#table-of-contents)
### Polygon operations
`RPolygon` instances support the following operations:
- `p.intersection(other)` and `p & other` return the intersection of two rectilinear polygons.
```python
>>> rp.rclosed(0, 2, 0, 2) & rp.rclosed(1, 3, 0, 1)
(x=[1,2], y=[0,1])
```
- `p.union(other)` and `p | other` return the union of two rectilinear polygons.
```python
>>> rp.rclosed(0, 2, 0, 2) | rp.rclosed(1, 3, 0, 1)
(x=[0,3], y=[0,1]) | (x=[0,2], y=[0,2])
```
Note that the resulting polygon is represented by the union of all maximal rectangles contained in
in the polygon, see [Maximum rectangle iterators](#maximum-rectangle-iterators).
- `p.complement()` and `~p` return the complement of the rectilinear polygon.
```python
>>> ~rp.ropen(-P.inf, 0, -P.inf, P.inf)
((x=[0,+inf), y=(-inf,+inf))
```
- `p.difference(other)` and `p - other` return the difference of two rectilinear polygons.
```python
rp.rclosed(0, 3, 0, 2) - rp.ropen(2, 4, 1, 3)
(x=[0,3], y=[0,1]) | (x=[0,2], y=[0,2])
```
Note that the resulting polygon is represented by the union of all maximal rectangles contained in
in the polygon, see [Maximum rectangle iterators](#maximum-rectangle-iterators).
[↑ back to top](#table-of-contents)
### Rectangle partitioning iterator
The method `rectangle_partitioning` of a `RPolygon` instance returns an iterator
over rectangles contained in the rectilinear polygon which disjunctively cover it. I.e.
```python
>>> poly = rp.rclosedopen(2, 5, 1, 4) | rp.rclosedopen(1, 8, 2, 3) | rp.rclosedopen(6, 8, 1, 3)
>>> poly = poly - rp.rclosedopen(4, 7, 2, 4)
>>> list(poly.rectangle_partitioning())
[(x=[1,4), y=[2,3)), (x=[2,5), y=[1,2)), (x=[6,8), y=[1,2)), (x=[2,4), y=[3,4)), (x=[7,8), y=[2,3))]
```
which can be visualized as follows:
**Left:** Simple Rectilinear polygon. The red areas are part of the polygon.
**Right:** Rectangles in the portion are shown with black borderlines. As it is visible
`rectangle_partitioning` prefers rectangles with long x-interval over
rectangles with long y-interval.
[↑ back to top](#table-of-contents)
### Maximum rectangle iterator
The method `maximal_rectangles` of a `RPolygon` instance returns an iterator over all maximal rectangles contained
in the rectilinear polygon.
A maximal rectangle is rectangle in the polygon which is not a real subset of any other rectangle contained in
the rectilinear polygon. I.e.
```python
>>> poly = rp.rclosedopen(2, 5, 1, 4) | rp.rclosedopen(1, 8, 2, 3) | rp.rclosedopen(6, 8, 1, 3)
>>> poly = poly - rp.rclosedopen(4, 7, 2, 4)
>>> list(poly.maximal_rectangles())
[(x=[1, 4), y = [2, 3)), (x=[2, 5), y = [1, 2)), (x=[6, 8), y = [1, 2)), (x=[2, 4), y = [1, 4)), (x=[7, 8), y = [1, 3))]
```
which can be visualized as follows:
**Left:** Simple Rectilinear polygon. The red areas are part of the polygon.
**Right:** Maximal contained rectangles are drawn above each other transparently.
[↑ back to top](#table-of-contents)
## Boundary
The method `boundary` of a `RPolygon` instance returns another `RPolygon` instance representing the boundary of
the polygon. I.e.
```python
>>> poly = rp.closed(0, 1, 2, 3)
>>> poly.boundary()
(x=[1,2], y=[3]) | (x=[1,2], y=[4]) | (x=[1], y=[3,4]) | (x=[2], y=[3,4])
```
[↑ back to top](#table-of-contents)
## Internal data structure
The polygon is internally stored using an [interval tree](https://en.wikipedia.org/wiki/Interval_tree). Every
node of the tree corresponds to an interval in x-dimension which is representable by boundaries (in x-dimension)
present in the polygon. Each node contains an 1D-interval (by using the library
[`portion`](https://github.com/AlexandreDecan/portion)) in y-dimension. Combining those 1D-intervals
yields a rectangle contained in the polygon.
I.e. for the rectangle `(x=[0, 2), y=[1, 3))` this can be visualized as follows.
```
interval tree with x-interval corresponding y-interval stored in
a lattice-like shape to each node each node
┌─x─┐ ┌─(-∞,+∞)─┐ ┌─()──┐
│ │ │ │ │ │
┌─x─┬─x─┐ ┌─(-∞,2)──┬──[0,+∞)─┐ ┌─()──┬──()─┐
│ │ │ │ │ │ │ │ │
x x x (-∞,0] [0,2) [2,+∞) () [1,3) ()
```
The class `RPolygon` used this model by holding three data structures.
- `_x_boundaries`: Sorted list of necessary boundaries in x-dimension with type (`OPEN` or `CLOSED`)
- `_used_y_ranges`: List of lists in a triangular shape representing the interval tree for the
space occupied by the rectilinear polygon.
- `_free_y_ranges`: List of list in a triangular shape representing the interval tree of
for the space not occupied by the rectilinear polygon.
Note that a separate data structure for the area outside the polygon is kept.
This is done in order to be able to obtain the complement of a polygon efficiently.
For the example shown above this is:
```python
>>> poly = rp.rclosedopen(0, 2, 1, 3)
>>> poly._x_boundaries
SortedList([(-inf, OPEN), (0, OPEN), (2, OPEN), (+inf, OPEN)])
>>> poly._used_y_ranges
[[(), (), ()],
[(), [1,3)],
[()]]
>>> poly._free_y_ranges
[[(-inf,1) | [3,+inf), (-inf,1) | [3,+inf), (-inf,+inf)],
[(-inf,1) | [3,+inf), (-inf,1) | [3,+inf)],
[(-inf,+inf)]]
```
You can use the function `data_tree_to_string` as noted below to print the internal data structure in a tabular format:
```python
>>> poly = rp.rclosedopen(0, 2, 1, 3)
>>> print(data_tree_to_string(poly._x_boundaries, poly._used_y_ranges, 6))
| +inf 2 0
----------------+------------------
-inf (OPEN)| () () ()
0 (CLOSED)| () [1,3)
2 (CLOSED)| ()
```
```python
>>> poly = rp.rclosedopen(2, 5, 1, 4) | rp.rclosedopen(1, 8, 2, 3) | rp.rclosedopen(6, 8, 1, 3)
>>> poly = poly - rp.rclosedopen(4, 7, 2, 4)
>>> print(data_tree_to_string(poly._x_boundaries, poly._used_y_ranges, 6))
| +inf 8 7 6 5 4 2 1
----------------+------------------------------------------------
-inf (OPEN)| () () () () () () () ()
1 (CLOSED)| () () () () () [2,3) [2,3)
2 (CLOSED)| () () () () [1,2) [1,4)
4 (CLOSED)| () () () () [1,2)
5 (CLOSED)| () () () ()
6 (CLOSED)| () [1,2) [1,2)
7 (CLOSED)| () [1,3)
```
```python
def data_tree_to_string(x_boundaries,
y_intervals,
spacing: int):
col_space = 10
n = len(y_intervals)
msg = " " * (spacing + col_space) + "|"
for x_b in x_boundaries[-1:0:-1]:
msg += f"{str(x_b.val):>{spacing}}"
msg += "\n" + f"-" * (spacing+col_space) + "+"
for i in range(n):
msg += f"-" * spacing
msg += "\n"
for i, row in enumerate(y_intervals):
x_b = x_boundaries[i]
msg += f"{str((~x_b).val) + ' (' + str((~x_b).btype) + ')':>{spacing+ col_space}}|"
for val in row:
msg += f"{str(val):>{spacing}}"
msg += "\n"
return msg
```
[↑ back to top](#table-of-contents)
## Changelog
This library adheres to a [semantic versioning](https://semver.org/) scheme.
See [CHANGELOG.md](https://github.com/tilmann-bartsch/rportion/blob/master/CHANGELOG.md) for the list of changes.
## Contributions
Contributions are very welcome! Feel free to report bugs or suggest new features using GitHub issues and/or pull requests.
## License
Distributed under [MIT License](https://github.com/tilmann-bartsch/rportion/blob/master/LICENSE).