pyraxial

An algebraic take on axis-aligned rectangles.

This module aims to simplify working with bounding boxes.

It defines the class Rect with the following methods and attributes:

• Two binary operators | ("join") and & ("meet").
• Two identity elements with respect to | and &, Rect.EMPTY and Rect.PLANE.
• Two polyadic class methods Rect.enclose(rects) and Rect.overlap(rects) as generalizations of | and & over arbitrary numbers of rectangles.
• A set of operators that define containment relations between rectangles.
• A class method Rect.closed_regions(rects) that computes the bounding boxes for all subsets of "transitively" overlapping rectangles in a given set of rectangles.

The Rect class together with the | and & operations and the identity elements forms a complete lattice so that for all Rect objects a, b and c the following laws hold:

Identity Elements:

    a | Rect.EMPTY  ==  a
    a & Rect.PLANE  ==  a

Absorbing Elements:

    a | Rect.PLANE  ==  Rect.PLANE
    a & Rect.EMPTY  ==  Rect.EMPTY

Idempotency:

    a | a  ==  a
    a & a  ==  a

Commutativity:

    a | b  ==  b | a
    a & b  ==  b & a

Associativity:

    (a | b) | c  ==  a | (b | c)
    (a & b) & c  ==  a & (b & c)

Absorption:

    a | (a & b)  ==  a
    a & (a | b)  ==  a

Since these laws already define a partially ordered set, the following laws also hold:

Least Element:

    Rect.EMPTY ≦ a

Greatest Element:

    a ≦ Rect.PLANE

Reflexivity:

    a ≦ a

Transitivity:

    a ≦ b  and  b ≦ c   🡒   a ≦ c

Antisymmetry:

    a ≦ b  and  b ≦ a   🡘   a = b

Monotonicity:

    a1 ≦ a2  and  b1 ≦ b2   🡒   a1 | b1  ≦  a2 | b2
    a1 ≦ a2  and  b1 ≦ b2   🡒   a1 & b1  ≦  a2 & b2

Semidistributivity:

    (a & b) | (a & c)  ≦  a & (b | c)
    a | (b & c)  ≦  (a | b) & (a | c)

Notice the absence of the laws of distribution and modularity.

A rectangle is created like so:

    r = Rect(box)

where box is an already existing Rect object, tuple, list, iterator or other iterable, provided it is either empty or contains/yields four numbers that denote the left, top, right and bottom coordinates (in that order). Otherwise, a ValueError is raised.

Coordinate values increase from left to right and from top to bottom. Therefor, if left ≦ right and top ≦ bottom the resulting rectangle will be a Rect with the specified coordinates. If left > right or top > bottom the resulting rectangle will equal Rect.EMPTY.

Rect objects are immutable and the properties have no setters.

All method results are covariant under subtyping.

Rect() and closed_regions() accept any type of iterable. The operators however work reliably only on sequence-like objects, but not iterators. If you pass an iterator as an argument, the behavior will be undefined, probably raising an exception, or worse, causing inexplicably wrong results.

Rects can be used as a drop-in in contexts where axis-aligned rectangles are represented by 4-tuples, like e.g. Pillow's Image.crop() method. For contexts where such rectangles are represented as pairs of point coordinates the class method Rect.from_points and the Rect.points property can be used.

See API documentation here:

https://pillmuncher.github.io/pyraxial