Intersection and difference of two rectangles

Question

Searching the internet has not given a satisfactory solution for the following problem. Given a class Rectangle defined as the following:

class Rectangle:

    def __init__(self, x1, y1, x2, y2):
        if x1 > x2 or y1 > y2:
            raise ValueError('coordinates are invalid')
        self.x1, self.y1, self.x2, self.y2 = x1, y1, x2, y2

    @property
    def width(self):
        return self.x2 - self.x1

    @property
    def height(self):
        return self.y2 - self.y1

    def bounds(self, other):
        return Rectangle(min(self.x1, other.x1), min(self.y1, other.y1),
                         max(self.x2, other.x2), max(self.y2, other.y2))

    def intersect(self, other):
        return self.x1 < other.x2 and self.x2 > other.x1 and \
               self.y1 < other.y2 and self.y2 > other.y1

How would you create a method to get the intersection and a generator to get the difference of two rectangles? Presumably, a more complete implementation of the following methods are needed, but it is not clear to me what should be written.

def __and__(self, other):
    if self.intersect(other):
        # return a new rectangle that provides
        # the intersection between self and other
    return None

def __sub__(self, other):
    take_away = self & other
    if take_away is None:
        return self
    if take_away is self:
        return None
    return self.get_partitions(take_away)

def get_partitions(self, take_away):
    # yield 1 or 3 rectangles that are not part of take_away
    # alternative:
    # yield 1 or 2 overlapping rectangles not part of take_away

Does anyone have an elegant implementation for these methods? My hope is to avoid writing code for every conceivable case that might be encountered.

Solution 1

Here is a complete solution for you.
Methods in the class are ordered illogically so that the important parts are visible without scrolling.

import itertools

class Rectangle:
    def intersection(self, other):
        a, b = self, other
        x1 = max(min(a.x1, a.x2), min(b.x1, b.x2))
        y1 = max(min(a.y1, a.y2), min(b.y1, b.y2))
        x2 = min(max(a.x1, a.x2), max(b.x1, b.x2))
        y2 = min(max(a.y1, a.y2), max(b.y1, b.y2))
        if x1<x2 and y1<y2:
            return type(self)(x1, y1, x2, y2)
    __and__ = intersection

    def difference(self, other):
        inter = self&other
        if not inter:
            yield self
            return
        xs = {self.x1, self.x2}
        ys = {self.y1, self.y2}
        if self.x1<other.x1<self.x2: xs.add(other.x1)
        if self.x1<other.x2<self.x2: xs.add(other.x2)
        if self.y1<other.y1<self.y2: ys.add(other.y1)
        if self.y1<other.y2<self.y2: ys.add(other.y2)
        for (x1, x2), (y1, y2) in itertools.product(
            pairwise(sorted(xs)), pairwise(sorted(ys))
        ):
            rect = type(self)(x1, y1, x2, y2)
            if rect!=inter:
                yield rect
    __sub__ = difference

    def __init__(self, x1, y1, x2, y2):
        if x1>x2 or y1>y2:
            raise ValueError("Coordinates are invalid")
        self.x1, self.y1, self.x2, self.y2 = x1, y1, x2, y2

    def __iter__(self):
        yield self.x1
        yield self.y1
        yield self.x2
        yield self.y2

    def __eq__(self, other):
        return isinstance(other, Rectangle) and tuple(self)==tuple(other)
    def __ne__(self, other):
        return not (self==other)

    def __repr__(self):
        return type(self).__name__+repr(tuple(self))


def pairwise(iterable):
    # https://docs.python.org/dev/library/itertools.html#recipes
    a, b = itertools.tee(iterable)
    next(b, None)
    return zip(a, b)


# 1.
a = Rectangle(0, 0, 1, 1)
b = Rectangle(0.5, 0.5, 1.5, 1.5)
print(a&b)
# Rectangle(0.5, 0.5, 1, 1)
print(list(a-b))
# [Rectangle(0, 0, 0.5, 0.5), Rectangle(0, 0.5, 0.5, 1), Rectangle(0.5, 0, 1, 0.5)]

# 2.
b = Rectangle(0.25, 0.25, 1.25, 0.75)
print(a&b)
# Rectangle(0.25, 0.25, 1, 0.75)
print(list(a-b))
# [Rectangle(0, 0, 0.25, 0.25), Rectangle(0, 0.25, 0.25, 0.75), Rectangle(0, 0.75, 0.25, 1), Rectangle(0.25, 0, 1, 0.25), Rectangle(0.25, 0.75, 1, 1)]

# 3.
b = Rectangle(0.25, 0.25, 0.75, 0.75)
print(a&b)
# Rectangle(0.25, 0.25, 0.75, 0.75)
print(list(a-b))
# [Rectangle(0, 0, 0.25, 0.25), Rectangle(0, 0.25, 0.25, 0.75), Rectangle(0, 0.75, 0.25, 1), Rectangle(0.25, 0, 0.75, 0.25), Rectangle(0.25, 0.75, 0.75, 1), Rectangle(0.75, 0, 1, 0.25), Rectangle(0.75, 0.25, 1, 0.75), Rectangle(0.75, 0.75, 1, 1)]

# 4.
b = Rectangle(5, 5, 10, 10)
print(a&b)
# None
print(list(a-b))
# [Rectangle(0, 0, 1, 1)]

# 5.
b = Rectangle(-5, -5, 10, 10)
print(a&b)
# Rectangle(0, 0, 1, 1)
print(list(a-b))
# []

Intersection is based on SFML's implementation. It is proven correct and is not interesting to explain.

The difference, however, was a lot of fun to make.

Consider the following cases and compare them with corresponding examples at the bottom of the code. The method may return from 0 to 8 rectangles!

It works by finding all the vertical (xs) and horizontal (ys) lines that go through our rectangle (all the black and grey lines on the picture).

The coordinate sets are turned into sorted lists and taken pairwise ([a, b, c] becomes [(a, b), (b, c)]).

The product of such horizontal and vertical segments gives us all the rectangles that we divided the original one into by these lines.

All that remains is to yield all of these rectangles except the intersection.

Solution 2

Oleh Prypin was extremely helpful with the provided code. The following is a refactored version of the same.

from itertools import product, tee

def test():
    print('Example 1:')
    a = Rectangle(1, 1, 5, 5)
    b = Rectangle(3, 3, 7, 7)
    print(a & b)
    print(list(a - b))
    ##########################
    print('Example 2:')
    b = Rectangle(3, 2, 7, 4)
    print(a & b)
    print(list(a - b))
    ##########################
    print('Example 3:')
    b = Rectangle(2, 2, 4, 4)
    print(a & b)
    print(list(a - b))
    ##########################
    print('Example 4:')
    b = Rectangle(6, 2, 10, 6)
    print(a & b)
    print(list(a - b))
    ##########################
    print('Example 5:')
    b = Rectangle(0, 0, 6, 6)
    print(a & b)
    print(list(a - b))
    ##########################
    print('Example 6:')
    b = Rectangle(2, 0, 4, 6)
    print(a & b)
    print(list(a - b))

def pairwise(iterable):
    "s -> (s0, s1), (s1, s2), (s2, s3), ..."
    a, b = tee(iterable)
    next(b, None)
    return zip(a, b)

class Rectangle:

    __slots__ = '__x1', '__y1', '__x2', '__y2'

    def __init__(self, x1, y1, x2, y2):
        self.__setstate__((min(x1, x2), min(y1, y2), max(x1, x2), max(y1, y2)))

    def __repr__(self):
        return '{}({})'.format(type(self).__name__, ', '.join(map(repr, self)))

    def __eq__(self, other):
        return self.data == other.data

    def __ne__(self, other):
        return self.data != other.data

    def __hash__(self):
        return hash(self.data)

    def __len__(self):
        return 4

    def __getitem__(self, key):
        return self.data[key]

    def __iter__(self):
        return iter(self.data)

    def __and__(self, other):
        x1, y1, x2, y2 = max(self.x1, other.x1), max(self.y1, other.y1), \
                         min(self.x2, other.x2), min(self.y2, other.y2)
        if x1 < x2 and y1 < y2:
            return type(self)(x1, y1, x2, y2)

    def __sub__(self, other):
        intersection = self & other
        if intersection is None:
            yield self
        else:
            x, y = {self.x1, self.x2}, {self.y1, self.y2}
            if self.x1 < other.x1 < self.x2:
                x.add(other.x1)
            if self.y1 < other.y1 < self.y2:
                y.add(other.y1)
            if self.x1 < other.x2 < self.x2:
                x.add(other.x2)
            if self.y1 < other.y2 < self.y2:
                y.add(other.y2)
            for (x1, x2), (y1, y2) in product(pairwise(sorted(x)),
                                              pairwise(sorted(y))):
                instance = type(self)(x1, y1, x2, y2)
                if instance != intersection:
                    yield instance

    def __getstate__(self):
        return self.x1, self.y1, self.x2, self.y2

    def __setstate__(self, state):
        self.__x1, self.__y1, self.__x2, self.__y2 = state

    @property
    def x1(self):
        return self.__x1

    @property
    def y1(self):
        return self.__y1

    @property
    def x2(self):
        return self.__x2

    @property
    def y2(self):
        return self.__y2

    @property
    def width(self):
        return self.x2 - self.x1

    @property
    def height(self):
        return self.y2 - self.y1

    intersection = __and__

    difference = __sub__

    data = property(__getstate__)

if __name__ == '__main__':
    test()

Comments

[Noctis Skytower]

Assuming your implementation is correct, I should be able to use your code to implement the __and__ method in my class. Any ideas for the get_partitions method?

[Oleh Prypin]

@NoctisSkytower Updated with complete class.

[Noctis Skytower]

Thanks for your help! Being able to rely on the smarts of other people is rather nice sometimes.

[Noctis Skytower]

After finding some extra time, your code finally got refactored as seen down below.