Calculate the area of intersection of two rotated rectangles in python

Question

I have two 2D rotated rectangles, defined as an (center x,center y, height, width) and an angle of rotation (0-360°). How would I calculate the area of intersection of these two

Answer 1

Here is a solution that does not use any libraries outside of Python's standard library.

Determining the area of the intersection of two rectangles can be divided in two subproblems:

• Finding the intersection polygon, if any;
• Determine the area of the intersection polygon.

Both problems are relatively easy when you work with the vertices (corners) of the rectangles. So first you have to determine these vertices. Assuming the coordinate origin is in the center of the rectangle, the vertices are, starting from the lower left in a counter-clockwise direction: (-w/2, -h/2), (w/2, -h/2), (w/2, h/2), and (-w/2, h/2). Rotating this over the angle a, and translating them to the proper position of the rectangle's center, these become: (cx + (-w/2)cos(a) - (-h/2)sin(a), cy + (-w/2)sin(a) + (-h/2)cos(a)), and similar for the other corner points.

A simple way to determine the intersection polygon is the following: you start with one rectangle as the candidate intersection polygon. Then you apply the process of sequential cutting (as described here. In short: you take each edges of the second rectangle in turn, and remove all parts from the candidate intersection polygon that are on the "outer" half plane defined by the edge (extended in both directions). Doing this for all edges leaves the candidate intersection polygon with only the parts that are inside the second rectangle or on its boundary.

The area of the resulting polygon (defined by a series of vertices) can be calculated from the coordinates of the vertices. You sum the cross products of the vertices of each edge (again in counter-clockwise order), and divide that by two. See e.g. www.mathopenref.com/coordpolygonarea.html

Enough theory and explanation. Here is the code:

from math import pi, cos, sin


class Vector:
    def __init__(self, x, y):
        self.x = x
        self.y = y

    def __add__(self, v):
        if not isinstance(v, Vector):
            return NotImplemented
        return Vector(self.x + v.x, self.y + v.y)

    def __sub__(self, v):
        if not isinstance(v, Vector):
            return NotImplemented
        return Vector(self.x - v.x, self.y - v.y)

    def cross(self, v):
        if not isinstance(v, Vector):
            return NotImplemented
        return self.x*v.y - self.y*v.x


class Line:
    # ax + by + c = 0
    def __init__(self, v1, v2):
        self.a = v2.y - v1.y
        self.b = v1.x - v2.x
        self.c = v2.cross(v1)

    def __call__(self, p):
        return self.a*p.x + self.b*p.y + self.c

    def intersection(self, other):
        # See e.g.     https://en.wikipedia.org/wiki/Line%E2%80%93line_intersection#Using_homogeneous_coordinates
        if not isinstance(other, Line):
            return NotImplemented
        w = self.a*other.b - self.b*other.a
        return Vector(
            (self.b*other.c - self.c*other.b)/w,
            (self.c*other.a - self.a*other.c)/w
        )


def rectangle_vertices(cx, cy, w, h, r):
    angle = pi*r/180
    dx = w/2
    dy = h/2
    dxcos = dx*cos(angle)
    dxsin = dx*sin(angle)
    dycos = dy*cos(angle)
    dysin = dy*sin(angle)
    return (
        Vector(cx, cy) + Vector(-dxcos - -dysin, -dxsin + -dycos),
        Vector(cx, cy) + Vector( dxcos - -dysin,  dxsin + -dycos),
        Vector(cx, cy) + Vector( dxcos -  dysin,  dxsin +  dycos),
        Vector(cx, cy) + Vector(-dxcos -  dysin, -dxsin +  dycos)
    )

def intersection_area(r1, r2):
    # r1 and r2 are in (center, width, height, rotation) representation
    # First convert these into a sequence of vertices

    rect1 = rectangle_vertices(*r1)
    rect2 = rectangle_vertices(*r2)

    # Use the vertices of the first rectangle as
    # starting vertices of the intersection polygon.
    intersection = rect1

    # Loop over the edges of the second rectangle
    for p, q in zip(rect2, rect2[1:] + rect2[:1]):
        if len(intersection) <= 2:
            break # No intersection

        line = Line(p, q)

        # Any point p with line(p) <= 0 is on the "inside" (or on the boundary),
        # any point p with line(p) > 0 is on the "outside".

        # Loop over the edges of the intersection polygon,
        # and determine which part is inside and which is outside.
        new_intersection = []
        line_values = [line(t) for t in intersection]
        for s, t, s_value, t_value in zip(
            intersection, intersection[1:] + intersection[:1],
            line_values, line_values[1:] + line_values[:1]):
            if s_value <= 0:
                new_intersection.append(s)
            if s_value * t_value < 0:
                # Points are on opposite sides.
                # Add the intersection of the lines to new_intersection.
                intersection_point = line.intersection(Line(s, t))
                new_intersection.append(intersection_point)

        intersection = new_intersection

    # Calculate area
    if len(intersection) <= 2:
        return 0

    return 0.5 * sum(p.x*q.y - p.y*q.x for p, q in
                     zip(intersection, intersection[1:] + intersection[:1]))


if __name__ == '__main__':
    r1 = (10, 15, 15, 10, 30)
    r2 = (15, 15, 20, 10, 0)
    print(intersection_area(r1, r2))

Answer 2

Such tasks are solved using computational geometry packages, e.g. Shapely:

import shapely.geometry
import shapely.affinity

class RotatedRect:
    def __init__(self, cx, cy, w, h, angle):
        self.cx = cx
        self.cy = cy
        self.w = w
        self.h = h
        self.angle = angle

    def get_contour(self):
        w = self.w
        h = self.h
        c = shapely.geometry.box(-w/2.0, -h/2.0, w/2.0, h/2.0)
        rc = shapely.affinity.rotate(c, self.angle)
        return shapely.affinity.translate(rc, self.cx, self.cy)

    def intersection(self, other):
        return self.get_contour().intersection(other.get_contour())


r1 = RotatedRect(10, 15, 15, 10, 30)
r2 = RotatedRect(15, 15, 20, 10, 0)

from matplotlib import pyplot
from descartes import PolygonPatch

fig = pyplot.figure(1, figsize=(10, 4))
ax = fig.add_subplot(121)
ax.set_xlim(0, 30)
ax.set_ylim(0, 30)

ax.add_patch(PolygonPatch(r1.get_contour(), fc='#990000', alpha=0.7))
ax.add_patch(PolygonPatch(r2.get_contour(), fc='#000099', alpha=0.7))
ax.add_patch(PolygonPatch(r1.intersection(r2), fc='#009900', alpha=1))

pyplot.show()

Answer 3

intersection, pnt = contourIntersection(rect1, rect2)

After looking at the possible duplicate page for this problem I couldn't find a completed answer for python so here is my solution using masking. This function will work with complex shapes on any angle, not just rectangles

You pass in the 2 contours of your rotated rectangles as parameters and it returns 'None' if no intersection occurs or an image of the intersected area and the left/top position of that image in relation to the original image the contours were taken from

Uses python, cv2 and numpy

import cv2
import math
import numpy as np


def contourIntersection(con1, con2, showContours=False):

    # skip if no bounding rect intersection
    leftmost1 = tuple(con1[con1[:, :, 0].argmin()][0])
    topmost1 = tuple(con1[con1[:, :, 1].argmin()][0])
    leftmost2 = tuple(con2[con2[:, :, 0].argmin()][0])
    topmost2 = tuple(con2[con2[:, :, 1].argmin()][0])

    rightmost1 = tuple(con1[con1[:, :, 0].argmax()][0])
    bottommost1 = tuple(con1[con1[:, :, 1].argmax()][0])
    rightmost2 = tuple(con2[con2[:, :, 0].argmax()][0])
    bottommost2 = tuple(con2[con2[:, :, 1].argmax()][0])

    if rightmost1[0] < leftmost2[0] or rightmost2[0] < leftmost1[0] or bottommost1[1] < topmost2[1] or bottommost2[1] < topmost1[1]:
        return None, None

    # reset top / left to 0
    left = leftmost1[0] if leftmost1[0] < leftmost2[0] else leftmost2[0]
    top = topmost1[1] if topmost1[1] < topmost2[1] else topmost2[1]

    newCon1 = []
    for pnt in con1:

        newLeft = pnt[0][0] - left
        newTop = pnt[0][1] - top

        newCon1.append([newLeft, newTop])
    # next
    con1_new = np.array([newCon1], dtype=np.int32)

    newCon2 = []
    for pnt in con2:

        newLeft = pnt[0][0] - left
        newTop = pnt[0][1] - top

        newCon2.append([newLeft, newTop])
    # next
    con2_new = np.array([newCon2], dtype=np.int32)

    # width / height
    right1 = rightmost1[0] - left
    bottom1 = bottommost1[1] - top
    right2 = rightmost2[0] - left
    bottom2 = bottommost2[1] - top

    width = right1 if right1 > right2 else right2
    height = bottom1 if bottom1 > bottom2 else bottom2

    # create images
    img1 = np.zeros([height, width], np.uint8)
    cv2.drawContours(img1, con1_new, -1, (255, 255, 255), -1)

    img2 = np.zeros([height, width], np.uint8)
    cv2.drawContours(img2, con2_new, -1, (255, 255, 255), -1)

    # mask images together using AND
    imgIntersection = cv2.bitwise_and(img1, img2)

    if showContours:
        img1[img1 > 254] = 128
        img2[img2 > 254] = 100

        imgAll = cv2.bitwise_or(img1, img2)
        cv2.imshow('Merged Images', imgAll)

    # end if

    if not imgIntersection.sum():
        return None, None

    # trim
    while not imgIntersection[0].sum():
        imgIntersection = np.delete(imgIntersection, (0), axis=0)
        top += 1

    while not imgIntersection[-1].sum():
        imgIntersection = np.delete(imgIntersection, (-1), axis=0)

    while not imgIntersection[:, 0].sum():
        imgIntersection = np.delete(imgIntersection, (0), axis=1)
        left += 1

    while not imgIntersection[:, -1].sum():
        imgIntersection = np.delete(imgIntersection, (-1), axis=1)

    return imgIntersection, (left, top)
# end function

To complete the answer so you can use the above function with the values of CenterX, CenterY, Width, Height and Angle of 2 rotated rectangles I have added the below functions. Simple change the Rect1 and Rect2 properties at the bottom of the code to your own

def pixelsBetweenPoints(xy1, xy2):
    X = abs(xy1[0] - xy2[0])
    Y = abs(xy1[1] - xy2[1])

    return int(math.sqrt((X ** 2) + (Y ** 2)))
# end function


def rotatePoint(angle, centerPoint, dist):
    xRatio = math.cos(math.radians(angle))
    yRatio = math.sin(math.radians(angle))
    xPotted = int(centerPoint[0] + (dist * xRatio))
    yPlotted = int(centerPoint[1] + (dist * yRatio))
    newPoint = [xPotted, yPlotted]

    return newPoint
# end function


def angleBetweenPoints(pnt1, pnt2):
    A_B = pixelsBetweenPoints(pnt1, pnt2)

    pnt3 = (pnt1[0] + A_B, pnt1[1])
    C = pixelsBetweenPoints(pnt2, pnt3)

    angle = math.degrees(math.acos((A_B * A_B + A_B * A_B - C * C) / (2.0 * A_B * A_B)))

    # reverse if above horizon
    if pnt2[1] < pnt1[1]:
        angle = angle * -1
    # end if

    return angle
# end function


def rotateRectContour(xCenter, yCenter, height, width, angle):
    # calc positions
    top = int(yCenter - (height / 2))
    left = int(xCenter - (width / 2))
    right = left + width

    rightTop = (right, top)
    centerPoint = (xCenter, yCenter)

    # new right / top point
    rectAngle = angleBetweenPoints(centerPoint, rightTop)
    angleRightTop = angle + rectAngle
    angleRightBottom = angle + 180 - rectAngle
    angleLeftBottom = angle + 180 + rectAngle
    angleLeftTop = angle - rectAngle

    distance = pixelsBetweenPoints(centerPoint, rightTop)
    rightTop_new = rotatePoint(angleRightTop, centerPoint, distance)
    rightBottom_new = rotatePoint(angleRightBottom, centerPoint, distance)
    leftBottom_new = rotatePoint(angleLeftBottom, centerPoint, distance)
    leftTop_new = rotatePoint(angleLeftTop, centerPoint, distance)

    contourList = [[leftTop_new], [rightTop_new], [rightBottom_new], [leftBottom_new]]
    contour = np.array(contourList, dtype=np.int32)

    return contour
# end function


# rect1
xCenter_1 = 40
yCenter_1 = 20
height_1 = 200
width_1 = 80
angle_1 = 45

rect1 = rotateRectContour(xCenter_1, yCenter_1, height_1, width_1, angle_1)

# rect2
xCenter_2 = 80
yCenter_2 = 25
height_2 = 180
width_2 = 50
angle_2 = 123

rect2 = rotateRectContour(xCenter_2, yCenter_2, height_2, width_2, angle_2)

intersection, pnt = contourIntersection(rect1, rect2, True)

if intersection is None:
    print('No intersection')
else:
    print('Area of intersection = ' + str(int(intersection.sum() / 255)))
    cv2.imshow('Intersection', intersection)
# end if

cv2.waitKey(0)