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

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)