How can you rotate an image around an off center pivot in Pygame

Question

I want to rotate an image around a pivot, which is not in the center of the Surface in Pygame .
The pivot is the green cross in the image:

I know the position of the pivot in the game window. How do I track the image at this point and I rotate it around this point simultaneously.

image = pygame.image.load("boomerang64.png")
pos = (200, 200)
angle = 0

while True:
    # [...]

    rotate_rect, rotate_image = ???? # rotate around green cross by angle 
    surf.blit(rotated_image, rotate_rect)
    angle += 1

    # [...]

Answer 1

First the position of the pivot on the Surface has to be defined:

image = pygame.image.load("boomerang64.png") # 64x64 surface
pivot = (48, 21)                             # position of the pivot on the image

When an image is rotated, then its size increase. We have to compare the axis aligned bounding box of the image before the rotation and after the rotation.
For the following math pygame.math.Vector2 is used. Note in screen coordinates the y points down the screen, but the mathematical y axis points form the bottom to the top. This causes that the y axis has to be "flipped" during calculations

Set up a list with the 4 corner points of the bounding box and rotate the vectors to the corner points by pygame.math.Vector2.rotate . Finally find the minimum of the rotated box. Since the y axis in pygame points downwards, this has to be compensated by finding the maximum of the rotated inverted height (" max(rotate(-h)) "):

w, h = image.get_size()
box = [pygame.math.Vector2(p) for p in [(0, 0), (w, 0), (w, -h), (0, -h)]]
box_rotate = [p.rotate(angle) for p in box]
min_x, min_y = min(box_rotate, key=lambda p: p[0])[0], max(box_rotate, key=lambda p: p[1])[1]

The computation of min_x and min_y can be improved by directly computing the x and y component of the rotated vectors by trigonometric functions:

w, h         = image.get_size()
sin_a, cos_a = math.sin(math.radians(angle)), math.cos(math.radians(angle)) 
min_x, min_y = min([0, sin_a*h, cos_a*w, sin_a*h + cos_a*w]), max([0, sin_a*w, -cos_a*h, sin_a*w - cos_a*h])

Calculate the "compensated" origin of the upper left point of the image by adding the minimum of the rotated box to the position in relation to the pivot on the image:

origin = (pos[0] - originPos[0] + min_x - pivot_move[0], pos[1] - originPos[1] - min_y + pivot_move[1])
rotated_image = pygame.transform.rotate(image, angle)
screen.blit(rotated_image, origin)

In the following example program, the function blitRotate(surf, image, pos, originPos, angle) does all the above steps and blit a rotated image to the Surface which is associated to the display:

• surf is the target Surface
• image is the Surface which has to be rotated and blit
• pos is the position of the pivot on the target Surface surf (relative to the top left of surf )
• originPos is position of the pivot on the image Surface (relative to the top left of image )
• angle is the angle of rotation in degrees

import pygame
import math

def blitRotate(surf, image, pos, originPos, angle):

    # calcaulate the axis aligned bounding box of the rotated image
    w, h         = image.get_size()
    sin_a, cos_a = math.sin(math.radians(angle)), math.cos(math.radians(angle)) 
    min_x, min_y = min([0, sin_a*h, cos_a*w, sin_a*h + cos_a*w]), max([0, sin_a*w, -cos_a*h, sin_a*w - cos_a*h])

    # calculate the translation of the pivot 
    pivot        = pygame.math.Vector2(originPos[0], -originPos[1])
    pivot_rotate = pivot.rotate(angle)
    pivot_move   = pivot_rotate - pivot

    # calculate the upper left origin of the rotated image
    origin = (pos[0] - originPos[0] + min_x - pivot_move[0], pos[1] - originPos[1] - min_y + pivot_move[1])

    # get a rotated image
    rotated_image = pygame.transform.rotate(image, angle)

    # rotate and blit the image
    surf.blit(rotated_image, origin)
  
pygame.init()
size = (400,400)
screen = pygame.display.set_mode(size)
clock = pygame.time.Clock()

image = pygame.image.load('boomerang64.png')
pivot = (48, 21)

angle, frame = 0, 0
done = False
while not done:
    clock.tick(60)
    for event in pygame.event.get():
        if event.type == pygame.QUIT:
            done = True
    screen.fill(0)

    pos = (200 + math.cos(frame * 0.05)*100, 200 + math.sin(frame * 0.05)*50)
    blitRotate(screen, image, pos, pivot, angle)

    pygame.draw.line(screen, (0, 255, 0), (pos[0]-20, pos[1]), (pos[0]+20, pos[1]), 3)
    pygame.draw.line(screen, (0, 255, 0), (pos[0], pos[1]-20), (pos[0], pos[1]+20), 3)
    pygame.display.flip()
    frame += 1
    angle += 10
    
pygame.quit()

---

The same algorithm can be used for a Sprite , too.
In that case the position ( self.pos ), pivot ( self.pivot ) and angle ( self.angle ) are instance attributes of the class. In the update method the self.rect and self.image attributes are computed.

Minimal example: repl.it/@Rabbid76/PyGame-RotateSpriteAroundOffCenterPivot

import pygame
import math

class SpriteRotate(pygame.sprite.Sprite):

    def __init__(self, imageName, pos, pivot):
        super().__init__() 
        self.image = pygame.image.load(imageName)
        self.original_image = self.image
        self.rect = self.image.get_rect(topleft = (pos[0]-pivot[0], pos[1]-pivot[1]))
        self.pos   = pos
        self.pivot = pivot
        self.angle = 0

    def update(self):

        # calcaulate the axis aligned bounding box of the rotated image
        w, h         = self.original_image.get_size()
        sin_a, cos_a = math.sin(math.radians(self.angle)), math.cos(math.radians(self.angle)) 
        min_x, min_y = min([0, sin_a*h, cos_a*w, sin_a*h + cos_a*w]), max([0, sin_a*w, -cos_a*h, sin_a*w - cos_a*h])

        # calculate the translation of the pivot 
        pivot        = pygame.math.Vector2(self.pivot[0], -self.pivot[1])
        pivot_rotate = pivot.rotate(self.angle)
        pivot_move   = pivot_rotate - pivot

        # calculate the upper left origin of the rotated image
        origin = (self.pos[0] - self.pivot[0] + min_x - pivot_move[0], self.pos[1] - self.pivot[1] - min_y + pivot_move[1])

        # get a rotated image
        self.image  = pygame.transform.rotate(self.original_image, self.angle)
        self.rect   = self.image.get_rect(topleft = (round(origin[0]), round(origin[1])))
        self.angle += 10
  
pygame.init()
size = (400,400)
screen = pygame.display.set_mode(size)
clock = pygame.time.Clock()

boomerang = SpriteRotate('boomerang64.png', (200, 200), (48, 21))
all_sprites = pygame.sprite.Group(boomerang)

frame = 0
done = False
while not done:
    clock.tick(60)
    for event in pygame.event.get():
        if event.type == pygame.QUIT:
            done = True

    pos = (200 + math.cos(frame * 0.05)*100, 200 + math.sin(frame * 0.05)*50)
    boomerang.pos = pos
    all_sprites.update()

    screen.fill(0)
    all_sprites.draw(screen)
    pygame.display.flip()
    frame += 1

pygame.quit()

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