The bounding box of a rotated object

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Circle
DPI = 72

# Rotation centre. It is helpful to have this with shape (2,1)
cx, cy = 150, 50
rc = np.array(((cx, cy),)).T

# Initial width and height of the bounding rectangle we will fit the object in.
rw, rh = 200, 300
# Initial corner positions of the bounding rectangle 
x1, y1, x2, y2 = 250, 50, 250+rw, 50+rh

def rotate_points(pts, theta, rc):
    """Rotate the (x,y) points pts by angle theta about centre rc."""
    c, s = np.cos(theta), np.sin(theta)
    R = np.array(((c,-s), (s, c)))
    return rc + R @ (pts - rc)


def plot_poly(pts, colour='tab:blue', lw=2, opacity=1, ls='-'):
    """Plot a closed polygon with vertices at pts."""

    plot_pts = np.vstack((pts.T, pts[:,0]))
    ax.plot(*zip(*plot_pts), c=colour, lw=lw, alpha=opacity, ls=ls)


def plot_obj(pts, colour='tab:green', lw=2):
    """Draw the object we are rotating: a circle and polygon."""

    plot_poly(pts[:,1:], colour, lw=lw, opacity=0.5)
    circle = Circle(pts[:,0], obj_cr, edgecolor=colour, fill=False, lw=lw,
                    alpha=0.5)
    ax.add_patch(circle)


def get_boundary_pts(pts):
    """Get the vertices of the bounding rectangle for the points pts."""

    xmin, xmax = np.min(pts[0]), np.max(pts[0])
    ymin, ymax = np.min(pts[1]), np.max(pts[1])
    return np.array(((xmin,ymin), (xmax,ymin), (xmax, ymax), (xmin, ymax))).T


def get_obj_boundary(obj_pts):
    """Get the vertices of the bounding rectangle for the rotated object."""

    fcx, fcy = obj_pts[:,0]
    # Get the boundary from the triangle coordinates and the circle limits
    _obj_boundary = np.vstack((obj_pts.T[1:], (fcx-obj_cr, fcy),
                (fcx+obj_cr, fcy), (fcx, fcy-obj_cr), (fcx, fcy+obj_cr))).T
    return get_boundary_pts(_obj_boundary)


fig, ax = plt.subplots(figsize=(8.33333333, 8.33333333), dpi=DPI)

# Initial bounding rectangle of unrotated object.
pts = np.array( ((x1,y1), (x2,y1), (x2,y2), (x1,y2)) ).T
# The radius of the circle in our plotted object.
obj_cr = (rh - rw*np.sqrt(3)/2)/2
# The coordinates defining our object.
obj_pts = np.array( ((x1 + rw/2, y1 + rh - obj_cr),    # circle centre
                     (x1, y1), (x2, y1),               #
                     (x1+rw/2, y2-2*obj_cr),           # triangle
                    )).T
# Plot the unrotated object and its bounding rectangle
plot_obj(obj_pts)
plot_poly(pts)

nrots = 60
theta = np.radians(360 // nrots)
boundary_trail_pts, obj_boundary_trail_pts = [], []
for i in range(nrots):
    fig, ax = plt.subplots(figsize=(8.33333333, 6.25), dpi=DPI)
    ax.set_xlim(-600,600)
    ax.set_ylim(-600,600)
    # Indicate the centre of rotation
    ax.add_patch(Circle((cx,cy), 10))

    # Plot the object
    plot_obj(obj_pts)
    # Plot the rotated object's boundary
    boundary_pts = get_obj_boundary(obj_pts)
    plot_poly(boundary_pts, colour='tab:purple', ls='--')
    obj_boundary_trail_pts.append(np.mean(boundary_pts, axis=1))
    ax.plot(*zip(*obj_boundary_trail_pts), c='tab:purple', ls='--')

    # Plot the original boundary, rotated
    plot_poly(pts, colour='tab:orange')
    # Plot the boundary to the original rotated boundary
    boundary_pts = get_boundary_pts(pts)
    plot_poly(boundary_pts, colour='tab:orange', ls=':')
    boundary_trail_pts.append(np.mean(boundary_pts, axis=1))
    ax.plot(*zip(*boundary_trail_pts), c='tab:orange', ls=':')

    plt.savefig('frames/bbrot-{:03d}.png'.format(i+1), dpi=DPI)

    obj_pts = rotate_points(obj_pts, theta, rc)
    pts = rotate_points(pts, theta, rc)
    plt.close(fig)