如何旋转一组二维点以最小化 numpy 中的垂直跨度？

Question

I have a collection of 2D points and I want to rotate them so as to display them spanning the least vertical height . Is there a way to find the angle without using optimization ?

As a brief illustration:

import numpy as np

# some initial points spread out vertically
xy = np.random.randn(2, 100)
xy[1, :] *= 2

def rotate(xy, theta):
    """Return a rotated set of points.
    """
    s = np.sin(theta)
    c = np.cos(theta)
    
    xyr = np.empty_like(xy)
    xyr[0, :] = c * xy[0, :] - s *  xy[1, :]
    xyr[1, :] = s * xy[0, :] + c *  xy[1, :]
    
    return xyr

def span(xy):
    """Return the vertical span of the points.
    """
    return xy[1, :].max() - xy[1, :].min()

def plot(xy):
    """2D plot with fixed aspect ratio.
    """
    import matplotlib.pyplot as plt
    fig, ax = plt.subplots(figsize=(3, 3))
    ax.scatter(*xy, alpha=0.3)
    ax.set_aspect(1)
    plt.show()
    plt.show()

Have a look at the original points:

>>> span(xy)
11.503342270923472
>>> plot(xy)

Have a look at some rotated points:

>>> xyr = rotate(xy, np.pi / 2)
>>> span(xyr)
4.594620173868735
>>> plot(xyr)

The optimal answer (which should here be around pi / 2) is easy to find by invoking scipy :

>>> from scipy.optimize import minimize_scalar
>>> minimize_scalar(lambda theta: span(rotate(xy, theta)))
     fun: 4.523188831276214
    nfev: 38
     nit: 34
 success: True
       x: 1.590391370976612

But surely there is a simpler way that doesn't require scipy - and 38 function evaluations!

Answer 1

This is probably not necessarily optimal, but a good enough simple solution may be to take the main components of the data and rotate the data to align largest main component with the horizontal axis.

import numpy as np

# Posted code...

# Find eigenvector with largest eigenvalue
c = (xy @ xy.T) / (xy.shape[1] - 1)
evals, evecs = np.linalg.eig(c)
imax = np.abs(evals).argmax()
v = evecs[:, imax]
# Correction angle is opposite of vector angle
theta = -np.arctan2(v[1], v[0])
# Rotate
xy2 = rotate(xy, theta)
# Plot
plot(xy2)

Output: