使用 matplotlib、python 从三个正交向量和幅度绘制椭球体

我有三个正交向量

(-0.460021, -0.241295, -0.854493), 
(-0.641846, 0.755344, 0.132244),
(-0.613526, -0.609288, 0.502348) 

以及沿每个方向的幅度值1.44, 3.45 and 2.02.

如何绘制代表沿正交向量的轴方向以及由其大小给出的轴长度的椭球体（3D 表面）？

如果我正确理解你的问题，你想要创建一个椭球体并旋转它，使其长轴与正交向量对齐。

Note：由于 Matplotlib 的 3D 功能相当有限，我将使用 Plotly：它会让我们的生活变得更加轻松，因为它为 3D 绘图提供了“相等”的纵横比。您可以轻松地将绘图库更改为 Matplotlib。

椭球体的参数方程可以在这里找到 https://mathworld.wolfram.com/Ellipsoid.html.

import numpy as np
import matplotlib.pyplot as plt
import plotly.graph_objects as go

# compute ellipsoid coordinates on standard basis
# e1=(1, 0, 0), e2=(0, 1, 0), e3=(0, 0, 1)
a, b, c = 1.44, 3.45, 2.02
u, v = np.mgrid[0:2*np.pi:40j, 0:np.pi:20j]
x1 = a * np.cos(u) * np.sin(v)
y1 = b * np.sin(u) * np.sin(v)
z1 = c * np.cos(v)
# points on the ellipsoid
points = np.stack([t.flatten() for t in [x1, y1, z1]])

v1 = np.array([-0.460021, -0.241295, -0.854493])
v2 = np.array([-0.641846, 0.755344, 0.132244])
v3 = np.array([-0.613526, -0.609288, 0.502348])
# 3x3 transformation matrix
T = np.array([v1, v2, v3]).T

# transform coordinates to the new orthonormal basis
new_points = T @ points
x2 = new_points[0, :]
y2 = new_points[1, :]
z2 = new_points[2, :]
x2, y2, z2 = [t.reshape(x1.shape) for t in [x2, y2, z2]]

# scale vector for better visualization
scale = 5
v1, v2, v3 = [scale * t for t in [v1, v2, v3]]

fig = go.Figure([
    # axis on the standard base
    go.Scatter3d(x=[0, 5], y=[0, 0], z=[0, 0], mode="lines", name="x1", line=dict(width=5, color="red")),
    go.Scatter3d(x=[0, 0], y=[0, 5], z=[0, 0], mode="lines", name="y1", line=dict(width=5, color="green")),
    go.Scatter3d(x=[0, 0], y=[0, 0], z=[0, 5], mode="lines", name="z1", line=dict(width=5, color="blue")),
    # axis on the new orthonormal base
    go.Scatter3d(x=[0, v1[0]], y=[0, v1[1]], z=[0, v1[2]], mode="lines", name="x2", line=dict(width=2, color="red")),
    go.Scatter3d(x=[0, v2[0]], y=[0, v2[1]], z=[0, v2[2]], mode="lines", name="y2", line=dict(width=2, color="green")),
    go.Scatter3d(x=[0, v3[0]], y=[0, v3[1]], z=[0, v3[2]], mode="lines", name="z2", line=dict(width=2, color="blue")),
    # original ellipsoid aligned to the standard base
    go.Surface(x=x1, y=y1, z=z1, opacity=0.35, colorscale="plotly3", surfacecolor=y1, cmin=y1.min(), cmax=y1.max(), colorbar=dict(len=0.6, yanchor="bottom", y=0)),
    # final ellipsoid aligned to the new orthonormal base
    go.Surface(x=x2, y=y2, z=z2, opacity=1, colorscale="aggrnyl", surfacecolor=y1, cmin=y1.min(), cmax=y1.max(), colorbar=dict(len=0.6, yanchor="bottom", y=0, x=0.95))
])
fig.update_layout({"scene": {"aspectmode": "auto"}})
fig