使用matplotlib，python绘制三个正交矢量和大小的椭圆形

Question

我有三个正顺序向量

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

，并且沿着这些方向的每个方向1.44、3.45和2.02沿每个方向的亮度值。

我如何绘制代表沿正顺序向量的轴方向以及其大小给出的轴长的椭圆形（3D表面）？

Answer 1

如果我正确理解您的问题，您想创建一个椭圆形并旋转它，以使其主要轴与正统向量对齐。

注释：由于Matplotlib的3D功能相当有限，因此我将使用绘图：这将使我们的生活变得容易得多，因为它为3D图提供了“相等”的长宽比。您可以轻松地将绘图库更改为matplotlib。

椭圆形的参数方程可以在这里找到。

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