Convention for Rotation: Passive / Active, Extrinsic / Intrinsic

We will compare all possible combinations of passive and active rotations and extrinsic and intrinsic concatenation of rotations.

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import proj3d
from pytransform3d.rotations import (
    passive_matrix_from_angle, active_matrix_from_angle, plot_basis)

Passive Extrinsic Rotations

plt.figure(figsize=(8, 3))
axes = [plt.subplot(1, 3, 1 + i, projection="3d") for i in range(3)]
for ax in axes:
    plt.setp(ax, xlim=(-1, 1), ylim=(-1, 1), zlim=(-1, 1),
             xlabel="X", ylabel="Y", zlabel="Z")

Rx45 = passive_matrix_from_angle(0, np.deg2rad(45))
Rz45 = passive_matrix_from_angle(2, np.deg2rad(45))

axes[0].set_title("Passive Extrinsic Rotations", y=0.95)
plot_basis(ax=axes[0], R=np.eye(3))
axes[1].set_title("$R_x(45^{\circ})$", y=0.95)
plot_basis(ax=axes[1], R=Rx45)
axes[2].set_title("$R_z(45^{\circ}) R_x(45^{\circ})$", y=0.95)
plot_basis(ax=axes[2], R=Rz45.dot(Rx45))

plt.tight_layout()

Passive Intrinsic Rotations

plt.figure(figsize=(8, 3))
axes = [plt.subplot(1, 3, 1 + i, projection="3d") for i in range(3)]
for ax in axes:
    plt.setp(ax, xlim=(-1, 1), ylim=(-1, 1), zlim=(-1, 1),
             xlabel="X", ylabel="Y", zlabel="Z")

axes[0].set_title("Passive Intrinsic Rotations", y=0.95)
plot_basis(ax=axes[0], R=np.eye(3))
axes[1].set_title("$R_x(45^{\circ})$", y=0.95)
plot_basis(ax=axes[1], R=Rx45)
axes[2].set_title("$R_x(45^{\circ}) R_{z'}(45^{\circ})$", y=0.95)
plot_basis(ax=axes[2], R=Rx45.dot(Rz45))

plt.tight_layout()

Active Extrinsic Rotations

plt.figure(figsize=(8, 3))
axes = [plt.subplot(1, 3, 1 + i, projection="3d") for i in range(3)]
for ax in axes:
    plt.setp(ax, xlim=(-1, 1), ylim=(-1, 1), zlim=(-1, 1),
             xlabel="X", ylabel="Y", zlabel="Z")

Rx45 = active_matrix_from_angle(0, np.deg2rad(45))
Rz45 = active_matrix_from_angle(2, np.deg2rad(45))

axes[0].set_title("Active Extrinsic Rotations", y=0.95)
plot_basis(ax=axes[0], R=np.eye(3))
axes[1].set_title("$R_x(45^{\circ})$", y=0.95)
plot_basis(ax=axes[1], R=Rx45)
axes[2].set_title("$R_z(45^{\circ}) R_x(45^{\circ})$", y=0.95)
plot_basis(ax=axes[2], R=Rz45.dot(Rx45))

plt.tight_layout()

Active Intrinsic Rotations

plt.figure(figsize=(8, 3))
axes = [plt.subplot(1, 3, 1 + i, projection="3d") for i in range(3)]
for ax in axes:
    plt.setp(ax, xlim=(-1, 1), ylim=(-1, 1), zlim=(-1, 1),
             xlabel="X", ylabel="Y", zlabel="Z")

axes[0].set_title("Active Intrinsic Rotations", y=0.95)
plot_basis(ax=axes[0], R=np.eye(3))
axes[1].set_title("$R_x(45^{\circ})$", y=0.95)
plot_basis(ax=axes[1], R=Rx45)
axes[2].set_title("$R_x(45^{\circ}) R_{z'}(45^{\circ})$", y=0.95)
plot_basis(ax=axes[2], R=Rx45.dot(Rz45))

plt.tight_layout()

plt.show()