# Interpolate Between Axis-Angle RepresentationsΒΆ

We can interpolate between two orientations that are represented by an axis and an angle either linearly or with slerp (spherical linear interpolation). Here we compare both methods and measure the angular velocity between two successive steps. We can see that linear interpolation results in a non-constant angular velocity. Usually it is a better idea to interpolate with slerp.

```print(__doc__)

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import axes3d
import matplotlib.animation as animation
from pytransform3d import rotations as pr

velocity = None
last_a = None
rotation_axis = None

def interpolate_linear(start, end, t):
return (1 - t) * start + t * end

def update_lines(step, start, end, n_frames, rot, profile):
global velocity
global last_a

if step == 0:
velocity = []
last_a = start

if step <= n_frames / 2:
t = step / float(n_frames / 2 - 1)
a = pr.axis_angle_slerp(start, end, t)
else:
t = (step - n_frames / 2) / float(n_frames / 2 - 1)
a = interpolate_linear(end, start, t)

R = pr.matrix_from_axis_angle(a)

# Draw new frame
rot[0].set_data(np.array([0, R[0, 0]]), [0, R[1, 0]])
rot[0].set_3d_properties([0, R[2, 0]])

rot[1].set_data(np.array([0, R[0, 1]]), [0, R[1, 1]])
rot[1].set_3d_properties([0, R[2, 1]])

rot[2].set_data(np.array([0, R[0, 2]]), [0, R[1, 2]])
rot[2].set_3d_properties([0, R[2, 2]])

# Update vector in frame
test = R.dot(np.ones(3) / np.sqrt(3.0))
rot[3].set_data(
np.array([test[0] / 2.0, test[0]]), [test[1] / 2.0, test[1]])
rot[3].set_3d_properties([test[2] / 2.0, test[2]])

velocity.append(
pr.angle_between_vectors(a[:3], last_a[:3]) + a[3] - last_a[3])
last_a = a
profile.set_data(np.linspace(0, 1, n_frames)[:len(velocity)], velocity)

return rot

if __name__ == "__main__":
# Generate random start and goal
np.random.seed(3)
start = np.array([0, 0, 0, np.pi])
start[:3] = pr.norm_vector(np.random.randn(3))
end = np.array([0, 0, 0, np.pi])
end[:3] = pr.norm_vector(np.random.randn(3))
n_frames = 100

fig = plt.figure(figsize=(12, 5))

ax = fig.add_subplot(121, projection="3d")
ax.set_xlim((-1, 1))
ax.set_ylim((-1, 1))
ax.set_zlim((-1, 1))
ax.set_xlabel("X")
ax.set_ylabel("Y")
ax.set_zlabel("Z")

Rs = pr.matrix_from_axis_angle(start)
Re = pr.matrix_from_axis_angle(end)

rot = [ax.plot([0, 1], [0, 0], [0, 0], c="r", lw=3)[0],
ax.plot([0, 0], [0, 1], [0, 0], c="g", lw=3)[0],
ax.plot([0, 0], [0, 0], [0, 1], c="b", lw=3)[0],
ax.plot([0, 1], [0, 1], [0, 1], c="gray", lw=3)[0],

ax.plot([0, Rs[0, 0]], [0, Rs[1, 0]], [0, Rs[2, 0]], c="r", lw=3,
alpha=0.5)[0],
ax.plot([0, Rs[0, 1]], [0, Rs[1, 1]], [0, Rs[2, 1]], c="g", lw=3,
alpha=0.5)[0],
ax.plot([0, Rs[0, 2]], [0, Rs[1, 2]], [0, Rs[2, 2]], c="b", lw=3,
alpha=0.5)[0],

ax.plot([0, Re[0, 0]], [0, Re[1, 0]], [0, Re[2, 0]], c="orange",
lw=3, alpha=0.5)[0],
ax.plot([0, Re[0, 1]], [0, Re[1, 1]], [0, Re[2, 1]], c="turquoise",
lw=3, alpha=0.5)[0],
ax.plot([0, Re[0, 2]], [0, Re[1, 2]], [0, Re[2, 2]], c="violet",
lw=3, alpha=0.5)[0]]

ax.set_xlim((0, 1))
ax.set_ylim((0, 1))
profile = ax.plot(0, 0)[0]

anim = animation.FuncAnimation(fig, update_lines, n_frames,
fargs=(start, end, n_frames, rot, profile),
interval=50, blit=False)

plt.show()
```

Total running time of the script: ( 0 minutes 0.150 seconds)

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