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Naive Interpolation of Euler Angles

This method for interpolating 3D rotations is not recommended at all!

Since 3D rotations can be represented by a list of three angles, it might be tempting to simply interpolate those angles independently.

Let’s try it and see what happens, shall we?

import numpy as np

import splines

As always, we use a few helper functions from helper.py:

from helper import angles2quat, animate_rotations, display_animation

We are using splines.CatmullRom to interpolate the Euler angles independently and splines.quaternion.CatmullRom to interpolate the associated quaternions for comparison:

def plot_interpolated_angles(angles):
    s1 = splines.CatmullRom(angles, endconditions='closed')
    times = np.linspace(s1.grid[0], s1.grid[-1], 100)
    s2 = splines.quaternion.CatmullRom(
        [angles2quat(azi, ele, roll) for azi, ele, roll in angles],
        endconditions='closed')
    ani = animate_rotations({
        'Euler angles': [angles2quat(*abc) for abc in s1.evaluate(times)],
        'quaternions': s2.evaluate(times),
    })
    display_animation(ani, default_mode='loop')

plot_interpolated_angles([
    (0, 0, 0),
    (45, 0, 0),
    (90, 45, 0),
    (90, 90, 0),
    (180, 0, 90),
])

Once Loop Reflect

There is clearly a difference between the two, but the Euler angles don’t look that bad.

Let’s try another example:

plot_interpolated_angles([
    (-175, 0, 0),
    (175, 0, 0),
])

Once Loop Reflect

Here we see that the naive interpolation isn’t aware that the azimuth angle is supposed to wrap around at 180 degrees.

This could be fixed with a less naive implementation, but there are also unfixable problems, as this example shows:

plot_interpolated_angles([
    (45, 45, 0),
    (45, 90, 0),
    (-135, 45, 180),
])

Once Loop Reflect

Even though all involved rotations are supposed to happen around a single rotation axis, The Euler angles interpolation is all over the place.