How to compute velocity and acceleration from a stream of points

acceleration_annotation.py

"""
Code discussed in this video:

https://youtu.be/qA58ggk9fPA

And here's the links mentioned:

Torque-free motion gif's:
Integrated motion:
https://gfycat.com/costlyhardtofinddolphin & 
https://gfycat.com/warmblindbear
Like in the video:
https://gfycat.com/commonshorttermaegeancat
https://gfycat.com/wanannualfeline
https://gfycat.com/deadhopefuldogwoodtwigborer
https://gfycat.com/baggydownrightbongo

Johnsons animation:
https://youtu.be/s9wiRjUKctU

Sperical pendulum gif's:
https://gfycat.com/highscalyhogget
https://gfycat.com/bleakcarefulindianspinyloach

Wikipedia:
https://en.wikipedia.org/wiki/Moment_of_inertia#Angular_momentum_2
https://en.wikipedia.org/wiki/Finite_difference
"""


def finite_backward_difference(ys, h):
    # https://en.wikipedia.org/wiki/Finite_difference
    assert h < len(ys), f"{h}, {len(ys)}"

    FRONT_IDX = -1
    end_idx = FRONT_IDX - h

    y_front = ys[FRONT_IDX]
    y_end = ys[end_idx]

    dy = np.array(y_front) - y_end

    return dy


def backward_derivatrive(ys, h):
    dy = finite_backward_difference(ys, h)

    return dy / h


def backward_second_derivatrive(ys, h):
    assert 2 * h < len(ys), f"{2 * h}, {len(ys)}"

    # Take the best available two backwards first derivatives which are h apart
    # (those are (-1) to (-1)-h resp. (-1)-h to (-1)-2*h
    dy_front = backward_derivatrive(ys, h)
    dy_end = backward_derivatrive(ys[:-h], h)

    padding = (h - 1) * [None]
    dys = [dy_end] + padding + [dy_front]

    return backward_derivatrive(dys, h)