Fixed there_and_back_with_pause() rate function behaviour with different pause_ratio values

[jkjkil4]

Overview: What does this pull request change?

fix the behaviour of there_and_back_with_pause when setting pause_ratio with other values.

Further Information and Comments

the picture below shows the bug of there_and_back_with_pause (left), and the corrected result (right)

this picture is generated by the following code:

# In[]
import matplotlib.pyplot as plt
import numpy as np
from matplotlib.axes import Axes


def sigmoid(x: float) -> float:
    return 1.0 / (1 + np.exp(-x))


def smooth(t: float, inflection: float = 10.0) -> float:
    error = sigmoid(-inflection / 2)
    return min(
        max((sigmoid(inflection * (t - 0.5)) - error) / (1 - 2 * error), 0),
        1,
    )


def there_and_back_with_pause(t: float, pause_ratio: float = 1.0 / 3) -> float:
    a = 1.0 / pause_ratio
    if t < 0.5 - pause_ratio / 2:
        return smooth(a * t)
    elif t < 0.5 + pause_ratio / 2:
        return 1
    else:
        return smooth(a - a * t)


def there_and_back_with_pause_corrected(t: float, pause_ratio: float = 1.0 / 3) -> float:
    a = 2.0 / (1.0 - pause_ratio)
    if t < 0.5 - pause_ratio / 2:
        return smooth(a * t)
    elif t < 0.5 + pause_ratio / 2:
        return 1
    else:
        return smooth(a - a * t)


def plot(axes: Axes, func, pause_ratio: float, aspect_eq=True) -> None:
    axes.set_title(f'...{func.__name__[14:]}(t, pause_ratio={pause_ratio:.3f})',
                   fontsize=9)
    axes.set_aspect('equal')
    x = np.linspace(0, 1, 100)
    y = [func(v, pause_ratio=pause_ratio) for v in x]
    axes.scatter(x, y, s=3)


fig = plt.figure(figsize=(10, 10))

plot(fig.add_subplot(4, 2, 1), there_and_back_with_pause, 0.1)
plot(fig.add_subplot(4, 2, 3), there_and_back_with_pause, 1 / 3)
plot(fig.add_subplot(4, 2, 5), there_and_back_with_pause, 0.5)
plot(fig.add_subplot(4, 2, 7), there_and_back_with_pause, 0.8)

plot(fig.add_subplot(4, 2, 2), there_and_back_with_pause_corrected, 0.1)
plot(fig.add_subplot(4, 2, 4), there_and_back_with_pause_corrected, 1 / 3)
plot(fig.add_subplot(4, 2, 6), there_and_back_with_pause_corrected, 0.5)
plot(fig.add_subplot(4, 2, 8), there_and_back_with_pause_corrected, 0.8)

fig.tight_layout()
fig.subplots_adjust(wspace=-0.6)
fig.show()

# %%

Reviewer Checklist

• The PR title is descriptive enough for the changelog, and the PR is labeled correctly
• If applicable: newly added non-private functions and classes have a docstring including a short summary and a PARAMETERS section
• If applicable: newly added functions and classes are tested

[behackl]

LGTM, thanks for your contribution!