Optimized interpolate and bezier in manim.utils.bezier

[chopan050]

Overview: What does this pull request change?

Slightly optimized interpolate and bezier in manim.utils.bezier.
Related PR: #3281

interpolate

Rewriting this function to use 1 product instead of 2, makes a speedup of around 30%:

def interpolate(start, end, alpha):
    # return (1 - alpha) * start + alpha * end # original
    return start + alpha * (end - start) # new

bezier

First of all, I added the cases for Bézier curves of grade 0 and 1. This prevents them from falling back to the expensive general nth grade case:

def bezier(points):
    P = np.asarray(points)
    degree = P.shape[0] - 1

    if degree == 0:

        def zero_bezier(t):
            return np.ones_like(t) * P[0]

        return zero_bezier

    if degree == 1:

        def linear_bezier(t):
            return P[0] + t * (P[1] - P[0])

        return linear_bezier

    # ...

Then, I made a very slight optimization for the cubic Bézier case:

    if degree == 3:

        def cubic_bezier(t):
            t2 = t * t
            t3 = t2 * t
            mt = 1 - t
            mt2 = mt * mt
            mt3 = mt2 * mt
            return mt3 * P[0] + 3 * t * mt2 * P[1] + 3 * t2 * mt * P[2] + t3 * P[3]

        return cubic_bezier

It reported a small speedup of around 10% - 15%. Not too bad! I did something similar with the quadratic Bézier:

    if degree == 2:

        def quadratic_bezier(t):
            t2 = t * t
            mt = 1 - t
            mt2 = mt * mt
            return mt2 * P[0] + 2 * t * mt * P[1] + t2 * P[2]

        return quadratic_bezier

Finally, the general nth grade Bézier case. I avoided the extensive use of the choose function which can be expensive, and instead used the recursive definition of Béziers I used for functions like partial_bezier_points and subdivide_bezier.

Also, bezier now returns named functions instead of lambdas, which is good for debugging and profiling: now you can know which function is taking what part of the time when profiling your code.

Links to added or changed documentation pages

bezier
https://manimce--3960.org.readthedocs.build/en/3960/reference/manim.utils.bezier.html#manim.utils.bezier.bezier

interpolate
https://manimce--3960.org.readthedocs.build/en/3960/reference/manim.utils.bezier.html#manim.utils.bezier.interpolate

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, thank you!