Fix shared reference issue in traced_points initialization

[mathsvisualization]

Motivation

While using TracingTail, I observed incorrect trail behavior.
The issue was caused by list multiplication during initialization:

self.traced_points = n_points * [curr_point]

This creates multiple references to the same NumPy array instead of independent copies. As a result, updating one point updated all stored points, producing visual artifacts and incorrect tracing behavior.

This PR fixes the issue by ensuring each traced point is an independent copy.

Proposed changes

• Replaced list multiplication with list comprehension.
• Ensured each stored traced point is created using .copy().
• Prevented shared mutable reference bug in TracingTail.

Before

self.traced_points: list[np.ndarray] = n_points * [curr_point]

After

self.traced_points: list[np.ndarray] = [
    curr_point.copy() for _ in range(n_points)
]

Test

The issue can be reproduced using a simple pendulum example.

Code

from manimlib import *
import numpy as np

class SimplePendulum(Scene):
    def construct(self):
        # Parameters
        length = 3
        angle = -PI / 4
        origin = UP * 2

        # Pivot point
        pivot = Dot(origin)

        # Bob
        bob = Dot(radius=0.15, stroke_color=GREY)
        bob.move_to(origin + length * np.array([
            np.sin(angle),
            -np.cos(angle),
            0
        ]))

        # String
        string = always_redraw(
            lambda: Line(origin, bob.get_center(), stroke_color=WHITE)
        )

        # Trail
        trail = TracingTail(
            bob.get_center,
            time_traced=2.0,
            stroke_color=WHITE,
        )

        self.add(pivot, string, bob, trail)

        # Animation (oscillation)
        def update_bob(mob, dt):
            nonlocal angle
            angle += -0.8 * np.sin(angle) * dt
            mob.move_to(origin + length * np.array([
                np.sin(angle),
                -np.cos(angle),
                0
            ]))

        bob.add_updater(update_bob)

        self.wait(6)

Result

Before the fix:

• Trail points overlapped incorrectly.
• Multiple segments collapsed due to shared references.

SimplePendulum.mp4

After the fix:

• Trail updates correctly.
• Each traced point is independent.
• No visual artifacts appear.

SimplePendulum.mp4