How to arrange and animate a frilled line along a torus shape in Manim?

Target: Arrange and Animate a Frilled Line along a Torus in Manim

Answer:

The primary challenge here is mapping a 1D parametric function (the frilled line) onto a 3D surface (the torus) while maintaining the "frill" (high-frequency oscillation) relative to the surface normal. In Manim, the most deterministic approach is to define a composite parametric function.

Use the following implementation. It uses a Surface to define the torus and a ParametricFunction that adds a sine-wave displacement (the frill) to the torus coordinates.

Python

# [Logic Hardening] Parametric Torus Mapping with Frequency Displacement from manim import * import numpy as np class FrilledTorus(ThreeDScene): def construct(self): # Torus parameters R, r = 3, 1 # Major and minor radii frill_amplitude = 0.2 frill_frequency = 40 # Number of frills along the loop # [Logic Point] Composite Parametric Mapping def frilled_torus_func(t): # Base torus coordinates # t moves from 0 to 2*PI x = (R + r * np.cos(t)) * np.cos(t * 0.1) # Slowly rotating loop y = (R + r * np.cos(t)) * np.sin(t * 0.1) z = r * np.sin(t) # Adding the 'frill' displacement # We oscillate the minor radius 'r' to create the frilled effect distorted_r = r + frill_amplitude * np.sin(frill_frequency * t) x_f = (R + distorted_r * np.cos(t)) * np.cos(t) y_f = (R + distorted_r * np.cos(t)) * np.sin(t) z_f = distorted_r * np.sin(t) return np.array([x_f, y_f, z_f]) frilled_line = ParametricFunction( frilled_torus_func, t_range=[0, TAU], color=YELLOW ) # Scene Setup axes = ThreeDAxes() self.set_camera_orientation(phi=75 * DEGREES, theta=-45 * DEGREES) self.add(axes) # [Logic Point] Animating the creation along the 3D path self.play(Create(frilled_line), run_time=4, rate_func=linear) self.begin_ambient_camera_rotation(rate=0.1) self.wait(2)

Note:

Coordinate Logic: The distorted_r variable injects the "frill" directly into the minor radius calculation, ensuring the frills point outward from the torus center.

Frequency: Adjust frill_frequency to increase the density of the frills.

Refactor: Ensure your R and r match the scale of your existing Manim scene.