Animation Specification: 3D Sine‑to‑Cosine Transformation (Defaults)

1. Overview

Create a Manim animation that displays a sine wave plotted in a 3‑D coordinate system and smoothly morphs it into a cosine wave. All parameters will use sensible defaults.

2. Mathematical Elements

• Domain: $x \in [0, 2\pi]$
• Initial Curve: $y = \sin(x)$ (plotted on the $xy$-plane, with $z = 0$)
• Final Curve: $y = \cos(x)$ (same plane)
• Interpolation: For the morph, use a parameter $t \in [0,1]$ and define the intermediate curve as:
  $$y_t = (1-t)\sin(x) + t\cos(x)$$
  This will be animated by Manim’s Transform.

3. Visual Elements

• Axes: 3‑D axes (ThreeDAxes) with range:
  • $x$: $[-0.5, 2\pi + 0.5]$
  • $y$: $[-1.5, 1.5]$
  • $z$: $[-0.5, 0.5]$ (kept flat for visual balance)
• Grid: Enabled on the $xy$-plane for reference.
• Curves:
  • Sine curve: ParametricFunction colored BLUE, stroke width 4.
  • Cosine curve: Same style, colored RED.
• Title: "Sine $\r\rightarrow$ Cosine Transformation" placed at the top, using Text with a semi‑transparent background.
• Labels:
  • Axis labels: "x", "y", "z".
  • Optional small markers at the start and end points of the curves (optional, default: none).

4. Animation Timing & Transitions

Segment	Duration	Description
1. Intro (axes fade‑in)	1.0 s	Create the axes and grid.
2. Plot sine curve	1.0 s	Create the blue sine curve.
3. Pause	0.5 s	Hold the sine curve.
4. Morph to cosine	2.5 s	Transform the sine curve into the red cosine curve using the interpolation defined above.
5. Hold final curve	0.5 s	Pause on the cosine curve.
6. Fade out	0.5 s	Fade out all objects.

Total runtime ≈ 6 seconds.

5. Camera Settings

• Perspective: Static camera positioned to view the 3‑D axes at an angle of $30^{\circ}$ elevation and $45^{\circ}$ azimuth (Manim default for ThreeDAxes).
• No movement during the morph; the camera remains fixed.

6. Rendering Settings (Defaults)

• Resolution: 1920×1080 (HD)
• Background color: White (#FFFFFF)
• Frame rate: 30 fps
• Quality: high_quality (Manim default)

7. Implementation Sketch (Manim Community Edition)

from manim import *

class SineToCosine3D(ThreeDScene):
    def construct(self):
        # Axes
        axes = ThreeDAxes(
            x_range=[0, 2*PI, PI/2],
            y_range=[-1.5, 1.5, 0.5],
            z_range=[-0.5, 0.5, 0.5],
            axis_config={"include_tip": True},
        )
        axes.add_coordinate_labels()
        self.set_camera_orientation(phi=30 * DEGREES, theta=45 * DEGREES)
        self.play(Create(axes), run_time=1)

        # Sine curve
        sine = ParametricFunction(
            lambda t: axes.c2p(t, np.sin(t), 0),
            t_range=[0, 2*PI],
            color=BLUE,
            stroke_width=4,
        )
        self.play(Create(sine), run_time=1)
        self.wait(0.5)

        # Cosine curve (target)
        cosine = ParametricFunction(
            lambda t: axes.c2p(t, np.cos(t), 0),
            t_range=[0, 2*PI],
            color=RED,
            stroke_width=4,
        )

        # Morph
        self.play(Transform(sine, cosine), run_time=2.5)
        self.wait(0.5)

        # Title
        title = Text("Sine → Cosine Transformation", font_size=36).to_edge(UP)
        self.add(title)

        # Fade out
        self.play(FadeOut(VGroup(axes, sine, title)), run_time=0.5)

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Note: This specification follows the user’s instruction to use the defaults. If any parameter should be changed (e.g., longer domain, camera motion, different colors), please let me know.