Sine to Cosine: 3D Phase Shift Morph

Title: 3‑D Transformation of a Sine Wave into a Cosine Wave

Purpose: Visually demonstrate the phase shift relationship between the sine and cosine functions by morphing a 3‑D sine surface into a cosine surface.

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1. Overall Structure & Timing

• Total duration: ~10 seconds (well under the 40‑second limit).
• Scene breakdown:
  1. Intro (0‑1 s): Camera pans to the 3‑D axes, labels appear.
  2. Sine wave appearance (1‑3 s): Plot the sine surface, animate its drawing.
  3. Transformation (3‑8 s): Continuous morph from sine to cosine, with accompanying phase‑shift annotation.
  4. Conclusion (8‑10 s): Freeze on the cosine surface, fade out axes labels, and display a final caption.

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2. Mathematical Elements

• Functions:
  • Sine surface: $$f_s(x, y) = \sin\bigl(\sqrt{x^2 + y^2}\bigr)$$
  • Cosine surface: $$f_c(x, y) = \cos\bigl(\sqrt{x^2 + y^2}\bigr)$$
• Domain: $x, y \in [-4\pi, 4\pi]$
• Phase‑shift note: Display the identity $\cos\theta = \sin\bigl(\theta + \tfrac{\pi}{2}\bigr)$ during the morph.

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3. Visual Elements

• Axes: 3‑D Cartesian axes with tick marks every $\pi$. Labels: $x$, $y$, $z$.
• Surfaces:
  • Sine surface: semi‑transparent teal (RGB 0, 150, 200) with a subtle gradient based on height.
  • Cosine surface: semi‑transparent orange (RGB 255, 140, 0) using the same gradient scheme.
• Grid: Light gray grid on the $xy$-plane for reference.
• Annotations:
  • Inline LaTeX formula for the phase shift appears at 14 s.
  • Small rotating arrow indicating the direction of the phase shift.
• Caption (final 5 s): "A sine wave shifted by $\frac{\pi}{2}$ becomes a cosine wave."

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4. Animation Timing & Transitions

Time (s)	Action
0‑0.5	Fade‑in axes and grid.
0.5‑1	Camera slowly rotates 30° around the $z$-axis to give a 3‑D perspective.
1‑2	Plot the sine surface using a "draw surface" effect (vertices appear from the origin outward).
2‑3	Slight pulse (scale up → down) to draw attention to the sine surface.
3‑4	Display the phase‑shift formula with a fade‑in.
4‑8	Morph the sine surface into the cosine surface via a smooth interpolation of the height function; simultaneously rotate the camera 45° around the vertical axis for dynamic view.
8‑9	Freeze the cosine surface; highlight it with a brief glow.
9‑10	Fade out axes labels, keep the surface, and show the final caption.

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5. Camera Angles & Perspectives

• Initial view: Slightly elevated angle (≈30° elevation) looking toward the origin, with a 45° azimuth.
• During morph: Continuous slow orbit: azimuth increases from 45° to 90° while maintaining the same elevation, creating a gentle 3‑D rotation.
• Final view: Hold the last camera position for the caption.

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6. Additional Details

• Lighting: Soft ambient light plus a directional light from above‑right to give depth to the surfaces.
• Background: Dark navy gradient to make the teal/orange surfaces stand out.
• Sound (optional): A subtle rising tone synced with the morph, ending on a soft chord when the cosine appears.
• Accessibility: All text (formulas, caption) uses a high‑contrast white font.

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End of Specification