Heart‑Wave Oscillation

Overview

A 12‑second animation visualizes a high‑frequency sinusoidal fill inside a heart‑shaped envelope defined by $y = x^{2/3} + 0.9\sin(kx+\omega t)\sqrt{3-x^2}$. The curve oscillates rapidly ($k=48$, $\omega=6$) while the envelope remains static, creating a glowing, “filled” heart effect.

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Phases

#	Phase Name	Duration	Description
1	Intro	~2 s	Axes fade in from slightly above, the constant label $k = 48$ and the formula $y = x^{2/3} + 0.9\sin(kx)\sqrt{3-x^2}$ write onto the top, and the watermark $@lingualin$ fades in near the origin.
2	Oscillation	~8 s	The time tracker $t$ animates from 0 to 20, causing the sinusoidal component to sweep across the envelope. Three overlaid copies of the curve (wide, faint glow → thin, bright core) produce a soft glowing fill.
3	Hold	~2 s	The final frame is held static to give a seamless loop feel.

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Layout

┌─────────────────────────────────────────────┐
│                TOP AREA                     │
│   k‑label  •  Formula (centered)            │
├───────────────────────┬─────────────────────┤
│                       │                     │
│        LEFT AREA      │      RIGHT AREA     │
│   (main visual –      │   (optional spare   │
│    axes + curves)     │    space)           │
│                       │                     │
├───────────────────────┴─────────────────────┤
│                BOTTOM AREA                 │
│               Watermark (@lingualin)       │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	k‑label (k = 48) and formula (y = x^{2/3} + 0.9\sin(kx)\sqrt{3 - x^2})	Both fade/write in during Phase 1; white color, font sizes 42 and 36 respectively; positioned at the top edge with a small vertical buffer.
Left	Axes (black background, white axes, subtle ticks) and the oscillating heart curve (glow group)	Central focus; axes span $[-2,2]$ in both directions; no camera movement.
Right	Unused – kept empty for possible future annotations.	No content required for this animation.
Bottom	Watermark (@lingualin)	Fades in with Phase 1; gray color #AAAAAA, opacity 0.6, font size 24, positioned slightly below the origin (DOWN*1.2).

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Notes

• Background: pure black (#000000).
• Axes: created with Axes (not NumberPlane), ranges $x\in[-2,2]$, $y\in[-2,2]$, axis color white, stroke opacity 0.4, arrow tips enabled, tick marks small and subtle.
• Curve definition:
  $$y(x, t) = x^{2/3} + 0.9\sin\bigl(k x + \omega t\bigr)\sqrt{\max\bigl(0, 3 - x^2\bigr)}$$
  with $k=48$, $\omega=6$, domain $x\in[-\sqrt{3},\sqrt{3}]$.
• Sampling: 1200 points via np.linspace(-np.sqrt(3), np.sqrt(3), 1200) for a smooth ParametricFunction.
• Glow effect: three overlaid ParametricFunctions with stroke widths 3.5, 2.5, 1.6 and opacities 0.08, 0.15, 0.95 respectively, all white (#FFFFFF).
• Symmetry: a mirrored copy of the curve with negative $y$ values creates vertical symmetry across the x‑axis.
• Animation timeline:
  • Phase 1 uses FadeIn for axes (shifted up 0.2) and Write for the two top texts, plus FadeIn for the watermark; run_time=2, rate_func=smooth.
  • Phase 2 animates t from 0 to 20 with linear rate over 8 seconds; the curve redraws continuously via always_redraw.
  • Phase 3 simply wait(2) to hold the final frame.
• No camera movement; the view remains centered on the origin throughout.
• Anti‑aliasing should be enabled in the final render to preserve the soft glow edges.
• The total runtime is 12 seconds, well under the 30‑second guideline.