Spiral Motion Animation

Overview

A 20‑second animation visualizing the parametric spiral defined by

$$x(t) = (20 - t)\cos(2\pi t), \qquad y(t) = (20 - t)\sin(2\pi t)$$

A point travels along the curve, leaving a visible fading trail that illustrates the path history. The key takeaway is how the radius shrinks linearly while the angle rotates uniformly, producing a classic inward spiral.

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Phases

#	Phase Name	Duration	Description
1	Intro	~2 s	Axes fade in, the spiral curve is drawn faintly, and the defining equations appear at the bottom.
2	Motion	~16 s	A bright point moves from $t=0$ to $t=20$ following the parametric equations, leaving a semi‑transparent trailing line that gradually fades.
3	Outro	~2 s	Motion stops, the trail solidifies into the full spiral, and the point fades out, leaving the final curve on screen.

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Layout

┌─────────────────────────────────────────────┐
│                 TOP AREA                    │
├─────────────────────────────────────────────┤
│                 MAIN AREA                   │
├─────────────────────────────────────────────┤
│                BOTTOM AREA                  │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title: "Parametric Spiral Motion" (optional)	Fades in with Intro phase
Main	Coordinate axes, the spiral curve, moving point, and its trail	Central visual focus
Bottom	Equations $x(t) = (20 - t)\cos(2\pi t)$, $y(t) = (20 - t)\sin(2\pi t)$ displayed in LaTeX	Appear after axes, stay for entire animation

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Notes

• The animation runs at 20 fps, giving a smooth visual over the 20‑second duration.
• The trail should use a fading opacity gradient so earlier segments become more transparent, emphasizing recent motion.
• No additional text is required beyond the title and equations; all concepts are conveyed visually.
• The scene will contain a single Scene subclass in the final Manim code.
• Rendering & Downloading:
  • Render the scene with Manim using a command such as:

    manim -pql spiral_motion.py SpiralMotionAnimation
    

    (-p opens the video automatically, -ql renders in low quality for quick preview; replace with -qh or -qm for higher quality).
  • The output video will be saved in the media/videos directory under a folder named after the script (e.g., media/videos/spiral_motion/1080p60/SpiralMotionAnimation.mp4).
  • To download the video, locate this .mp4 file and copy it to your desired location, or use a file‑sharing service if you need to transfer it elsewhere.