Circular Motion and Sine Wave

Overview

An animation that visualizes the relationship between uniform circular motion and the sine function. A point travels counter‑clockwise around a unit circle on the left side of the screen while its vertical projection simultaneously draws a sinusoidal wave on the right side. The key takeaway is that the sine wave is the y‑coordinate of a point moving uniformly on a unit circle.

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Phases

#	Phase Name	Duration	Description
1	Setup	~2s	The unit circle appears on the left, the horizontal axis for the wave appears on the right, and a thin vertical line connects the point on the circle to the wave start point.
2	Motion & Tracing	~8s	The point moves smoothly around the circle at constant angular speed. Its vertical coordinate is continuously plotted on the right, creating a sine wave that grows from left to right. A moving dot on the wave indicates the current time.
3	Pause & Highlight	~3s	Motion pauses; a brief highlight shows the correspondence: the point’s y‑position equals the wave’s height at the same time. Optional label "y = \sin(\ heta)" appears near the dot.
4	Fade Out	~2s	All objects fade, leaving the final static picture of the circle with the completed sine wave.

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Layout

┌───────────────────────┬───────────────────────┐
│                       │                       │
│      LEFT (circle)    │   RIGHT (wave)        │
│                       │                       │
├───────────────────────┴───────────────────────┤
│               Caption (optional)            │
└───────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Left	Unit circle of radius 1 centered at the origin; a moving point on the circumference; a thin radial line indicating the current angle $\ heta$.	Primary visual for circular motion.
Right	Horizontal axis representing time; a growing sinusoidal curve $y = \sin(\ heta)$; a moving dot that follows the tip of the curve.	Shows the projection of the point’s y‑coordinate.
Caption	Short label such as "Circular motion ↔ Sine wave" or the equation $y = \sin(\ heta)$ during Phase 3.	Optional; appears only in Phase 3.

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Notes

• Assumptions: The animation runs at a moderate pedagogical pace (~13 seconds total) and uses a constant angular speed of one full revolution per 4 seconds, giving a clear, repeatable pattern.
• Only one Manim Scene class is required; all elements are created within that scene.
• No persistent text is used except the optional caption in Phase 3, keeping the visual focus on the geometry.
• The vertical connecting line from the point to the wave start point helps the viewer see the instantaneous correspondence.
• Colors: circle and point in a calm blue, wave in orange, connecting line in light gray.
• All transitions are smooth fades or linear motions; no abrupt jumps.