Visualizing Euler's Formula

Overview

A brief animation that visualizes Euler's identity $e^{ix}=\cos x + i\sin x$ by drawing the unit circle in the complex plane, tracing the exponential curve, and showing the correspondence between the angle $x$, the point on the circle, and the cosine/sine components. The key takeaway is that rotating a unit vector by angle $x$ in the complex plane is equivalent to the complex exponential.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Title "Euler's Formula" fades in at the top. The unit circle appears faintly in the left area.
2	Build Unit Circle	~5s	The circle is drawn, axes labeled real (horizontal) and imaginary (vertical). A point at (1,0) is highlighted.
3	Angle Sweep	~6s	A rotating radius sweeps from 0 to $2\pi$. As it rotates, a trailing arc shows the angle $x$. The point on the circle moves accordingly.
4	Exponential Trace	~6s	Simultaneously, a curve representing $e^{ix}$ is traced in the complex plane (identical to the circle). A small label follows the point showing the current coordinates $(\cos x, \sin x)$.
5	Component Highlight	~5s	Dashed lines drop from the moving point to the real and imaginary axes, visually representing $\cos x$ and $\sin x$. Small markers on the axes grow/shrink with the values.
6	Summary Overlay	~3s	The equation $e^{ix}=\cos x + i\sin x$ fades in at the bottom, with arrows pointing to the corresponding visual elements.
7	Outro	~2s	All elements fade out, leaving only the title which then fades out.

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Layout

┌───────────────────────────────────────┐
│               TOP AREA                 │
├───────────────────────┬───────────────┤
│                       │               │
│        LEFT AREA      │   RIGHT AREA  │
│  (Complex plane with │   (Optional   │
│   unit circle, axes) │   minimal    │
│                       │   notes)      │
├───────────────────────┴───────────────┤
│               BOTTOM AREA              │
└───────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title "Euler's Formula"	Fades in at start of Intro, stays until Outro
Left	Main visual: complex plane, unit circle, rotating radius, exponential trace, component lines	Central focus throughout all phases
Right	Optional brief note (e.g., "Angle $x$ in radians") that appears during Phase 3 and fades out	Keeps the left side uncluttered
Bottom	Equation $e^{ix}=\cos x + i\sin x$ and small caption	Appears in Phase 6, fades out with Outro

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Notes

• Keep total runtime under 30 seconds (≈30 s). All durations are approximate and can be slightly adjusted for smooth pacing.
• Use smooth continuous rotations for the radius; ease‑in at the start and ease‑out at the end for a natural feel.
• Colors: real axis – blue, imaginary axis – orange, rotating radius – green, exponential trace – red, dashed component lines – gray.
• No textual explanations are needed beyond the title, optional brief note, and final equation; the visual correspondence should be self‑explanatory.
• The scene must be implemented as a single Manim Scene class.