Evaluating the Gaussian Integral with Polar Coordinates

Animation Specification for the Gaussian Integral

Scene: Single Scene class (e.g., GaussianIntegralScene). Total runtime ≈ 25 seconds.

1. Animation Description & Purpose

• Visually demonstrate the classic evaluation
  $$\int_{-\infty}^{\infty} e^{-x^{2}}\,dx = \sqrt{\pi}$$
• Highlight the key steps: Gaussian curve, symmetry, squaring the integral, conversion to polar coordinates, and final result.
• Emphasize geometric intuition (area under the curve ↔ quarter‑circle in the plane).

2. Mathematical Elements & Formulas

Step	Formula (LaTeX)	Visual Representation
0	$$I = \int_{-\infty}^{\infty} e^{-x^{2}}\,dx$$	Appears as a centered equation with an opaque dark‑gray background.
1	I^{2} = \left(\int_{-\infty}^{\infty} e^{-x^{2}}\,dx\r\right)\!\left(\int_{-\infty}^{\infty} e^{-y^{2}}\,dy\r\right) = \iint_{\mathbb{R}^{2}} e^{-(x^{2}+y^{2})}\,dx\,dy	Fade‑in after the curve is shown; the product sign expands into a 2‑D integral label.
2	$$\iint_{\mathbb{R}^{2}} e^{-(x^{2}+y^{2})}\,dx\,dy = \int_{0}^{2\pi}\!\int_{0}^{\infty} e^{-r^{2}}\,r\,dr\,d\theta$$	Polar‑coordinate transformation appears with arrows indicating the change of variables.
3	$$\int_{0}^{\infty} e^{-r^{2}}\,r\,dr = \tfrac{1}{2}$$	Result of the radial integral shown after the radial integration animation.
4	$$I^{2} = 2\pi \cdot \tfrac{1}{2} = \pi \quad\R\rightarrow\quad I = \sqrt{\pi}$$	Final equation appears with a brief highlight on the square‑root.

3. Visual Elements

• Axes: Standard Cartesian axes (gray) spanning $[-4,4]$ on both axes.
• Gaussian Curve: Plot of $y = e^{-x^{2}}$ in teal, thickness 3.0. The curve is drawn from left to right.
• Shaded Area: Semi‑transparent teal fill under the curve from $-R$ to $+R$ where $R$ expands from 0 to 4 (simulating the limit $R\to\infty$).
• Symmetry Highlight: A vertical dashed line at $x=0$ (light gray) appears after the curve is drawn.
• 2‑D Plane for Squared Integral: After the curve, a duplicate Gaussian is drawn on the y‑axis (rotated 90°) to form a surface; the product region $\mathbb{R}^{2}$ is indicated by a faint grid.
• Polar Grid: Concentric circles (radius steps 0.5) and radial lines every $30^{\circ}$ fade in to illustrate polar coordinates.
• Arrows & Brackets: Curved arrows show the substitution $(x,y)\to(r,\theta)$. Brackets label the radial and angular integrals.
• Result Highlight: The final $\sqrt{\pi}$ appears in bold white text on a dark‑gray rectangular background (opacity 0.85) centered on the screen.

4. Animation Timing & Transitions (seconds)

Time (s)	Action
0.0‑0.8	Fade‑in the initial equation $I = \int_{-\infty}^{\infty} e^{-x^{2}}dx$ with background.
0.8‑2.5	Draw axes, then animate the Gaussian curve from left to right.
2.5‑4.0	Expand shaded area: radius $R$ grows from 0 to 4, simultaneously fading the fill to illustrate the limit.
4.0‑4.6	Appear vertical dashed symmetry line.
4.6‑6.0	Fade‑out the 1‑D curve, fade‑in the duplicated curve on the y‑axis, and overlay a faint 2‑D grid to suggest $\mathbb{R}^{2}$.
6.0‑7.5	Show the product integral formula $I^{2}=\iint_{\mathbb{R}^{2}} e^{-(x^{2}+y^{2})}dxdy$.
7.5‑9.5	Fade‑in polar grid; animate arrows converting $(x,y)$ to $(r,\theta)$. Display the polar‑coordinate integral formula.
9.5‑11.5	Animate the radial integral: a small dot moves outward along a radius while the integrand $e^{-r^{2}}r$ is highlighted; then the dot collapses to the origin, indicating the evaluation $\int_{0}^{\infty} e^{-r^{2}}r dr = \tfrac12$.
11.5‑13.0	Show the angular integral (full rotation) as a sweeping arc completing a 360° sweep, then fade‑out the polar grid.
13.0‑14.5	Present the simplified product $I^{2}=\pi$ with a brief “square both sides” visual cue (a small square appearing around $I$).
14.5‑16.0	Reveal the final result $I = \sqrt{\pi}$ with the opaque background box.
16.0‑20.0	Optional pause for viewer absorption; slowly zoom out to show the whole derivation as a single static picture.
20.0‑25.0	Fade‑out everything to black (scene end).

5. Camera Angles & Perspectives

• Static Camera: The camera remains centered on the origin throughout; only subtle zooms are used (e.g., slight zoom‑out during the final pause). No rotation or pan is required.
• Zoom: From 14.5 s to 20 s, a smooth 1.2× zoom‑out reveals the entire derivation.

6. Additional Details

• Colors:
  • Axes & grid: light gray.
  • Gaussian curve & fill: teal (RGB #009688).
  • Polar grid: soft orange (RGB #FF9800) with low opacity.
  • Text/background: white text on dark‑gray (RGB #333333) rectangle.
• Easing: Use smooth easing for all draws; there_and_back for the radial dot motion.
• Opacity: Shaded area opacity 0.35; polar grid opacity 0.2.
• No extraneous text: Only the essential formulas listed above appear, each with an opaque background when displayed.
• Scene Length: Approximately 25 seconds, well within the 30‑second guideline.