video/.agents/skills/manimgl-best-practices/references/rotating_exponentials.md

Rotating Exponentials - Reference Guide

Example file: examples/rotating_exponentials.py

User Query Scenarios

This example addresses queries like:

• "Visualize e^(it) on the complex plane"
• "Show Euler's formula animation"
• "Demonstrate how cosine comes from rotating exponentials"
• "Create a complex plane with rotating vector"
• "Show e^(iπ) = -1 visually"

Scene Thinking Process (3b1b Style)

1. Core Concept

Euler's Formula: e^(it) = cos(t) + i·sin(t) - a rotating unit vector in the complex plane. Two counter-rotating exponentials sum to give real cosine.

2. Visual Design Decisions

Why use ComplexPlane?

• Natural coordinate system for complex numbers
• Built-in grid and labels
• n2p() method converts complex to point

Why show the traced path?

• Reveals the unit circle emerges naturally
• Shows the relationship between angle and position

3. Technical Implementation

Rotating Vector with TracedPath

time_tracker = ValueTracker(0)

vector = Vector(RIGHT, color=YELLOW)
vector.add_updater(lambda v: v.put_start_and_end_on(
    ORIGIN,
    plane.n2p(np.exp(1j * time_tracker.get_value()))
))

tip_dot = Dot(color=YELLOW)
tip_dot.add_updater(lambda d: d.move_to(vector.get_end()))

traced = TracedPath(tip_dot.get_center, stroke_color=BLUE)

Counter-Rotating for Cosine

# e^(it) rotates counter-clockwise
v1.add_updater(lambda v: v.put_start_and_end_on(
    ORIGIN, plane.n2p(np.exp(1j * t))
))
# e^(-it) rotates clockwise
v2.add_updater(lambda v: v.put_start_and_end_on(
    ORIGIN, plane.n2p(np.exp(-1j * t))
))
# Sum is always real: 2cos(t)

4. Scene Variants

Scene	Purpose
RotatingExponential	Basic e^(it) visualization
CounterRotatingExponentials	Shows e^(it) + e^(-it) = 2cos(t)
EulersFormula	Famous e^(iπ) = -1
ComplexExponentialSpiral	Decaying spiral e^((a+bi)t)

Key Patterns

Pattern: always_redraw for Arcs

angle_arc = always_redraw(lambda: Arc(
    start_angle=0,
    angle=time_tracker.get_value() % TAU,
    radius=0.3,
    color=GREEN
))

Pattern: Complex Number to Point

# Using ComplexPlane.n2p() (number to point)
point = plane.n2p(1 + 2j)  # Complex number
point = plane.n2p(np.exp(1j * theta))  # Euler form

Run Commands

manimgl rotating_exponentials.py RotatingExponential -w
manimgl rotating_exponentials.py CounterRotatingExponentials -w
manimgl rotating_exponentials.py EulersFormula -w
manimgl rotating_exponentials.py ComplexExponentialSpiral -w