Mapping the Unit Circle | Math Animation with Manim Python

Beyond the Pulse: The Geometry of the Sine Wave

Defining Amplitude through the Geometry of Projection

While our first experiment established that the wave and circle are linked in The Pulse of a Circle , today we prove that the circle’s physical size is the “DNA” that determines the wave’s height.

Unit Circle to Sine Wave Projection

Mathematics of Waves: Unit circle animation mapping the rotation of a point to a sine wave.

Lab Results: The Radius-Amplitude Equivalence

If you look closely at the peak of each wave, you’ll notice it aligns perfectly with the furthest vertical reach of the circle. This isn’t a coincidence; it’s a geometric law.

In this experiment, we’ve moved from a standard unit circle \(R=1\) to a customized radius of 1.8. This change in the “Source” (the circle) dictates the “Pulse” (the wave).

The Key Takeaway:

The Input: The Radius of the circle.
The Output: The Amplitude of the sine wave.

By stretching the radius, we are essentially telling the wave how much “energy” or “height” it is allowed to have.

What’s Next? In our next blog, we’ll see what happens when we manipulate the Speed—turning this slow rotation into a high-frequency vibration.

Note: This relationship is a fundamental concept in Trigonometric Projections
and is a classic example of Simple Harmonic Motion.

The Mathematical Model: Scaling the Pulse

The vertical displacement y is now scaled by the radius A.

y = A sin(θ)

By setting the radius to 1.8, the height y oscillates between 1.8 and -1.8, physically locking the wave’s peak to the circle’s edge.

The radius “DNA” is shown in Yellow.
The resulting wave traces a path in Electric Cyan.

Real-Life Use Examples

Audio Engineering & Sound Design: Sound waves are sinusoidal pressure variations. The amplitude dictates how far air molecules are displaced, which directly translates to the volume or loudness of a sound (higher amplitude equals louder audio).

Alternating Current (AC) Power Grids: Household electricity flows in a sinusoidal wave governed by generator rotation. The peak voltage delivered to your wall outlet is dictated by the physical design scale of the commercial generators at the power plant.

Mechanical Suspension Systems: When a car drives over a bump, the shock absorber executes simple harmonic motion. The maximum vertical distance the wheel travels from equilibrium matches the amplitude parameter modeled by our scaled radius.

Frequently Asked Questions

Why does changing the circle’s radius directly change the sine wave’s amplitude?

The y-coordinate of any point on a circle of radius A is defined trigonometrically as y = A sin(theta). Because the maximum value of the sine function is 1, scaling the circle by a factor of A scales the maximum vertical reach to A, locking the wave’s peak directly to the circle’s outer edge.

What is the difference between a standard unit circle and a custom radius circle?

A standard unit circle strictly has a radius of R = 1, restricting output values between 1 and -1. Customizing the radius transforms the circle into a generalized geometric template, allowing you to model real-world systems where maximum displacement exceeds standard mathematical bounds.

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Name: Source Code: Manim Implementation *

from manim import *
import numpy as np

class SineAmplitudeProjection(Scene):
    def construct(self):

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Beyond the Pulse: The Geometry of the Sine Wave

Defining Amplitude through the Geometry of Projection

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