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.
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.
The Mathematical Model: Scaling the Pulse
This relationship is a fundamental concept in Trigonometric Projections
and is a classic example of Simple Harmonic Motion.
The vertical displacement y is now scaled by the radius A.
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 resulting wave traces a path in Electric Cyan.
Name: Source Code: Manim Implementation *