Euler's formula, rotating

Parametric equations

Euler’s formula $e^{i\theta }=\cos \theta +i\sin \theta$ says that multiplying by $e^{i\theta }$ is a counterclockwise rotation by $\theta$. In the figure, the tip of the rotating vector stays in step with the sine wave of its imaginary part on the right, tying the exponential to the trigonometric functions.

In general, $r{e}^{i\theta }$ is the complex number of modulus $r$ and argument $\theta$.

Interaction

• Amplitude A: change the length of the rotating vector and the amplitude of the sine wave on the right
• Angular frequency ω: control how fast the vector turns and how tightly the wave is packed
• Initial phase δ: shift where the rotation starts and the phase of the wave

What to notice

• The imaginary part of the rotating vector on the complex plane is the height of the sine wave in time.
• $e^{i(\omega t+\delta )}$ writes the circular motion and the trigonometric functions in one expression.
• The angular frequency sets both the period of the rotation and the spatial period of the wave.

Related

• Polar form of a complex number
• Complex numbers and Euler’s formula
• Julia set

Further reading

• Euler’s formula (Wikipedia)