Projection of Circular Motion

Interpreting the real and imaginary parts of the complex sinusoid,

$$\begin{eqnarray*} \mbox{re}\left\{e^{j\omega t}\right\} &=& \cos(\omega t) \\ \mbox{im}\left\{e^{j\omega t}\right\} &=& \sin(\omega t), \end{eqnarray*}$$

in the complex plane, we see that sinusoidal motion is the projection of circular motion onto any straight line. Thus, the sinusoidal motion $\cos(\omega t)$ is the projection of the circular motion $e^{j\omega t}$ onto the $x$ (real-part) axis, while $\sin(\omega t)$ is the projection of $e^{j\omega t}$ onto the $y$ (imaginary-part) axis.

Figure 4.7 shows a plot of a complex sinusoid versus time, along with its projections onto coordinate planes. This is a 3D plot showing the $z$-plane versus time. The axes are the real part, imaginary part, and time. (Or we could have used magnitude and phase versus time.)

Figure: A complex sinusoid and its projections.

Note that the left projection (onto the $z$ plane) is a circle, the lower projection (real-part vs. time) is a cosine, and the upper projection (imaginary-part vs. time) is a sine. A point traversing the plot projects to uniform circular motion in the $z$ plane, and sinusoidal motion on the two other planes.