Projection of Circular Motion
Interpreting the real and imaginary parts of the complex sinusoid,
in the complex plane, we see that sinusoidal motion is the projection of circular motion onto any straight line. Thus, the sinusoidal motion is the projection of the circular motion onto the (real-part) axis, while is the projection of onto the (imaginary-part) axis.
Figure 4.9 shows a plot of a complex sinusoid versus time, along with its projections onto coordinate planes. This is a 3D plot showing the -plane versus time. The axes are the real part, imaginary part, and time. (Or we could have used magnitude and phase versus time.)
Note that the left projection (onto the 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 plane, and sinusoidal motion on the two other planes.
Positive and Negative Frequencies