Computer Animation AUP

There are some interesting surfaces that exist in the Complex Plane (in the s-plane). For example, Mount Nichols is described by the equation

$$M = \left\vert\frac{L}{L+1}\right\vert$$

where M is the height of the surface and L is a complex independent variable. Mount Nichols is usually plotted on ``gain/phase'' coordinates, where the vertical axis is the magnitude of the complex number L and the horizontal axis is the phase of L, as in Figure 1. Notice the peak as the magnitude goes to infinity when L=-1 (when the magnitude is 0dB and the phase is -180$^\circ$).

  

Figure 1: Two step animation: from a cut of Mount Nichols to the magnitude Bode Plot

The interesting thing about Mount Nichols is that when you cut this surface along a trajectory described by some function $G(j\omega)$, the resulting slice is the magnitude portion of the Bode Plot of the closed loop feedback system containing the transfer function G(s). Notice how the shape of the cliff is (nearly) the flipped version of the magnitude plot.

The major focus of this AUP would encompass producing short animated cartoons of this process including plotting the function G(s), cutting Mount Nichols along the specified trajectory, and extracting, flipping and rescaling the mountain slice to get the magnitude portion of the Bode plot. This process could be carried out for any G(s) and could also be done on other coordinate systems.

Some background in computer animation is required, and a background in complex variables (like 18.04) would be helpful. If you are interested in this project, or if you would like more information, please contact Kent Lundberg (klund(at)mit.edu).