What's going on here? This applet dynamically demonstrates a
mathematical transformation called conformal mapping. Simply put,
this takes a point in the complex plane and moves ("maps") it to a
different place. This has all sorts of interesting applications,
from
aerodynamics to fractals. I'm fooling
around with them for user interfaces, because I think they do
interesting things dynamically (some of which you can try
here).
How it works: drag the mouse to move the source
location, which is a grid of points two units square. This is
transformed (using the equation shown at upper right) into the
green points displayed on the graph. You can optionally turn off
the axis, shown in black, which displays the unit circle and the
real and imaginary axes. You can optionally display the source
region, shown as a red square. The slider controls the zoom factor:
sliding it to the right zooms in. The "Next Map" button cycles through
several available transformations; the last one is the identity
[f(z) = z] that does not change the source.