(obsolete FLI version)


Douglas N. Arnold

The squaring map Exponential, sine, and cosine Conformality M�bius transformations How to?


If your browser supports Java, the latest version of this page is preferable to this one. There is also a version based on animated GIFs. These later versions have more and better quality graphics.


The animations on this page are in FLI/FLC format. Viewers for this format are available for most platforms. You'll get the most from the animations by controlling the frame advance manually (which is possible with most viewers). If you can't control the frame advance manually, it is best to put the viewer in shuttle, or ping-pong, mode and adjust the frame advance speed to rather slow.

The squaring map

As a first example, consider the simple function f(z)=z^2. We shall examine how a square with corners at +/- 3/2 +/- 3/2i is transformed under this map. First we just look at the half of the square in the right half plane, which is shown in the first frame (3K).This region mapped one-to-one onto the region contained between two parabolas, exhibited in final frame (5K).The lines of constant real part are mapped onto parabolas opening along the negative real axis, and those with constant imaginary part to parabolas opening along the positive real axis. Note that these parabolas meet at right angles, a consequence of conformality (see below). The frames of the animation (196K) show the image of the half-square under a linear combination of the identity map and the squaring map: z -> (1-t) z + t z^2, with t increasing from 0 to 1. In other word, this is an animation of the linear homotopy taking the identity to the squaring map. Note that the intermediate maps are again quadratic in z and hence conformal at all but one point.

The next animation (148K) shows the deformation of the half of the square lying in the left half plane. This region is mapped one-to-one onto the same region and the right half of the square. Therefore the deformation of the entire square (251K) shows the square being folded over onto itself as it distorts, so as to cover the parabolically bounded region twice. If you pause the animation on the penultimate frame, the two-to-one character of the map is evident.

The squaring map behaves more simply when applied to a polar grid, since it takes circles of constant modulus to circles of constant modulus, and rays of constant argument to rays of constant argument. We animate the deformation of the disc of radius 2 centered at the origin and overlayed with a polar grid. The animations show the deformation of the half disc in the right half plane (215K), the deformation of the half disc in the left half plane (174K), and the deformation of the entire disc (316K).

A better view of the same map comes when the disc is colored, here by a hue which varies with the argument (so the color is constant on each ray emanating from the origin). An animation of the colored disc (536K), displays clearly how the disc is folded over on itself to cover the image disc twice (again, pause a few frames from the end). The discontinuity in the coloring in the final frame reflect the impossibility of defining a continuous square root function on the disc. By looking at the last few frames, it is also possible to get a pretty clear idea of the Riemann surface for the squaring map (a surface which, when realized in 3 dimensional space, intersects itself along the negative real axis).

The exponential, sine, and cosine


The next animations show two-to-one coverings of a disc by the complex cosine (653K) and complex sine (655K) restricted to suitable rectangles.




M�bius Transformations

M�bius transformations are widely studied as a useful example of conformal maps. A key property is that they map circles in the Riemann sphere into circles in the Riemann sphere, or, equivalently, that the image of a line or circle in the plane is another line or circle. This animation (1M) shows the identity map (a M�bius transformation!) being deformed into a less trivial M�bius transformation. The homotopy is chosen so that all the intermediate steps are M�bius transformations as well. The animation shows the unit square deformed into a closed curve composed of four circular arcs. The inside of the square is mapped to the region outside of the curve, and a region about infinity is mapped to the interior of the curve. The specific family of M�bius transformations displayed is

<displayed math>

How were these graphics made?



Last modified January 29, 1997 by Douglas N. Arnold,