A selection of animations, implemented both as Java applets and as animated GIFs, illustrating complex mappings: the map z -> z2; the complex exponential, sine and cosine; Mobius transformations; Schwartz-Cristoffel transformations; and the nature of conformal vs. non-conformal maps. The review of the related site Graphics for the Calculus Classroom may also be of interest.
To build intuition for the geometric properties of mappings of the plane produced by functions of a complex variable.
Target Student Population:
Students in an introductory class on complex variables or conformal mappings.
Prerequisite Knowledge or Skills:
Basic understanding of functions of a complex variable considered as mappings of the plane. Exposure to the complex exponential, sine and cosine maps.
Type of Material:
The animations are all available either as animated GIFs or as Java applets. (The Java versions allow some user interactivity, such as varying the speed of the animation; the GIF versions do not.)
These animations could be used for demonstrations in a classroom, or as a supplement to a reading assignment.
Any web browser. (To use the Java versions of the animations, Java must be supported.)
Evaluation and Observation
The general idea of the site is simple but effective: each animation consists of a "morph" from the identity map z -> z to a mapping z -> f(z), for some complex function f. In each case, a finite region of the plane is morphed and a coloured, conformal grid is used to show how the region distorts. (This basic pattern is changed in some of the later animations, but the morphing idea is retained.) Sometimes, with a 2-to-1 map, for example, it can be hard to follow the region through the morph; in such cases, intuition is built up by first looking at sub-regions on which the mapping is 1-to-1.
Some of the images are unnecessarily inscrutable, and a brief explanation would be very useful. For example, in the deformation from f(z)=z to f(z)=z^2, illustrated on Re(z)< 0, it is very hard to see what is happening, because the grid lines match up almost perfectly when the images overlap. In fact as soon as t>0 in the homotopy there is a branch point which shows up in the square as soon as t>1/4. It moves from the edge of the square to the origin during the rest of the homotopy. This can be seen once you know to look for it. Another problem which could be remedied by a little more text: for the exp,sin and cos examples, it is not stated explicitly that what we are seeing is a (linear?) homotopy between f(z)=z and the map in question. The presentation is not consistent. For the "power map" demo, the homotopy applies to the exponent; for the conformality-nonconformality demo it is again implicitly the (linear?) homotopy from f(z)=z to the map in question; while for the Moebius transformations t is explicit.
Potential Effectiveness as a Teaching Tool
Each item on the site illustrates its point clearly. The comparison between conformal and non-conformal mappings is particularly nice. (The nature of the map is indicated by following small squares through the morphing process,
and noting that in the conformal case they remain squares, to first order, while in the non-conformal case they distort much more dramatically.)
As it stands, the site is only appropriate for use in classroom demonstrations by a teacher (the original purpose of this material). With additional text, it would be suitable for ``standalone'' use, and would be potentially much more valuable. For example, none of the JAVA animations have any explanatory text at all. But there is no obstruction to putting text on the HTML pages that carry JAVA animations, so that the text can be consulted while the animation is running.
Ease of Use for Both Students and Faculty
The site is simply organised, with clear instructions. The user need in fact never do anything more than click the mouse.
More than once, a Java animation froze one reviewer's browser. (The problem arises on closing the applets: closing the first applet causes no problems, but closing a second causes the browser to freeze.) The problem seems to be known to the author, because a note on the main page suggests that if this happens, the user should use the animated GIF versions. Unfortunately, as noted above, these are non-interactive.