Möbius transformations

This applet lets you draw points, lines, and circles, and see what happens to them under Möbius transformations. It is possible to use this applet to find the answers to most of the homework questions in Sections 7.2 and 7.3.

A Möbius transformation (also called a fractional linear transformation, projective linear transformation, or a bilinear transformation by some authors) is any map of the form w = (az+b)/(cz+d), where a,b,c,d are complex numbers. For technical reasons we exclude the case when ad-bc = 0 (for instance, the map w = (3z+6)/(z+2) is not a Möbius transform).

The applet is not as complicated as it first appears! A tutorial is given below.

Thanks to Hauke Reddmann for some suggestions.

• Move the mouse on the "z" grid (i.e. the left grid). If you've already played with the Complex map applet then you won't be surprised: right now we're displaying the boring old identity map. Unlike the previous applet, though, clicking the mouse produces red dots instead of freehand curves.

• Initially, the applet is set to draw "Red" "Points", but you can change that by using the two option bars at the top left of the applet. Try drawing some green lines or blue circles. (You will need to drag the mouse to do this).

• You can get rid of the last object drawn by pressing "Delete". To get rid of everything, press "Clear".

  You can also draw objects more precisely by using the buttons "Point", "Line", and "Circle". If you click on any one of these buttons, a window will appear. Fill in the numbers and click "Done" to enter a new object.

• So far, the "z" grids and "w" grids are looking identical. That's because they're currently linked by the equation "w=z", as you can see on the bottom of the applet. The purpose of the rest of the buttons on the applet is to change this equation to a different Möbius transform.

• Draw some stuff on the two grids. Then press one of the buttons on the "Translate w by:" row. What happened to the objects drawn? What happened to the equation connecting "w" and "z"? Draw some more objects on the grids. Note that you can draw objects on the "w" grid as well as the "z" grid.

• Repeat the above, but use the buttons on the "Dilate w by:" row. Can you understand what each button does, and why they are labelled the way they are? Some buttons reverse the actions of others: which ones?

• Repeat the above, but with the "Invert w" button. This replaces w by 1/w. You should remember this map from the previous applet; it inverts the magnitude, and flips the phase, of a complex number. To make the transition smoother, I've also added in some intermediate stages of evolution so you can see how w turns itself "inside out" to become 1/w.

• You may have noticed a blue cross appearing on the two grids. The one on the z grid is the "pole" (or "singularity") of the Möbius transformation. This is the one value of z for which the corresponding value of w is infinite. If you move your mouse close to the singularity you will see the corresponding point on the other grid go haywire. The cross on the w grid is similar, but applies to the inverse Möbius transformation.

• By now, the Möbius transformation linking w and z should be pretty messy, especially if you've been using the exp(pi i/4) or exp(-pi i/4) keys. To reset the map back to the identity, click on "Identity map".

• There are two other buttons which give off-the-shelf maps: the "Inversion map", which displays the map w = 1/z, and the "Smith map" (also called the Cayley map by mathematicians) w = (z-1)/(z+1), which comes up in electrical engineering.

• You can also go to a customized Möbius transformation by the "Enter new values" button at the bottom of the applet. A window will appear. Set the numbers to be whatever you please (probably most of the entries will be zero), and press "Done". If your transformation is degenerate (that is, if ad-bc=0) then it will be ignored.

• We're almost done with the tutorial - only two more buttons need to be explained. "Simplify w" doesn't do much, other than try to clean up the equation connecting w and z. "Swap w and z" does exactly what you would think to the z and w grids. Since z and w are interchanged, the current Möbius transform must be replaced with its inverse; for instance, a translation by +0.5 will become a translation by -0.5.

• Obviously, the applet can be used to see what happens to points, lines and circles under any Möbius transformation one pleases. But with a little bit of creativity, one can also see what happens to slightly more complicated objects: disks, line segments, arcs, squares, etc.

• Let's take an example. Suppose we wanted to know what happened to the disk {z: |z-1| < 2} under the inversion map w = 1/z. We don't have a button for "Disk", but we do have a button for "Circle". So we can draw the circle {z: |z-1| = 2}, and select "Inversion map".

• As you should be able to see, the circle gets mapped to another circle (in this case, the circle is {w: |w + 1/3| = 2/3}). So you might think that the disk will get mapped to the disk {w: |w + 1/3| < 2/3}. But actually, the disk gets turned inside out - it gets sent to {w: |w + 1/3| > 2/3}, which is the region outside the circle!

• How can you tell? The easiest way is to place a dot or two inside the disk, and see where they map to. If they map inside the target circle, then the disk maps to the inside of the circle. Otherwise it maps to the outside of the circle.

• If you want to get a better idea of what's happening, put dots of one color inside the disk and dots of another color outside the disk. Reset the map to "Identity map", and then press "Invert w". Notice how things get turned inside out.

• Now you should be able to use the applet to find the answer to most of the Homework Questions in 7.2 and 7.3. Note that the applet doesn't actually give answers in equation form, only in picture form. You still have to actually do the homework, of course! The applet is only there to help you check your answers, and get some idea of how Möbius transformations work.

Previous applet: Elementary complex maps

Next applet: Multiple-valued maps

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