möbius transforms [Sep. 9th, 2010|03:27 pm] Tobin Fricke's Lab Notebook





I learned a neat complex analysis trick yesterday, by virtue of opening the book Visual Complex Analysis randomly to page 156.



A Möbius transform is a function of the form:







where a, b, c, d are complex constants and z is the complex variable. Such transforms can invert, rotate, translate, and dilate the complex plane.



You can represent a Mobius transformation as a matrix acting on a two-component complex vector:







The trick is that you consider the vectors to represent the scalar complex numbers given by the ratio of the components:







Then matrix multiplication of these matrices acts the same way as composition of the Mobius transformations!



This is the trick of homogeneous coordinates. It's also useful for real-valued vectors; by adding an additional dummy coordinate you can represent translations as matrices. The trick shows up all the time once you see it. For example, in optics, this is how we do calculations of the complex beam parameter using matrices.