Maple code for mapping circles

Here we present some procedures for dealing with Möbius transformations applied to circles. In class, we created the procedures in mobius.mpl, which are:

mobius(m,z):

This applies the Möbius transformation corresponding to $m=\begin{bmatrix}a&b\\ c&d\end{bmatrix}$ to the point $z$, returning $\frac{az+b}{cz+d}$. The point $z$ may be $\infty$, and the answer could be $\infty$ as well.

fixmatrix(m):

This takes a matrix $m$ of nonzero determinant and returns $\frac{1}{\sqrt{\det m}}\, m$.

The circle mapping procedures are in MobiusonCircle.mpl:

Hcircle(z,r):

This returns the Hermitian matrix $H$ corresponding to a ``circle'' of center $z$ and radius $r$. If $r<0$, it assumes that the object is a straight line passing through $z$ with direction angle $-r$ degrees counterclockwise from the positive real axis.

CircleMap(m,H):

This applies the Möbius transformation corresponding to the matrix $m$ to the circle corresponding to the Hermitian matrix $H$, and returns the Hermitian matrix of the image circle.

Hcenter(H):

This returns the center of the circle corresponding to the Hermitian matrix $H$. If it's really a line, the output is just a point on the line.

Hradius(H):

This returns the radius of the circle corresponding to the Hermitian matrix $H$. If it's really a line, the output is just the negative of the direction angle of the line, in degrees.

PlotHcircle(H):

This plots the circle corresponding to the Hermitian matrix $H$. Standard plotting options may be given as well.

PlotHdisk(H):

This plots in a filled-in disk corresponding to the Hermitian matrix $H$. Standard plotting options may be given as well.