spheres.relativity.mobius

spheres.relativity.mobius(abcd)[source]

Given parameters \(\begin{pmatrix} a & b \\ b & c \end{pmatrix}\), arranged in a 2x2 matrix, returns a function which implements the corresponding Möbius transformation

\[f(z) = \frac{az+b}{cz+d}\]

which acts on the extended complex plane. Note that if \(c \neq 0\), we have:

\[ \begin{align}\begin{aligned}f(-\frac{d}{c}) = \infty\\f(\infty) = \frac{a}{c}\end{aligned}\end{align} \]

And if \(c = 0\), we have:

\[f(\infty) = \infty\]
Parameters

abcd (np.ndarray or qt.Qobj) – 2x2 matrix representing Möbius parameters.

Returns

mobius – A function which takes an extended complex coordinate as input, and returns an extended complex coordinate as output.

Return type

func

Raises

Exception – If \(ad = bc\), then \(f(z) = \frac{a}{c}\), a constant function, which doesn’t qualify as a Möbius transformation.