from cxregions import Mobius, Line, Circle, unitcircle
f = Mobius(Line(-1, 1), Circle(0, 1))
f(Line(-1, 1)).isapprox(unitcircle)Detected IPython. Loading juliacall extension. See https://juliapy.github.io/PythonCall.jl/stable/compat/#IPythonTrueMobius
A Möbius transformation (also called bilinear or fractional-linear transformation) is the ratio of two linear polynomials:
\[ f(z)=\frac{az+b}{cz+d}. \]
Among other notable properties, they map circles and lines to other circles and lines.
The package defines a Mobius type that can be constructed in a variety of ways:
| Construction | Description |
|---|---|
Mobius(a, b, c, d) |
Specify the coefficients as in the formula above. |
Mobius(A) |
Specify the coefficients as an array \(A=[a\;b;\;\,c\;d]\). |
Mobius(z, w) |
Construct the unique transformation that maps the three points z[1], z[2], z[3] to w[1], w[2], w[3], respectively. Either vector of points may include Inf. |
Mobius(C1, C2) |
Construct a transformation that maps the Line or Circle C1 to the Line or Circle C2. |
Methods
Suppose f is a value of type Mobius. Then f(z) evaluates the transformation at the number z. In addition, f(C), where C is a Circle or Line, returns the Circle or Line that is the image of C under f. Similarly, f(R), where R is a disk or half-plane, returns the appropriate type of image region. For example,
Two other methods are defined:
inv(f)Construct the inverse transformation.f.compose(g)Construct the composed map, \(z \mapsto f(g(z))\).