Möbius

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.

Alternate spelling

For convenience of typing on some keyboards, Mobius is defined as an alias for Möbius.

The package defines a Möbius type that can be constructed in a variety of ways:

Construction	Description
Möbius(a, b, c, d)	Specify the coefficients as in the formula above.
Möbius(A)	Specify the coefficients as the matrix $A=[ab;cd]$.
Möbius(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.
Möbius(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 Möbius. 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 an AbstractDisk or AbstractHalfplane, returns the appropriate type of image region. For example,

julia> f = Möbius(Line(-1, 1), Circle(0, 1))
Möbius transformation:

   (1.0 + 1.0im) z + (3.6666666666666665 - 1.6666666666666665im)
   ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
   (1.0 + 1.0im) z + (-1.6666666666666665 + 3.6666666666666665im)

julia> f(upperhalfplane)
Disk interior to:
   Circle(0.0+0.0im,1.0,ccw)

julia> isapprox(ans, unitdisk)
true

Two other methods are defined:

• inv(f) Construct the inverse transformation.

• f∘g (type "\circ" followed by Tab key) Construct the composed map, $z\mapsto f(g(z))$.