Möbius Transformation
Mathematically, perhaps the most interesting transformation available in Cinderella is the Möbius transformation. This is a transformation with the property that circles and lines are again mapped to circles and lines. Any
rotation,
translation, or
similarity can be considered a special case of a Möbius transformation, whereas neither an
affine transformation nor a
projective transformation is a Möbius transformation.
A Möbius transformation is defined by three preimage/image pairs of points. Defining a Möbius transformation in Cinderella is analogous to the creation of a
similarity, though with six required points instead of four.
The following two pictures demonstrate how a circle is mapped under a Möbius transformation. Since the Möbius transformation in the picture maps
A → D,
B → E,
C → F, a circle through
A, B, C is mapped to a circle through
D, E, F.
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Defining a Möbius transformation | |
Mapping a circle |
If one iterates a Möbius transformation, the sequence of mapped pictures forms a mathematically and aesthetically very interesting pattern that can be considered a generalization of a logarithmic spiral. The following picture shows the image of an iterated Möbius transformation applied to a single circle and its center.
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Iterating a Möbius transformation |
A Möbius transformation usually has two distinct fixed points. In the above drawing these fixed points have been constructed explicitly by making sure that the transformation maps
A to itself and
B to itself.
See also