Moebius_Preserving

Moebius transformation preserving a circle

Moebius Transform (t) preserving a circle (HI).
It is generated by three points X, Y, Z on the circle and their images FX, FY, FZ also on the circle. The circle and these six points on it may be taken arbitrarily. They define then a moebius transform F, which maps the circle onto itself. The [elliptic] transforms are those that preserve the inner region of the circle and define a short of [hyperbolic rotation]. By this we mean that a circle a of the bundle of all circles passing through A and E (fixed points of the transformation) is mapped onto a circle b of the same bundle, so that the angle of a and b is fixed (angle of rotation). The angle of rotation is given by the angle of the parallelogram at its pole.

A, E are the fixed points of the transformation. D maps via F to the point at infinity and F is the image (via F) of the point at infinity. ADEF is called the [characteristic parallelogram] of the Moebius transformation. In the present case (elliptic) the parallelogram is a Rhombus.

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