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To simplify the transformation, set c to zero (move it to the origin). Now a and b between them give a similarity transformation: a controls the scale and rotation while b controls the translation.

The identity transformation is given when b=c=0 and a=d.

A normalized Möbius transformation is one where ad - bc = 1.

A unitary Möbius transformation has c = -b̅ and d = a̅ (where z̅ is the complex conjugate of z).

A normalized Möbius transformation is loxodromic if a+d is not on the real line. Set Non-loxodromic: ON to enforce that a+d is real, and Normalize: ON to ensure that the transformation is normalized. Then if |a+d| < 2 the Möbius transformation is elliptic. If |a+b|=2 then it is parabolic. Otherwise, with |a+d| > 2, it is hyperbolic.

A Möbius transformation with |a|² - |b|² = 1, c = b̅, d = a̅ maps the unit circle to itself. Set Unit circle group: ON to enforce this contraint. When a is real this gives a translation of the hyperbolic plane in the Poincaré disk model.

With Drawing mode: ON, draw on the screen. Multiple copies of the drawing (blue) under iterations of the Möbius transformation (green) and its inverse (red) are shown.

Read the book Visual Complex Analysis by Tristan Needham. To see the relationship with the Riemann sphere, see this video. To explore the possibilities of iterating Möbius transforms, see the book Indra's Pearls, by David Mumford, Caroline Series and David Wright.

Source code: https://github.com/timhutton/mobius-transforms