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The **Möbius transformation** is **f( z ) = ( az + b ) / ( cz + d )** where **z**, **a**,
**b**, **c** and **d** are complex numbers. Drag the points to change the transformation.

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

If the unit circle is the Poincare disk model of the hyperbolic plane, then the Möbius transformation with **c = b̅** and **a = d = 1** is an isometry of the hyperbolic plane corresponding to a translation of the origin by **b**.

Read the book Visual Complex Analysis by Tristan Needham. To see the relationship with the Riemann sphere, see this amazing video.

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