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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.

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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 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.

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