Math
Möbius transformations
Transform complex grids and circles by translations, rotations, reciprocals and fractional linear maps, with a Riemann-sphere view.
Try this
Click the left plane to move the red probe. The right grid is its Möbius image; poles pass through infinity. The sphere view uses inverse stereographic projection. General Möbius maps deform the sphere rather than act as rigid rotations.
Equations used in the module
Equations w = (az+b)/(cz+d), ad−bc ≠ 0 z = (dw−b)/(a−cw) Generalized circles (circles or lines) map to generalized circles Sphere: (2Re z, 2Im z, |z|²−1)/(1+|z|²); ∞ ↔ north pole
