page_title: Möbius transformation
section_title: Stereographic projection
hierarchical_section_title: Möbius transformation / Geometric interpretation of the characteristic constant / Stereographic projection
caption_attribution_description: English: Mobius transformation, showing stereographic projection between plane and sphere. Generated with POVRAY. Povray file created with a java program.
context_page_description: In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
  of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.
  Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection to the plane.
  These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle.
  The Möbius transformations are the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL. Together with its subgroups, it has numerous applications in mathematics and physics.
  Möbius transformations are named in honor of August Ferdinand Möbius; they are also variously named homographies, homographic transformations, linear fractional transformations, bilinear transformations, or fractional linear transformations.
context_section_description: These images show Möbius transformations stereographically projected onto the Riemann sphere. Note in particular that when projected onto a sphere, the special case of a fixed point at infinity looks no different from having the fixed points in an arbitrary location.
caption: Loxodromic
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