page_title: Conformal geometry
section_title: Euclidean space
hierarchical_section_title: Conformal geometry / Möbius geometry / Two dimensions / Euclidean space
context_page_description: In mathematics, conformal geometry is the study of the set of angle-preserving transformations on a space.
  In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two dimensions, conformal geometry may refer either to the study of conformal transformations of what are called "flat spaces", or to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics that are defined up to scale. Study of the flat structures is sometimes termed Möbius geometry, and is a type of Klein geometry.
context_section_description: The group of conformal symmetries of the quadratic form
  is the group GL₁(C) = C×, the multiplicative group of the complex numbers.  Its Lie algebra is gl₁(C) = C.
  Consider the (Euclidean) complex plane equipped with the metric
  The infinitesimal conformal symmetries satisfy
  where f satisfies the Cauchy–Riemann equation, and so is holomorphic over its domain.  (See Witt algebra.)
  The conformal isometries of a domain therefore consist of holomorphic self-maps.  In particular, on the conformal compactification – the Riemann sphere – the conformal transformations are given by the Möbius transformations
  where ad − bc is nonzero.
caption: A coordinate grid prior to a Möbius transformation