Konformaal

Konformaal, or conformal in English, is an adjective used in mathematics and theoretical physics to describe structures or transformations that preserve angles locally. The term occurs in several languages, including Finnish and Estonian, where konformaal is the common translation of conformal.

In complex analysis, a map f from a domain in the complex plane to the plane is conformal if it is holomorphic and its derivative is nonzero at every point. Such maps preserve the shape of infinitesimal figures up to scaling, hence preserving angles. Classic examples include Möbius transformations, which are conformal on the extended complex plane. The Riemann mapping theorem asserts that any simply connected proper domain in the complex plane is conformally equivalent to the unit disk.

In differential geometry, a conformal map between Riemannian manifolds preserves angles but not necessarily lengths. If two metrics g and g' on a manifold satisfy g' = e^{2u}g for some smooth function u, they are conformally related; the common class of metrics up to such a scaling is called a conformal class.

Beyond two dimensions, conformal geometry studies structures invariant under angle-preserving transformations, forming a conformal group that acts on the manifold. The concept extends to physics, notably in conformal field theory and string theory, where angle-preserving symmetries constrain physical models and scale invariance plays a key role.

Other notable conformal examples include the stereographic projection, which is conformal from the sphere to the plane, and conformal projections in cartography that preserve local angles.

See also: complex analysis, differential geometry, conformal field theory, Riemann mapping, Möbius transformation.