Conformal map

A conformal map is a function between geometric domains that preserves angles locally. In the complex plane, holomorphic functions with nonzero derivative are conformal.

Holomorphic function: A complex-valued function that is complex differentiable at every point of an open domain. Holomorphicity supplies the local linear approximation that makes planar conformal maps angle-preserving.
Complex plane: The plane whose points are complex numbers, with coordinates given by real and imaginary parts. Planar conformal maps are commonly expressed as complex functions on this space.
Riemann mapping theorem: The theorem that every simply connected proper open subset of the complex plane is conformally equivalent to the unit disk. It guarantees a disk model for a broad class of planar domains.
Antiholomorphic function: A complex function locally expressible as a holomorphic function of the complex conjugate variable. It preserves angles but reverses orientation, unlike holomorphic conformal maps.
Conformal equivalence: An equivalence relation between domains when a bijective conformal map exists between them. It groups geometrically different-looking domains that share the same conformal structure.
Complex derivative: The limit defining a complex function’s rate of change, when that limit exists independently of direction. A nonzero derivative acts locally as a rotation and scaling, preserving angles.
Domain (mathematics): A connected open subset of a mathematical space on which a function is defined. Conformal maps act between domains, and their shapes constrain possible mappings.
Schwarz–Christoffel mapping: A conformal map from a half-plane or disk to a polygonal region, given by an integral formula. It provides explicit maps for polygonal geometries used in applied calculations.
Quasiconformal mapping: A homeomorphism whose distortion of infinitesimal circles into ellipses is bounded. It allows controlled angle distortion where exact conformality is too restrictive.
Conformal radius: A scale associated with a point in a simply connected planar domain through a normalized conformal map to the unit disk. It measures how the domain’s conformal geometry differs around a chosen point.

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