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In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.. More formally, let and be open subsets of .A function : is called conformal (or angle-preserving) at a point if it preserves angles between directed curves through , as well as preserving orientation.
In cartography, a conformal map projection is one in which every angle between two curves that cross each other on Earth (a sphere or an ellipsoid) is preserved in the image of the projection; that is, the projection is a conformal map in the mathematical sense. For example, if two roads cross each other at a 39° angle, their images on a map ...
A map cannot achieve that property for any area, no matter how small. It can, however, achieve constant scale along specific lines. Some possible properties are: The scale depends on location, but not on direction. This is equivalent to preservation of angles, the defining characteristic of a conformal map.
A rectangular grid (top) and its image under a conformal map f (bottom). It is seen that f maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°. A conformal map is a function which preserves angles locally. In the most common case the function has a domain and range in the complex plane. More formally, a map,
In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon.Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard Riemann in 1851); the Schwarz–Christoffel formula provides an explicit construction.
A conformal manifold is a pseudo-Riemannian manifold equipped with an equivalence class of metric tensors, in which two metrics g and h are equivalent if and only if =, where λ is a real-valued smooth function defined on the manifold and is called the conformal factor.
A rectangular grid (top) and its image under a conformal map (bottom). It is seen that maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°. In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
Conformal map, in mathematics; Conformal map projection, in cartography; Conformal prediction, in computer science This page was last edited on 24 ...