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A diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one. For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map.
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.
The straight-line distance between the central point on the map to any other point is the same as the straight-line 3D distance through the globe between the two points. c. 150 BC: Stereographic: Azimuthal Conformal Hipparchos* Map is infinite in extent with outer hemisphere inflating severely, so it is often used as two hemispheres.
Equivalent projections are widely used for thematic maps showing scenario distribution such as population, farmland distribution, forested areas, and so forth, because an equal-area map does not change apparent density of the phenomenon being mapped. By Gauss's Theorema Egregium, an equal-area projection cannot be conformal. This implies that ...
A conformal manifold is a Riemannian manifold (or 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.
If s = 0, this is simply the identity map. A homeomorphism is 1-quasiconformal if and only if it is conformal. Hence the identity map is always 1-quasiconformal. If f : D → D′ is K-quasiconformal and g : D′ → D′′ is K′-quasiconformal, then g o f is KK′-quasiconformal.
In the case of maps f : U → C defined on an open subset U of the complex plane C, some authors (e.g., Freitag 2009, Definition IV.4.1) define a conformal map to be an injective map with nonzero derivative i.e., f’(z)≠ 0 for every z in U. According to this definition, a map f : U → C is conformal if and only if f: U → f(U) is ...