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The Gauss map provides a mapping from every point on a curve or a surface to a corresponding point on a unit sphere. In this example, the curvature of a 2D-surface is mapped onto a 1D unit circle. In differential geometry , the Gauss map of a surface is a function that maps each point in the surface to a unit vector that is orthogonal to the ...
Cobweb plot of the Gauss map for = and =. This shows an 8-cycle. This shows an 8-cycle. In mathematics , the Gauss map (also known as Gaussian map [ 1 ] or mouse map ), is a nonlinear iterated map of the reals into a real interval given by the Gaussian function :
In mathematics, a chaotic map is a map (an evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions. Chaotic maps often occur in the study of dynamical systems.
Areas inflate with latitude, becoming so extreme that the map cannot show the poles. 2005 Web Mercator: Cylindrical Compromise Google: Variant of Mercator that ignores Earth's ellipticity for fast calculation, and clips latitudes to ~85.05° for square presentation. De facto standard for Web mapping applications. 1822 Gauss–Krüger = Gauss ...
The projection is known by several names: the (ellipsoidal) transverse Mercator in the US; Gauss conformal or Gauss–Krüger in Europe; or Gauss–Krüger transverse Mercator more generally. Other than just a synonym for the ellipsoidal transverse Mercator map projection, the term Gauss–Krüger may be used in other slightly different ways:
The differential dn of the Gauss map n can be used to define a type of extrinsic curvature, known as the shape operator [55] or Weingarten map. This operator first appeared implicitly in the work of Wilhelm Blaschke and later explicitly in a treatise by Burali-Forti and Burgati. [ 56 ]
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 ...
The open mapping theorem forces the inverse function (defined on the image of ) to be holomorphic. Thus, under this definition, a map is conformal if and only if it is biholomorphic. The two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative.