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Hyperbolic groups have a solvable word problem. They are biautomatic and automatic. [9] Indeed, they are strongly geodesically automatic, that is, there is an automatic structure on the group, where the language accepted by the word acceptor is the set of all geodesic words.
Some connector styles may combine pin and socket connection types in a single unit, referred to as a hermaphroditic connector. [6]: 56 These connectors includes mating with both male and female aspects, involving complementary paired identical parts each containing both protrusions and indentations. These mating surfaces are mounted into ...
One has a hyperboloid of revolution if and only if =. Otherwise, the axes are uniquely defined (up to the exchange of the x-axis and the y-axis). There are two kinds of hyperboloids. In the first case (+1 in the right-hand side of the equation): a one-sheet hyperboloid, also called a hyperbolic hyperboloid.
This page is a list of hyperboloid structures. These were first applied in architecture by Russian engineer Vladimir Shukhov (1853–1939). Shukhov built his first example as a water tower ( hyperbolic shell ) for the 1896 All-Russian Exposition .
Hyperbolic motions can also be described on the hyperboloid model of hyperbolic geometry. [ 1 ] This article exhibits these examples of the use of hyperbolic motions: the extension of the metric d ( a , b ) = | log ( b / a ) | {\displaystyle d(a,b)=\vert \log(b/a)\vert } to the half-plane and the unit disk .
Hyperboloid structure; Hyperbolic set This page was last edited on 28 December 2019, at 19:43 (UTC). Text is available under the Creative Commons ...
The Weierstrass coordinates of a point are the Cartesian coordinates of the point when the point is mapped in the hyperboloid model of the hyperbolic plane, the x-axis is mapped to the (half) hyperbola ( , , +) and the origin is mapped to the point (0,0,1). [1]
The concept has its roots in a notion expressed by F. L. Wachter in 1816 in a letter to his teacher Gauss.Noting that in Euclidean geometry the limit of a sphere as its radius tends to infinity is a plane, Wachter affirmed that even if the fifth postulate were false, there would nevertheless be a geometry on the surface identical with that of the ordinary plane. [1]