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hyperbolic sector : ウィキペディア英語版
hyperbolic sector

A hyperbolic sector is a region of the Cartesian plane bounded by rays from the origin to two points (''a'', 1/''a'') and (''b'', 1/''b'') and by the rectangular hyperbola ''xy'' = 1 (or the corresponding region when this hyperbola is rescaled and its orientation is altered by a rotation leaving the center at the origin, as with the unit hyperbola).
A hyperbolic sector in standard position has ''a'' = 1 and ''b'' > 1 .
Hyperbolic sectors are the basis for the hyperbolic functions.
==Area==
The area of a hyperbolic sector in standard position is ln ''b'' .
Proof: Integrate under 1/''x'' from 1 to ''b'', add triangle , and subtract triangle .
〔V.G. Ashkinuse & Isaak Yaglom (1962) ''Ideas and Methods of Affine and Projective Geometry'' (in Russian), page 151, Ministry of Education, Moscow〕
When in standard position, a hyperbolic sector corresponds to a positive hyperbolic angle at the origin, with the measure of the latter being defined as the area of the former.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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