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Hyperbolic angle
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==Definition== Consider the rectangular hyperbola <math>\textstyle\{(x,\frac 1 x): x>0\}</math>, and (by convention) pay particular attention to the ''branch'' <math>x > 1</math>. First define: * The hyperbolic angle in ''standard position'' is the [[angle]] at <math>(0, 0)</math> between the ray to <math>(1, 1)</math> and the ray to <math>\textstyle(x, \frac 1 x)</math>, where <math>x > 1</math>. * The magnitude of this angle is the [[area]] of the corresponding [[hyperbolic sector]], which turns out to be <math>\operatorname{ln}x</math>. Note that, because of the role played by the [[natural logarithm]]: * Unlike circular angle, the hyperbolic angle is ''unbounded'' (because <math>\operatorname{ln}x</math> is unbounded); this is related to the fact that the [[harmonic series (mathematics)|harmonic series]] is unbounded. * The formula for the magnitude of the angle suggests that, for <math>0 < x < 1</math>, the hyperbolic angle should be negative. This reflects the fact that, as defined, the angle is ''directed''. Finally, extend the definition of ''hyperbolic angle'' to that subtended by any interval on the hyperbola. Suppose <math>a, b, c, d</math> are [[positive real numbers]] such that <math>ab = cd = 1</math> and <math>c > a > 1</math>, so that <math>(a, b)</math> and <math>(c, d)</math> are points on the hyperbola <math>xy=1</math> and determine an interval on it. Then the [[squeeze mapping]] <math>\textstyle f:(x, y)\to(bx, ay)</math> maps the angle <math>\angle\!\left ((a, b), (0,0), (c, d)\right)</math> to the ''standard position'' angle <math>\angle\!\left ((1, 1), (0,0), (bc, ad)\right)</math>. By the result of [[Gregoire de Saint-Vincent]], the hyperbolic sectors determined by these angles have the same area, which is taken to be the magnitude of the angle. This magnitude is <math>\operatorname{ln}{(bc)}=\operatorname{ln}(c/a) =\operatorname{ln}c-\operatorname{ln}a</math>.
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