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Squeeze mapping
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===Bridge to transcendentals=== The area-preserving property of squeeze mapping has an application in setting the foundation of the [[transcendental function]]s [[natural logarithm]] and its inverse the [[exponential function]]: '''Definition:''' Sector(''a,b'') is the [[hyperbolic sector]] obtained with central rays to (''a'', 1/''a'') and (''b'', 1/''b''). '''Lemma:''' If ''bc'' = ''ad'', then there is a squeeze mapping that moves the sector(''a,b'') to sector(''c,d''). Proof: Take parameter ''r'' = ''c''/''a'' so that (''u,v'') = (''rx'', ''y''/''r'') takes (''a'', 1/''a'') to (''c'', 1/''c'') and (''b'', 1/''b'') to (''d'', 1/''d''). '''Theorem''' ([[Gregoire de Saint-Vincent]] 1647) If ''bc'' = ''ad'', then the quadrature of the hyperbola ''xy'' = 1 against the asymptote has equal areas between ''a'' and ''b'' compared to between ''c'' and ''d''. Proof: An argument adding and subtracting triangles of area {{frac|1|2}}, one triangle being {(0,0), (0,1), (1,1)}, shows the hyperbolic sector area is equal to the area along the asymptote. The theorem then follows from the lemma. '''Theorem''' ([[Alphonse Antonio de Sarasa]] 1649) As area measured against the asymptote increases in arithmetic progression, the projections upon the asymptote increase in geometric sequence. Thus the areas form ''logarithms'' of the asymptote index. For instance, for a standard position angle which runs from (1, 1) to (''x'', 1/''x''), one may ask "When is the hyperbolic angle equal to one?" The answer is the [[transcendental number]] x = [[e (mathematical constant)|e]]. A squeeze with ''r'' = e moves the unit angle to one between (''e'', 1/''e'') and (''ee'', 1/''ee'') which subtends a sector also of area one. The [[geometric progression]] : ''e'', ''e''<sup>2</sup>, ''e''<sup>3</sup>, ..., ''e''<sup>''n''</sup>, ... corresponds to the asymptotic index achieved with each sum of areas : 1,2,3, ..., ''n'',... which is a proto-typical [[arithmetic progression]] ''A'' + ''nd'' where ''A'' = 0 and ''d'' = 1 .
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