In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. Hyperbolic angle is a shuffled form of natural logarithm as they both are defined as an area against hyperbola xy = 1, and they both are preserved by squeeze mappings since those mappings preserve area. The hyperbola xy = 1 is rectangular with semi-major axis 2 {\displaystyle {\sqrt {2}}} , analogous to the circular angle equaling the area of a circular sector in a circle with radius 2 {\displaystyle {\sqrt {2}}} . Hyperbolic angle is used as the independent variable for the hyperbolic functions sinh, cosh, and tanh, because these functions may be premised on hyperbolic analogies to the corresponding circular (trigonometric) functions by regarding a hyperbolic angle as defining a hyperbolic triangle. The hyperbolic angle parametrizes the unit hyperbola, which has hyperbolic functions as coordinates.
Definition
Consider the rectangular hyperbola { ( x , 1 x ) : x > 0 } {\displaystyle \textstyle \{(x,{\frac {1}{x}}):x>0\}} , and (by convention) pay particular attention to the part with x > 1 {\displaystyle x>1} . First define:
The hyperbolic angle in standard position is the angle at ( 0 , 0 ) {\displaystyle (0,0)} between the ray to ( 1 , 1 ) {\displaystyle (1,1)} and the ray to ( x , 1 x ) {\displaystyle \textstyle (x,{\frac {1}{x}})} , where x > 1 {\displaystyle x>1} . The magnitude of this angle is the signed area of the corresponding hyperbolic sector, which turns out to be ln x {\displaystyle \operatorname {ln} x} . Note that by properties of natural logarithm:
Unlike circular angle, the hyperbolic angle is unbounded (because ln x {\displaystyle \operatorname {ln} x} is unbounded); this is related to the fact that the harmonic series is unbounded. The formula for the magnitude of the angle suggests that, for 0 < x < 1 {\displaystyle 0<x<1} , 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 a , b , c , d {\displaystyle a,b,c,d} are positive real numbers such that a b = c d = 1 {\displaystyle ab=cd=1} and c > a > 1 {\displaystyle c>a>1} , so that ( a , b ) {\displaystyle (a,b)} and ( c , d ) {\displaystyle (c,d)} are points on the hyperbola x y = 1 {\displaystyle xy=1} and determine an interval on it. Then the squeeze mapping f : ( x , y ) → ( b x , a y ) {\displaystyle \textstyle f:(x,y)\to (bx,ay)} maps the angle ∠ ( ( a , b ) , ( 0 , 0 ) , ( c , d ) ) {\displaystyle \angle \!\left((a,b),(0,0),(c,d)\right)} to the standard position angle ∠ ( ( 1 , 1 ) , ( 0 , 0 ) , ( b c , a d ) ) {\displaystyle \angle \!\left((1,1),(0,0),(bc,ad)\right)} . 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 ln ( b c ) = ln ( c / a ) = ln c − ln a {\displaystyle \operatorname {ln} {(bc)}=\operatorname {ln} (c/a)=\operatorname {ln} c-\operatorname {ln} a} .
Comparison with circular angle
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