Hyperbolic sector

Hyperbolic sector

A hyperbolic sector is a region of the Cartesian plane {("x","y")} bounded by rays from the origin to two points ("a", 1/"a") and ("b", 1/"b") and by the hyperbola "xy" = 1.

A hyperbolic sector in standard position has "a" = 1 and "b" > 1 .

The area of a hyperbolic sector in standard position is loge "b" . (Proof: Integrate under 1/"x" from 1 to "b", add triangle {(0, 0), (1, 0), (1, 1)}, and subtract triangle {(0, 0), ("b", 0), ("b", 1/"b")} )

When in standard position, a hyperbolic sector corresponds to a positive hyperbolic angle.


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