Hyperbolic coordinates information

In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane

\{(x,y)\ :\ x>0,\ y>0\ \}=Q

.

Hyperbolic coordinates take values in the hyperbolic plane defined as:

HP=\{(u,v):u\in \mathbb {R} ,v>0\}

.

These coordinates in HP are useful for studying logarithmic comparisons of direct proportion in Q and measuring deviations from direct proportion.

For $(x,y)$ in $Q$ take

u=\ln {\sqrt {\frac {x}{y}}}

and

v={\sqrt {xy}}

.

The parameter u is the hyperbolic angle to (x, y) and v is the geometric mean of x and y.

The inverse mapping is

x=ve^{u},\quad y=ve^{-u}

.

The function $Q\rightarrow HP$ is a continuous mapping, but not an analytic function.

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