Hyperbolic coordinates

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

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

Hyperbolic coordinates take values in the hyperbolic plane defined as:

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

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

For ${\displaystyle (x,y)}$ in ${\displaystyle Q}$ take

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

and

${\displaystyle v={\sqrt {xy}}}$.

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

The inverse mapping is

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

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