Editing Hyperbolic coordinates (section)

{{short description|Geometric mean and hyperbolic angle as coordinates in quadrant I}}
[[Image:Hyperbolic coordinates.svg|thumb|200px|right|Hyperbolic coordinates plotted on the Euclidean plane: all points on the same blue ray share the same coordinate value ''u'', and all points on the same red hyperbola share the same coordinate value ''v''.]]

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

:<math>\{(x, y) \ :\  x > 0,\ y > 0\ \} = Q</math>.

Hyperbolic coordinates take values in the [[hyperbolic plane]] defined as:

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

These coordinates in ''HP'' are useful for studying [[logarithmic scale|logarithmic]] comparisons of [[direct proportion]] in ''Q'' and measuring deviations from direct proportion.

For <math>(x,y)</math> in <math>Q</math> take

:<math>u = \ln \sqrt{\frac{x}{y}} </math>

and

:<math>v = \sqrt{xy}</math>.

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

The inverse mapping is

:<math>x = v e^u ,\quad y = v e^{-u}</math>.

The function <math>Q \rarr HP</math> is a [[continuous mapping]], but not an [[analytic function]].