Hyperbolic coordinates 
on the right-hand region 
of the Cartesian
plane are defined by
 |  |  |
(1)
|
 |  |
(2)
|
where 
and 
.
The inverse transformation is
 |  |  |
(3)
|
 |  |  |
(4)
|
Writing the same point in polar coordinates 
gives
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
where 
.
A three-dimensional extension obtained by adjoining an unchanged coordinate 
therefore converts to cylindrical coordinates 
by the same formulas, with 
unchanged. The coordinate curves 
are branches of rectangular hyperbolas,
while 
are rays from the origin. Analogous
charts, with signs or 
and 
interchanged, cover the other regions
separated by the lines 
. The term is also used for related coordinate
systems on hyperbolic space, so the convention
must be specified.
