Editing Hyperbolic spiral (section)

===Coordinate equations===
The hyperbolic spiral has the equation
<math display=block>r=\frac a \varphi ,\quad \varphi > 0</math>
for [[polar coordinates]] <math>(r,\varphi)</math> and [[Scaling (geometry)|scale]] coefficient <math>a</math>. It can be represented in Cartesian coordinates by applying the standard [[List of common coordinate transformations|polar-to-Cartesian conversions]] <math>x=r\cos\varphi</math> {{nowrap|and <math>y=r\sin\varphi</math>,}} obtaining a [[parametric equation]] for the Cartesian coordinates of this curve that treats <math>\varphi</math> as a parameter rather than as a coordinate:{{r|polezhaev}}
<math display=block>x = a \frac{\cos \varphi} \varphi, \qquad y = a \frac{\sin \varphi} \varphi ,\quad \varphi > 0.</math>
Relaxing the constraint that <math>\varphi>0</math> to <math>\varphi\ne0</math> and using the same equations produces a reflected copy of the spiral, and some sources treat these two copies as ''branches'' of a single curve.{{r|drabek|morris}}

{{multiple image
|image1=Hyperbol-spiral-1.svg|
|caption1=Hyperbolic spiral: branch for {{math|''φ'' > 0}}
|image2=Hyperbol-spiral-2.svg|
|caption2=Hyperbolic spiral: both branches
|total_width=600|align=center}}

The hyperbolic spiral is a [[transcendental curve]], meaning that it cannot be defined from a [[polynomial equation]] of its Cartesian coordinates.{{r|polezhaev}} However, one can obtain a [[trigonometric equation]] in these coordinates by starting with its polar defining equation in the form <math>r\varphi=a</math> and replacing its variables according to the Cartesian-to-polar conversions <math>\varphi=\tan^{-1}\tfrac{y}{x}</math> and {{nowrap|<math display=inline>r=\sqrt{x^2+y^2}</math>,}} giving:{{r|shikin}}
<math display=block>\sqrt{x^2+y^2}\tan^{-1}\frac{y}{x}=a.</math>

It is also possible to use the polar equation to define a spiral curve in the [[hyperbolic plane]], but this is different in some important respects from the usual form of the hyperbolic spiral in the Euclidean plane. In particular, the corresponding curve in the hyperbolic plane does not have an asymptotic line.{{r|dunham}}