Editing Hyperbolic spiral (section)

==Constructions==
===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}}

=== Inversion ===
[[File:Hyperbol-spiral-inv-arch-spir.svg|thumb|Hyperbolic spiral (blue) as image of an Archimedean spiral (green) by inversion through a circle (red)]]
[[Circle inversion]] through the [[unit circle]] is a transformation of the plane that, in polar coordinates, maps the point <math>(r,\varphi)</math> (excluding the origin) to <math>(\tfrac1r,\varphi)</math> and vice versa.{{r|indra}} The [[Image (mathematics)|image]] of an [[Archimedean spiral]] <math>r=\tfrac{\varphi}{a}</math> under this transformation (its [[inverse curve]]) is the hyperbolic spiral with {{nowrap|equation <math>r=\tfrac{a}{\varphi}</math>.{{r|mactutor}}}}

=== Central projection of a helix ===
[[File:Schraublinie-hyp-spirale.svg|thumb|upright=0.8|Hyperbolic spiral as central projection of a helix]]
The [[central projection]] of a helix onto a plane perpendicular to the axis of the helix describes the view that one would see of the guardrail of a [[spiral staircase]], looking up or down from a viewpoint on the axis of the staircase.{{r|hammer}} To model this projection mathematically, consider the central projection from point <math>(0,0,d)</math> onto the image {{nowrap|plane <math>z=0</math>.}} This will map a point <math>(x,y,z)</math> to  the {{nowrap|point <math>\tfrac{d}{d-z}(x,y)</math>.{{r|loria-roever}}}}

The image under this projection of the helix with parametric representation
<math display=block>(r\cos t, r\sin t, ct),\quad c\neq 0,</math>
is the curve
<math display=block>\frac{dr}{d-ct}(\cos t,\sin t)</math>
with the polar equation
<math display=block>\rho=\frac{dr}{d-ct},</math>
which describes a hyperbolic spiral.{{r|loria-roever}}