View source for Hyperbolic spiral

{{Short description|Spiral asymptotic to a line}}
[[File:Gustavino Spiral.jpg|thumb|A spiral staircase in the Cathedral of St. John the Divine. Several [[Helix|helical curves]] in the staircase project to hyperbolic spirals in its photograph.]]

A '''hyperbolic spiral''' is a type of [[Spiral|spiral]] with a [[Pitch angle of a spiral|pitch angle]] that increases with distance from its center, unlike the constant angles of [[Logarithmic spiral|logarithmic spiral]]s or decreasing angles of [[Archimedean spiral]]s. As this curve widens, it approaches an asymptotic line. It can be found in the view up a [[Engineering:Spiral staircase|spiral staircase]] and the starting arrangement of certain footraces, and is used to model [[Spiral galaxy|spiral galaxies]] and [[Volute|architectural volutes]].

As a [[Plane curve|plane curve]], a hyperbolic spiral can be described in polar coordinates <math>(r,\varphi)</math> by the equation
<math display=block>r=\frac{a}{\varphi},</math>
for an arbitrary choice of the [[Scale factor|scale factor]] <math>a.</math>

Because of the [[Multiplicative inverse|reciprocal]] relation between <math>r</math> and <math>\varphi</math> it is also called a '''reciprocal spiral'''.{{r|waud}} The same relation between Cartesian coordinates would describe a [[Hyperbola|hyperbola]], and the hyperbolic spiral was first discovered by applying the equation of a hyperbola to polar coordinates.{{r|maxwell}} Hyperbolic spirals can also be generated as the [[Inverse curve|inverse curve]]s of Archimedean spirals,{{r|bowser|drabek}} or as the central projections of [[Helix|helix]]es.{{r|hammer}}

Hyperbolic spirals are patterns in the [[Euclidean plane]], and should not be confused with other kinds of spirals drawn in the hyperbolic plane. In cases where the name of these spirals might be ambiguous, their alternative name, reciprocal spirals, can be used instead.{{r|dunham}}

==History and applications==
[[Biography:Pierre Varignon|Pierre Varignon]] first studied the hyperbolic spiral in 1704,{{r|varignon|mactutor}} as an example of the ''polar curve'' obtained from another curve (in this case the [[Hyperbola|hyperbola]]) by reinterpreting the Cartesian coordinates of points on the given curve as polar coordinates of points on the polar curve. Varignon and later [[Biography:James Clerk Maxwell|James Clerk Maxwell]] were interested in the [[Roulette (curve)|roulettes]] obtained by tracing a point on this curve as it rolls along another curve; for instance, when a hyperbolic spiral rolls along a straight line, its center traces out a [[Tractrix|tractrix]].{{r|maxwell}}

[[Biography:Johann Bernoulli|Johann Bernoulli]]<ref>Johann Bernoulli should not be confused with his older brother [[Biography:Jacob Bernoulli|Jacob Bernoulli]], who made extensive studies of the [[Logarithmic spiral|logarithmic spiral]].</ref> and [[Biography:Roger Cotes|Roger Cotes]] also wrote about this curve, in connection with [[Biography:Isaac Newton|Isaac Newton]]'s discovery that bodies that follow [[Conic section|conic section]] trajectories must be subject to an [[Inverse-square law|inverse-square law]], such as the one in [[Physics:Newton's law of universal gravitation|Newton's law of universal gravitation]]. Newton asserted that the reverse was true: that conic sections were the only trajectories possible under an inverse-square law. Bernoulli criticized this step, observing that in the case of an inverse-cube law, multiple trajectories were possible, including both a [[Logarithmic spiral|logarithmic spiral]] (whose connection to the inverse-cube law was already observed by Newton) and a hyperbolic spiral. Cotes found a family of spirals, the [[Physics:Cotes's spiral|Cotes's spiral]]s, including the logarithmic and hyperbolic spirals, that all required an inverse-cube law. By 1720, Newton had resolved the controversy by proving that inverse-square laws always produce conic-section trajectories.{{sfnp|Hammer|2016|pp=119–120}}{{r|guiccardini|bernoulli|cotes}}

{{CSS image crop|Image=Men 200 m French Athletics Championships 2013 t161532.jpg|bSize=360|cWidth=360|cHeight=180|oTop=60|Description=The staggered start of a 200m race}}
For a hyperbolic spiral with {{nowrap|equation <math>r=\tfrac{a}{\varphi}</math>,}} a circular arc centered at the origin, continuing clockwise for {{nowrap|length <math>a</math>}} from any of its points, will end on the {{nowrap|<math>x</math>-axis.{{r|bowser}}}} Because of this equal-length property, the starting marks of 200m and 400m footraces are placed in staggered positions along a hyperbolic spiral. This ensures that the runners, restricted to their concentric lanes, all have equal-length paths to the finish line. For longer races where runners move to the inside lane after the start, a different spiral (the [[Involute|involute]] of a circle) is used instead.{{r|haines}}

{{multiple image|total_width=460
|image1=NGC 4622HSTFull.jpg|caption1=The pitch angle of [[Astronomy:NGC 4622|NGC 4622]] increases with distance{{r|ngc4622}}
|image2=Corinthian capital, AM of Epidauros, 202545.jpg|caption2=[[Volute]]s on a Corinthian order capital in the Archaeological Museum of Epidaurus}}
The increasing pitch angle of the hyperbolic spiral, as a function of distance from its center, has led to the use of these spirals to model the shapes of some [[Spiral galaxy|spiral galaxies]], which in some cases have a similarly increasing pitch angle. However, this model does not provide a good fit to the shapes of all spiral galaxies.{{r|galaxy1|galaxy2}} In architecture, it has been suggested that hyperbolic spirals are a good match for the design of [[Volute|volute]]s from columns of the Corinthian order.{{r|volute}} It also describes the [[Perspective (graphical)|perspective view]] up the axis of a [[Engineering:Spiral staircase|spiral staircase]] or other [[Helix|helical]] structure.{{r|hammer}}

Along with the Archimedean and logarithmic spiral, the hyperbolic spiral has been used in [[Physics:Psychophysics#Experimentation|psychological experiments]] on the perception of rotation.{{r|scott-noland}}

==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 [[Physics: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|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|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|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|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 [[Engineering:Spiral staircase|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}}

== Properties ==
=== Asymptotes ===
The hyperbolic spiral approaches the origin as an asymptotic point.{{r|shikin}} Because
<math display=block>\lim_{\varphi\to 0}x = a\lim_{\varphi\to 0} \frac{\cos \varphi} \varphi =\infty,\qquad
\lim_{\varphi\to 0}y = a\lim_{\varphi\to 0} \frac{\sin \varphi} \varphi = a,</math>
the curve has an asymptotic line with {{nowrap|equation <math>y=a</math>.{{r|polezhaev}}}}

=== Pitch angle ===
[[File:Sektor-steigung-pk-def.svg|thumb|Definition of sector (light blue) and pitch angle {{mvar|α}}]]
From [[Polar coordinate system#Vector calculus|vector calculus in polar coordinates]] one gets the formula <math>\tan\alpha=\tfrac{r'}{r}</math> for the [[Pitch angle of a spiral|pitch angle]] <math>\alpha</math> between the tangent of any curve and the tangent of its corresponding polar circle.{{r|kepr}} For the hyperbolic spiral <math>r=\tfrac{a}{\varphi}</math> the pitch angle is{{r|scott-noland}}
<math display=block>\alpha=\tan^{-1}\left(-\frac{1}{\varphi}\right).</math>

=== Curvature ===
The [[Curvature|curvature]] of any curve with polar equation <math>r=r(\varphi)</math> is{{r|curvature}}
<math display=block>\kappa = \frac{r^2 + 2(r')^2 - r\, r''}{\left(r^2+(r')^2\right)^{3/2}} .</math>
From the equation <math>r=a/\varphi</math> and its derivatives <math>r'=-a/\varphi^2</math> and <math>r''=2a/\varphi^3</math> one gets the curvature of a hyperbolic spiral, in terms of the radius <math>r</math> or of the angle <math>\varphi</math> of any of its points:{{r|ganguli}}
<math display=block>\kappa = \frac{\varphi^4}{a \left(\varphi^2 + 1\right)^{3/2}}
= \frac{a^3}{r(a^2+r^2)^{3/2}}.
</math>

=== Arc length ===
The length of the arc of a hyperbolic spiral <math>r=a/\varphi</math> between the points <math>(r(\varphi_1),\varphi_1)</math> and <math>(r(\varphi_2),\varphi_2)</math> can be calculated by the integral:{{r|polezhaev}}
<math display=block>\begin{align}
L&=a \int_{\varphi_1}^{\varphi_2}\frac{\sqrt{1+\varphi^2}}{\varphi^2}\,d\varphi \\
&= a\left[-\frac{\sqrt{1+\varphi^2}}{\varphi}+\ln\left(\varphi+\sqrt{1+\varphi^2}\right)\right]_{\varphi_1}^{\varphi_2} .
\end{align}</math>
Here, the bracket notation means to calculate the formula within the brackets for both <math>\varphi_1</math> and <math>\varphi_2</math>, and to subtract the result for <math>\varphi_1</math> from the result for <math>\varphi_2</math>.

=== Sector area ===
The area of a sector (see diagram above) of a hyperbolic spiral with equation <math>r=a/\varphi</math> is:{{r|polezhaev}}
<math display=block>\begin{align}
A&=\frac12\int_{\varphi_1}^{\varphi_2} r(\varphi)^2\, d\varphi\\
&=\frac{a}{2}\bigl(r(\varphi_1)-r(\varphi_2)\bigr) .
\end{align}</math>
That is, the area is proportional to the difference in radii, with constant of proportionality {{nowrap|<math>a/2</math>.{{r|cotes|polezhaev}}}}

==References==
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}}

==External links==
* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Hyperbolic_spiral Online exploration using JSXGraph (JavaScript)]

{{Spirals}}

[[Category:Spirals]]

pt:Espiral logarítmica

{{Sourceattribution|Hyperbolic spiral}}