Editing Hyperbolic spiral (section)

==History and applications==
[[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]]) by reinterpreting the Cartesian coordinates of points on the given curve as polar coordinates of points on the polar curve. Varignon and later [[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]].{{r|maxwell}}

[[Johann Bernoulli]]<ref>Johann Bernoulli should not be confused with his older brother [[Jacob Bernoulli]], who made extensive studies of the [[logarithmic spiral]].</ref> and [[Roger Cotes]] also wrote about this curve, in connection with [[Isaac Newton]]'s discovery that bodies that follow [[conic section]] trajectories must be subject to an [[inverse-square law]], such as the one in [[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]] (whose connection to the inverse-cube law was already observed by Newton) and a hyperbolic spiral. Cotes found a family of spirals, the [[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]] of a circle) is used instead.{{r|haines}}

{{multiple image|total_width=460
|image1=NGC 4622HSTFull.jpg|caption1=The pitch angle of [[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]]s from columns of the [[Corinthian order]].{{r|volute}} It also describes the [[Perspective (graphical)|perspective view]] up the axis of a [[spiral staircase]] or other [[helix|helical]] structure.{{r|hammer}}

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