Editing Hyperbolic spiral (section)

{{Short description|Spiral asymptotic to a line}}
{{good article}}
{{Use dmy dates|date=January 2024}}
[[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]] with a [[Pitch angle of a spiral|pitch angle]] that increases with distance from its center, unlike the constant angles of [[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 [[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]], 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]] <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]], 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]]s of Archimedean spirals,{{r|bowser|drabek}} or as the [[central projection]]s of [[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}}