# Hyperbolic spiral

## Spiral asymptotic to a line / From Wikipedia, the free encyclopedia

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A **hyperbolic spiral** is a type of spiral with a pitch angle that increases with distance from its center, unlike the constant angles of logarithmic spirals or decreasing angles of Archimedean spirals. 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 galaxies and architectural volutes.

As a plane curve, a hyperbolic spiral can be described in polar coordinates $(r,\varphi )$ by the equation $r={\frac {a}{\varphi }},$ for an arbitrary choice of the scale factor $a.$

Because of the reciprocal relation between $r$ and $\varphi$ it is also called a **reciprocal spiral**.^{[1]} 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.^{[2]} Hyperbolic spirals can also be generated as the inverse curves of Archimedean spirals,^{[3]}^{[4]} or as the central projections of helixes.^{[5]}

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.^{[6]}