Spiral asymptotic to a line
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 formula_1 by the equation
formula_2
for an arbitrary choice of the scale factor formula_3
Because of the reciprocal relation between formula_4 and formula_5 it is also called a reciprocal spiral. 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. Hyperbolic spirals can also be generated as the inverse curves of Archimedean spirals, or as the central projections of helixes.
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.
History and applications.
Pierre Varignon first studied the hyperbolic spiral in 1704, 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 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.
Johann Bernoulli 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 spirals, 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.
The staggered start of a 200m race
For a hyperbolic spiral with a circular arc centered at the origin, continuing clockwise for length formula_6 from any of its points, will end on the formula_7-axis. 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.
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 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. In architecture, it has been suggested that hyperbolic spirals are a good match for the design of volutes from columns of the Corinthian order. It also describes the perspective view up the axis of a spiral staircase or other helical structure.
Along with the Archimedean and logarithmic spiral, the hyperbolic spiral has been used in psychological experiments on the perception of rotation.
Constructions.
Coordinate equations.
The hyperbolic spiral has the equation
formula_8
for polar coordinates formula_1 and scale coefficient formula_6. It can be represented in Cartesian coordinates by applying the standard polar-to-Cartesian conversions formula_11 obtaining a parametric equation for the Cartesian coordinates of this curve that treats formula_5 as a parameter rather than as a coordinate:
formula_13
Relaxing the constraint that formula_14 to formula_15 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.
The hyperbolic spiral is a transcendental curve, meaning that it cannot be defined from a polynomial equation of its Cartesian coordinates. However, one can obtain a trigonometric equation in these coordinates by starting with its polar defining equation in the form formula_16 and replacing its variables according to the Cartesian-to-polar conversions formula_17 and giving:
formula_18
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.
Inversion.
Circle inversion through the unit circle is a transformation of the plane that, in polar coordinates, maps the point formula_1 (excluding the origin) to formula_20 and vice versa. The image of an Archimedean spiral formula_21 under this transformation (its inverse curve) is the hyperbolic spiral with 
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. To model this projection mathematically, consider the central projection from point formula_22 onto the image This will map a point formula_23 to the point formula_24.
The image under this projection of the helix with parametric representation
formula_25
is the curve
formula_26
with the polar equation
formula_27
which describes a hyperbolic spiral.
Properties.
Asymptotes.
The hyperbolic spiral approaches the origin as an asymptotic point. Because
formula_28
the curve has an asymptotic line with 
Pitch angle.
From vector calculus in polar coordinates one gets the formula formula_29 for the pitch angle formula_30 between the tangent of any curve and the tangent of its corresponding polar circle. For the hyperbolic spiral formula_31 the pitch angle is
formula_32
Curvature.
The curvature of any curve with polar equation formula_33 is
formula_34
From the equation formula_35 and its derivatives formula_36 and formula_37 one gets the curvature of a hyperbolic spiral, in terms of the radius formula_4 or of the angle formula_5 of any of its points:
formula_40
Arc length.
The length of the arc of a hyperbolic spiral formula_35 between the points formula_42 and formula_43 can be calculated by the integral:
formula_44
Here, the bracket notation means to calculate the formula within the brackets for both formula_45 and formula_46, and to subtract the result for formula_45 from the result for formula_46.
Sector area.
The area of a sector (see diagram above) of a hyperbolic spiral with equation formula_35 is:
formula_50
That is, the area is proportional to the difference in radii, with constant of proportionality formula_51.
References.
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