Hyperbolic spiral
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
[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.
[5] Hyperbolic spirals are patterns in the Euclidean plane, and should not be confused with other kinds of spirals drawn in the hyperbolic plane.
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.
[2] Johann Bernoulli[9] 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.
By 1720, Newton had resolved the controversy by proving that inverse-square laws always produce conic-section trajectories.
, a circular arc centered at the origin, continuing clockwise for length
[3] 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.
[14] 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.
[16][17] In architecture, it has been suggested that hyperbolic spirals are a good match for the design of volutes from columns of the Corinthian order.
[18] It also describes the perspective view up the axis of a spiral staircase or other helical structure.
It can be represented in Cartesian coordinates by applying the standard polar-to-Cartesian conversions
, obtaining a parametric equation for the Cartesian coordinates of this curve that treats
[4][21] The hyperbolic spiral is a transcendental curve, meaning that it cannot be defined from a polynomial equation of its Cartesian coordinates.
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.
under this transformation (its inverse curve) is the hyperbolic spiral with equation
[8] 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.
[24] The image under this projection of the helix with parametric representation
[24] The hyperbolic spiral approaches the origin as an asymptotic point.
[20] From vector calculus in polar coordinates one gets the formula
one gets the curvature of a hyperbolic spiral, in terms of the radius
The area of a sector (see diagram above) of a hyperbolic spiral with equation
