Hyperbolic spiral

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Hyperbolic spiral: branch for
{\displaystyle \varphi >0} Hyperbol-spiral-1.svg
Hyperbolic spiral: branch for
Hyperbolic spiral: both branches Hyperbol-spiral-2.svg
Hyperbolic spiral: both branches

A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation


of a hyperbola. Because it can be generated by a circle inversion of an Archimedean spiral, it is called reciproke spiral, too. [1] [2]

Pierre Varignon first studied the curve in 1704. [2] Later Johann Bernoulli and Roger Cotes worked on the curve as well.

In cartesian coordinates

the hyperbolic spiral with the polar equation

can be represented in cartesian coordinates by

The hyperbola has in the --plane the coordinate axes as asymptotes. The hyperbolic spiral (in the --plane) approaches for the origin as asymptotic point. For the curve has an asymptotic line (see next section).

From the polar equation and one gets a representation by an equation:

Geometric properties


Because of

the curve has an

Polar slope

Definition of sector (light blue) and polar slope angle
{\displaystyle \alpha } Sektor-steigung-pk-def.svg
Definition of sector (light blue) and polar slope angle

From vector calculus in polar coordinates one gets the formula for the polar slope and its angle between the tangent of a curve and the tangent of the corresponding polar circle.

For the hyperbolic spiral the polar slope is


The curvature of a curve with polar equation is

From the equation and the derivatives and one gets the curvature of a hyperbolic spiral:

Arc length

The length of the arc of a hyperbolic spiral between can be calculated by the integral:

Sector area

The area of a sector (see diagram above) of a hyperbolic spiral with equation is:


Hyperbolic spiral (blue) as image of an Archimedean spiral (green) with an circle inversion Hyperbol-spiral-inv-arch-spir.svg
Hyperbolic spiral (blue) as image of an Archimedean spiral (green) with an circle inversion

The inversion at the unit circle has in polar coordinates the simple description: .

For the two curves intersect at a fixpoint on the unit circle.

The osculating circle of the Archimedean spiral at the origin has radius (see Archimedean spiral) and the center . The image of this circle is the line (see circle inversion). Hence:


The diagram shows an example with .

Central projection of a helix

Hyperbolic spiral as central projection of a helix Schraublinie-hyp-spirale.svg
Hyperbolic spiral as central projection of a helix

Consider the central projection from point onto the image plane . This will map a point to the point

The image under this projection of the helix with parametric representation

is the curve with the polar equation

which describes a hyperbolic spiral.

For parameter the hyperbolic spiral has a pole and the helix intersects the plane at a point . One can check by calculation that the image of the helix as it approaches is the asymptote of the hyperbolic spiral.

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  1. Bowser, Edward Albert (1880), An Elementary Treatise on Analytic Geometry: Embracing Plane Geometry and an Introduction to Geometry of Three Dimensions (4th ed.), D. Van Nostrand, p. 232
  2. 1 2 Lawrence, J. Dennis (2013), A Catalog of Special Plane Curves, Dover Books on Mathematics, Courier Dover Publications, p. 186, ISBN   9780486167664 .