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

- In cartesian coordinates
- Geometric properties
- Asymptote
- Polar slope
- Curvature
- Arc length
- Sector area
- Inversion
- Central projection of a helix
- References
- External links

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.

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*:

Because of

the curve has an

*asymptote*with equation

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:

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

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

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

- The image of an Archimedean spiral with a circle inversion is the hyperbolic spiral with equation

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 preimage of the asymptote of the hyperbolic spiral with the inversion of the Archimedean spiral is the osculating circle of the Archimedean spiral at the origin.

- Example

The diagram shows an example with .

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.

In physics and geometry, a **catenary** is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends.

**Euler's formula**, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:

In mathematics, a **hyperbola** is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

In mathematics, the **polar coordinate system** is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the *pole*, and the ray from the pole in the reference direction is the *polar axis*. The distance from the pole is called the *radial coordinate*, *radial distance* or simply *radius*, and the angle is called the *angular coordinate*, *polar angle*, or *azimuth*. The radial coordinate is often denoted by *r* or *ρ*, and the angular coordinate by *φ*, *θ*, or *t*. Angles in polar notation are generally expressed in either degrees or radians.

In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the *radial distance* of that point from a fixed origin, its *polar angle* measured from a fixed zenith direction, and the *azimuthal angle* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

A **logarithmic spiral**, **equiangular spiral**, or **growth spiral** is a self-similar spiral curve that often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it *Spira mirabilis*, "the marvelous spiral".

In mathematics, a **spiral** is a curve which emanates from a point, moving farther away as it revolves around the point.

A **Fermat's spiral** or **parabolic spiral** is a plane curve named after Pierre de Fermat. Its polar coordinate representation is given by

A **cardioid** is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion.

In geometry, a **nephroid** is a specific plane curve whose name means 'kidney-shaped'. Although the term *nephroid* was used to describe other curves, it was applied to the curve in this article by Proctor in 1878.

In trigonometry, **tangent half-angle formulas** relate the tangent of half of an angle to trigonometric functions of the entire angle. Among these are the following

**Arc length** is the distance between two points along a section of a curve.

In mathematics an **orthogonal trajectory** is

In mathematics, **Viviani's curve**, also known as **Viviani's window**, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder that is tangent to the sphere and passes through the center of the sphere. Before Viviani this curve was studied by Simon de La Loubère and Gilles de Roberval.

In geometry, the **trisectrix of Maclaurin** is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines, each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is 1:3 and the lines initially coincide with the line between the two points. A generalization of this construction is called a sectrix of Maclaurin. The curve is named after Colin Maclaurin who investigated the curve in 1742.

The main **trigonometric identities** between trigonometric functions are proved, using mainly the geometry of the right triangle. For greater and negative angles, see Trigonometric functions.

In geometry, the **sinusoidal spirals** are a family of curves defined by the equation in polar coordinates

In geometry, a **sectrix of Maclaurin** is defined as the curve swept out by the point of intersection of two lines which are each revolving at constant rates about different points called **poles**. Equivalently, a sectrix of Maclaurin can be defined as a curve whose equation in biangular coordinates is linear. The name is derived from the trisectrix of Maclaurin, which is a prominent member of the family, and their sectrix property, which means they can be used to divide an angle into a given number of equal parts. There are special cases are also known as **arachnida** or **araneidans** because of their spider-like shape, and **Plateau curves** after Joseph Plateau who studied them.

In the geometry of curves, an **orthoptic** is the set of points for which two tangents of a given curve meet at a right angle.

In mathematics, a **conical spiral** is a curve on a right circular cone, whose floor plan is a plane spiral. If the floor plan is a logarithmic spiral, it is called conchospiral.

- ↑ 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 - 1 2 Lawrence, J. Dennis (2013),
*A Catalog of Special Plane Curves*, Dover Books on Mathematics, Courier Dover Publications, p. 186, ISBN 9780486167664 .

- Hans-Jochen Bartsch, Michael Sachs:
*Taschenbuch mathematischer Formeln für Ingenieure und Naturwissenschaftler*, Carl Hanser Verlag, 2018, ISBN 3446457070, 9783446457072, S. 410. - Kinko Tsuji, Stefan C. Müller:
*Spirals and Vortices: In Culture, Nature, and Science*, Springer, 2019, ISBN 3030057984, 9783030057985, S. 96. - Pierre Varignon:
*Nouvelle formation de Spirales – exemple II*, Mémoires de l’Académie des sciences de l’Institut de France, 1704, S. 94–103. - Friedrich Grelle:
*Analytische Geometrie der Ebene*, Verlag F. Brecke, 1861 hyperbolische Spirale, S. 215. - Jakob Philipp Kulik:
*Lehrbuch der höhern Analysis, Band 2*, In Commiss. bei Kronberger u. Rziwnatz, 1844, Spirallinien, S. 222.

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