Hyperbolic Spiral

A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation
:$r=\frac$
of a hyperbola. Because it can be generated by a circle inversion of an Archimedean spiral, it is called Reciprocal spiral, too..

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

The hyperbolic spiral has a pitch angle that increases with distance from its center, unlike the logarithmic spiral (in which the angle is constant) or Archimedean spiral (in which it decreases with distance). For this reason, it has been used to model the shapes of spiral galaxies, which in some cases similarly have an increasing pitch angle. However, this model does not provide a good fit to the shapes of all spiral galaxies.

In cartesian coordinates

the hyperbolic spiral with the polar equation
:$r=\frac a \varphi ,\quad \varphi \ne 0$
can be represented in Cartesian coordinates by
:$x = a \frac \varphi, \qquad y = a \frac \varphi ,\quad \varphi \ne 0.$

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'':
: $\frac=\tan\left(\frac\right) .$

Geometric properties

Asymptote

Because
:$\lim_x = a\lim_ \frac \varphi =\infty,\qquad \lim_y = a\lim_ \frac \varphi = a$
the curve has an ''asymptote'' with equation .

Polar slope

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
: $\tan\alpha=-\frac.$

Curvature

The curvature of a curve with polar equation is
:$\kappa = \frac .$

From the equation and the derivatives and one gets the ''curvature'' of a hyperbolic spiral:
: $\kappa(\varphi) = \frac.$

Arc length

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

:\begin
L&=\int_^\sqrt\,d\varphi=\cdots \\
&=a \int_^\frac\,d\varphi \\
&= a\left \frac+\ln\left(\varphi+\sqrt\right)\right^ .
\end

Sector area

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

:$\begin A&=\frac12\int_^ r(\varphi)^2\, d\varphi\\ &=\frac12\int_^\frac\, d\varphi\\ &= \frac\left(\frac-\frac\right)\\ &=\frac\bigl(r(\varphi_1)-r(\varphi_2)\bigr) . \end$

Inversion

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 . At the two curves intersect at a fixed point on the unit circle.

The osculating circle of the Archimedean spiral at the origin has radius (see Archimedean spiral) and 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 .

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
:$(r\cos t, r\sin t, ct),\quad c\neq 0,$
is the curve
:$\frac(\cos t,\sin t)$
with the polar equation
:$\rho=\frac,$
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.

References

* Hans-Jochen Bartsch, Michael Sachs: ''Taschenbuch mathematischer Formeln für Ingenieure und Naturwissenschaftler'', Carl Hanser Verlag, 2018, , 9783446457072, S. 410.
* Kinko Tsuji, Stefan C. Müller: ''Spirals and Vortices: In Culture, Nature, and Science'', Springer, 2019, , 9783030057985, S. 96.
* Pierre Varignon
''Nouvelle formation de Spirales – exemple II'', Mémoires de l’Académie des sciences de l’Institut de France, 1704, pp. 94–103.
* Friedrich Grelle: ''Analytische Geometrie der Ebene'', Verlag F. Brecke, 186
hyperbolische Spirale
S. 215.
* Jakob Philipp Kulik: ''Lehrbuch der höhern Analysis, Band 2'', In Commiss. bei Kronberger u. Rziwnatz, 1844
Spirallinien
S. 222.

External links

*

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Spirals

pt:Espiral logarítmica