# Logarithmic spiral

Last updated

A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige lini").  More than a century later, the curve was discussed by Descartes (1638), and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral".

## Contents

The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.

## Definition

In polar coordinates $(r,\varphi )$ the logarithmic spiral can be written as 

$r=ae^{k\varphi },\quad \varphi \in \mathbb {R} ,$ or

$\varphi ={\frac {1}{k}}\ln {\frac {r}{a}},$ with $e$ being the base of natural logarithms, and $a>0,k\neq 0$ being real constants.

## In Cartesian coordinates

The logarithmic spiral with the polar equation

$\;r=ae^{k\varphi }$ can be represented in Cartesian coordinates $(x=r\cos \varphi ,\,y=r\sin \varphi )$ by

• $x=ae^{k\varphi }\cos \varphi ,\qquad y=ae^{k\varphi }\sin \varphi .$ In the complex plane $(z=x+iy,\,e^{i\varphi }=\cos \varphi +i\sin \varphi )$ :

• $z=ae^{(k+i)\varphi }.$ ## Spira mirabilis and Jacob Bernoulli

Spira mirabilis, Latin for "miraculous spiral", is another name for the logarithmic spiral. Although this curve had already been named by other mathematicians, the specific name ("miraculous" or "marvelous" spiral) was given to this curve by Jacob Bernoulli, because he was fascinated by one of its unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve, a property known as self-similarity. Possibly as a result of this unique property, the spira mirabilis has evolved in nature, appearing in certain growing forms such as nautilus shells and sunflower heads. Jacob Bernoulli wanted such a spiral engraved on his headstone along with the phrase "Eadem mutata resurgo" ("Although changed, I shall arise the same."), but, by error, an Archimedean spiral was placed there instead.  

## Properties

The logarithmic spiral $\;r=ae^{k\varphi }\;,\;k\neq 0,\;$ has the following properties (see Spiral):

• Polar slope: $\ \tan \alpha =k\quad ({\color {red}{\text{constan$ !}}})} with polar slope angle$\alpha$ (see diagram).
(In case of $k=0$ angle $\alpha$ would be 0 and the curve a circle with radius $a$ .)
• Curvature: $\;\kappa ={\frac {1}{r{\sqrt {1+k^{2}}}}}={\frac {\cos \alpha }{r}}$ • Arc length:$\ L(\varphi _{1},\varphi _{2})={\frac {\sqrt {k^{2}+1}}{k}}{\big (}r(\varphi _{2})-r(\varphi _{1}){\big )}={\frac {r(\varphi _{2})-r(\varphi _{1})}{\sin \alpha }}$ Especially: $\ L(-\infty ,\varphi _{2})={\frac {r(\varphi _{2})}{\sin \alpha }}\quad ({\color {red}{\text{finit$ !}}})\;} , if $k>0$ .
This property was first realized by Evangelista Torricelli even before calculus had been invented. 
• Sector area:$\ A={\frac {r(\varphi _{2})^{2}-r(\varphi _{1})^{2}}{4k}}$ • Inversion: Circle inversion ($r\to 1/r$ ) maps the logarithmic spiral $\;r=ae^{k\varphi }\;$ onto the logarithmic spiral $\;r={\tfrac {1}{a}}e^{-k\varphi }\ .$ • Rotating, scaling: Rotating the spiral by angle $\varphi _{0}$ yields the spiral $r=ae^{-k\varphi _{0}}e^{k\varphi }$ , which is the original spiral uniformly scaled (at the origin) by $e^{-k\varphi _{0}}$ .
Scaling by $\;e^{kn2\pi }\;,n=\pm 1,\pm 2,...,\;$ gives the same curve.
A scaled logarithmic spiral is congruent (by rotation) to the original curve.
Example: The diagram shows spirals with slope angle $\alpha =20^{\circ }$ and $a=1,2,3,4,5$ . Hence they are all scaled copies of the red one. But they can also be generated by rotating the red one by angles $-109^{\circ },-173^{\circ },-218^{\circ },-253^{\circ }$ resp.. All spirals have no points in common (see property on complex exponential function).
• Relation to other curves: Logarithmic spirals are congruent to their own involutes, evolutes, and the pedal curves based on their centers.
• Complex exponential function: The exponential function exactly maps all lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at $0$ :
$z(t)=\underbrace {(kt+b)\;+it} _{\text{line}}\quad \to \quad e^{z(t)}=e^{kt+b}\cdot e^{it}=\underbrace {e^{b}e^{kt}(\cos t+i\sin t)} _{\text{log. spiral}}\$ The polar slope angle $\alpha$ of the logarithmic spiral is the angle between the line and the imaginary axis.

## Special cases and approximations

The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (polar slope angle about 17.03239 degrees). It can be approximated by a "Fibonacci spiral", made of a sequence of quarter circles with radii proportional to Fibonacci numbers.

## In nature

An extratropical cyclone over Iceland shows an approximately logarithmic spiral pattern
The arms of spiral galaxies often have the shape of a logarithmic spiral, here the Whirlpool Galaxy Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral. The plotted spiral (dashed blue curve) is based on growth rate parameter b=0.1759{\displaystyle b=0.1759}, resulting in a pitch of arctan⁡b≈10∘{\displaystyle \arctan b\approx 10^{\circ }}.

In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follow some examples and reasons:

• The approach of a hawk to its prey in classical pursuit, assuming the prey travels in a straight line. Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral's pitch. 
• The approach of an insect to a light source. They are used to having the light source at a constant angle to their flight path. Usually the sun (or moon for nocturnal species) is the only light source and flying that way will result in a practically straight line. 
• The arms of spiral galaxies.  Our own galaxy, the Milky Way, has several spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees. 
• The nerves of the cornea (this is, corneal nerves of the subepithelial layer terminate near superficial epithelial layer of the cornea in a logarithmic spiral pattern). 
• The bands of tropical cyclones, such as hurricanes. 
• Many biological structures including the shells of mollusks.  In these cases, the reason may be construction from expanding similar shapes, as is the case for polygonal figures.
• Logarithmic spiral beaches can form as the result of wave refraction and diffraction by the coast. Half Moon Bay (California) is an example of such a type of beach. 

## In engineering applications

A kerf-canceling mechanism leverage the self similarity of the logarithmic spiral to lock in place under rotation, independent of the kerf of the cut. 
• Logarithmic spiral antennas are frequency-independent antennas, that is, antennas whose radiation pattern, impedance and polarization remain largely unmodified over a wide bandwidth. 
• When manufacturing mechanisms by subtractive fabrication machines (such as laser cutters), there can be a loss of precision when the mechanism is fabricated on a different machine due to the difference of material removed (that is, the kerf) by each machine in the cutting process. To adjust for this variation of kerf, the self-similar property of the logarithmic spiral has been used to design a kerf cancelling mechanism for laser cutters. 
• Logarithmic spiral bevel gears are a type of spiral bevel gear whose gear tooth centerline is a logarithmic spiral. A logarithmic spiral has the advantage of providing equal angles between the tooth centerline and the radial lines, which gives the meshing transmission more stability. 

## Related Research Articles 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 spiral is a curve which emanates from a point, moving farther away as it revolves around the point. A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation A Fermat's spiral or parabolic spiral is a plane curve named after Pierre de Fermat. Its polar coordinate representation is given by In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider by a factor of φ for every quarter turn it makes. 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. It's also the set of points of reflections of a fixed point on a circle through all tangents to the circle. 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

The rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the linear rotor requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water, ammonia, or methane. The rigid-rotor Schroedinger equation is discussed in Section 11.2 on pages 240-253 of the textbook by Bunker and Jensen. In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral. The theorem is named after the Greek astronomer and mathematician Ptolemy. Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy. The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways. In mathematics an orthogonal trajectory is a curve, which intersects any curve of a given pencil of (planar) curves orthogonally.

The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the fundamental theorem of calculus to any curve in a plane or space rather than just the real line.

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 mathematics, a mean of circular quantities is a mean which is sometimes better-suited for quantities like angles, daytimes, and fractional parts of real numbers. This is necessary since most of the usual means may not be appropriate on circular quantities. For example, the arithmetic mean of 0° and 360° is 180°, which is misleading because for most purposes 360° is the same thing as 0°. As another example, the "average time" between 11 PM and 1 AM is either midnight or noon, depending on whether the two times are part of a single night or part of a single calendar day. This is one of the simplest examples of statistics of non-Euclidean spaces. In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the x-axis. 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.

For a plane curve C and a given fixed point O, the pedal equation of the curve is a relation between r and p where r is the distance from O to a point on C and p is the perpendicular distance from O to the tangent line to C at the point. The point O is called the pedal point and the values r and p are sometimes called the pedal coordinates of a point relative to the curve and the pedal point. It is also useful to measure the distance of O to the normal even though it is not an independent quantity and it relates to as . 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.

1. Albrecht Dürer (1525). Underweysung der Messung, mit dem Zirckel und Richtscheyt, in Linien, Ebenen unnd gantzen corporen.
2. Priya Hemenway (2005). Divine Proportion: Φ Phi in Art, Nature, and Science. Sterling Publishing Co. ISBN   978-1-4027-3522-6.
3. Livio, Mario (2002). . New York: Broadway Books. ISBN   978-0-7679-0815-3.
4. Yates, R. C.: A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Evolutes". p. 206.
5. Carl Benjamin Boyer (1949). The history of the calculus and its conceptual development. Courier Dover Publications. p. 133. ISBN   978-0-486-60509-8.
6. Chin, Gilbert J. (8 December 2000), "Organismal Biology: Flying Along a Logarithmic Spiral", Science , 290 (5498): 1857, doi:10.1126/science.290.5498.1857c
7. John Himmelman (2002). Discovering Moths: Nighttime Jewels in Your Own Backyard. Down East Enterprise Inc. p. 63. ISBN   978-0-89272-528-1.
8. G. Bertin and C. C. Lin (1996). Spiral structure in galaxies: a density wave theory. MIT Press. p. 78. ISBN   978-0-262-02396-2.
9. David J. Darling (2004). The universal book of mathematics: from Abracadabra to Zeno's paradoxes. John Wiley and Sons. p. 188. ISBN   978-0-471-27047-8.
10. C. Q. Yu CQ and M. I. Rosenblatt, "Transgenic corneal neurofluorescence in mice: a new model for in vivo investigation of nerve structure and regeneration," Invest Ophthalmol Vis Sci. 2007 Apr;48(4):1535-42.
11. Andrew Gray (1901). Treatise on physics, Volume 1. Churchill. pp.  356–357.
12. Michael Cortie (1992). "The form, function, and synthesis of the molluscan shell". In István Hargittai and Clifford A. Pickover (ed.). Spiral symmetry. World Scientific. p. 370. ISBN   978-981-02-0615-4.
13. Allan Thomas Williams and Anton Micallef (2009). Beach management: principles and practice. Earthscan. p. 14. ISBN   978-1-84407-435-8.
14. "kerf-canceling mechanisms". hpi.de. Retrieved 2020-12-26.
15. Mayes, P.E. (1992). "Frequency-independent antennas and broad-band derivatives thereof". Proceedings of the IEEE. 80 (1): 103–112. doi:10.1109/5.119570.
16. Roumen, Thijs; Apel, Ingo; Shigeyama, Jotaro; Muhammad, Abdullah; Baudisch, Patrick (2020-10-20). "Kerf-Canceling Mechanisms: Making Laser-Cut Mechanisms Operate across Different Laser Cutters". Proceedings of the 33rd Annual ACM Symposium on User Interface Software and Technology. Virtual Event USA: ACM: 293–303. doi:10.1145/3379337.3415895. ISBN   978-1-4503-7514-6.
17. Jiang, Jianfeng; Luo, Qingsheng; Wang, Liting; Qiao, Lijun; Li, Minghao (2020). "Review on logarithmic spiral bevel gear". Journal of the Brazilian Society of Mechanical Sciences and Engineering. 42 (8): 400. doi:. ISSN   1678-5878.