Spiral

In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.^[1][2][3][4] It is a subtype of whorled patterns, a broad group that also includes concentric objects.

Helices

Two major definitions of "spiral" in the

1. a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.
2. a three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a helix.

The first definition describes a

• A conical or
  conical helix
  .
• Quite explicitly, definition 2 also includes a cylindrical coil spring and a strand of DNA, both of which are quite helical, so that "helix" is a more useful description than "spiral" for each of them; in general, "spiral" is seldom applied if successive "loops" of a curve have the same diameter.^[5]

In the side picture, the black curve at the bottom is an Archimedean spiral, while the green curve is a helix. The curve shown in red is a conical spiral.

Two-dimensional

A

${\displaystyle r}$ is a

of angle ${\displaystyle \varphi }$:

• ${\displaystyle r=r(\varphi )\;.}$

The circle would be regarded as a

• The Archimedean spiral: ${\displaystyle r=a\varphi }$
• The hyperbolic spiral: ${\displaystyle r=a/\varphi }$
• Fermat's spiral: ${\displaystyle r=a\varphi ^{1/2}}$
• The lituus: ${\displaystyle r=a\varphi ^{-1/2}}$
• The logarithmic spiral: ${\displaystyle r=ae^{k\varphi }}$
• The
  Cornu spiral
  or clothoid
• The
  Fibonacci spiral and golden spiral
• The Spiral of Theodorus: an approximation of the Archimedean spiral composed of contiguous right triangles
• The involute of a circle

• 
  Archimedean spiral
• 
  hyperbolic spiral
• 
  Fermat's spiral
• 
  lituus
• 
  logarithmic spiral
• 
  Cornu spiral
• 
  spiral of Theodorus
• 
  Fibonacci Spiral (golden spiral)
• 
  The involute of a circle (black) is not identical to the Archimedean spiral (red).

An Archimedean spiral is, for example, generated while coiling a carpet.^[6]

A hyperbolic spiral appears as image of a helix with a special central projection (see diagram). A hyperbolic spiral is some times called reciproke spiral, because it is the image of an Archimedean spiral with a circle-inversion (see below).^[7]

The name logarithmic spiral is due to the equation ${\displaystyle \varphi ={\tfrac {1}{k}}\cdot \ln {\tfrac {r}{a}}}$. Approximations of this are found in nature.

Spirals which do not fit into this scheme of the first 5 examples:

A Cornu spiral has two asymptotic points.
The spiral of Theodorus is a polygon.
The Fibonacci Spiral consists of a sequence of circle arcs.
The involute of a circle looks like an Archimedean, but is not: see Involute#Examples.

Geometric properties

The following considerations are dealing with spirals, which can be described by a polar equation ${\displaystyle r=r(\varphi )}$, especially for the cases ${\displaystyle r(\varphi )=a\varphi ^{n}}$ (Archimedean, hyperbolic, Fermat's, lituus spirals) and the logarithmic spiral ${\displaystyle r=ae^{k\varphi }}$.

Polar slope angle

The angle ${\displaystyle \alpha }$ between the spiral tangent and the corresponding polar circle (see diagram) is called angle of the polar slope and ${\displaystyle \tan \alpha }$ the polar slope.

From vector calculus in polar coordinates one gets the formula

${\displaystyle \tan \alpha ={\frac {r'}{r}}\ .}$

Hence the slope of the spiral ${\displaystyle \;r=a\varphi ^{n}\;}$ is

• ${\displaystyle \tan \alpha ={\frac {n}{\varphi }}\ .}$

In case of an Archimedean spiral (${\displaystyle n=1}$) the polar slope is ${\displaystyle \;\tan \alpha ={\tfrac {1}{\varphi }}\ .}$

The logarithmic spiral is a special case, because of ${\displaystyle \ \tan \alpha =k\ }$ constant !

curvature

The curvature ${\displaystyle \kappa }$ of a curve with polar equation ${\displaystyle r=r(\varphi )}$ is

${\displaystyle \kappa ={\frac {r^{2}+2(r')^{2}-r\;r''}{(r^{2}+(r')^{2})^{3/2}}}\ .}$

For a spiral with ${\displaystyle r=a\varphi ^{n}}$ one gets

• ${\displaystyle \kappa =\dotsb ={\frac {1}{a\varphi ^{n-1}}}{\frac {\varphi ^{2}+n^{2}+n}{(\varphi ^{2}+n^{2})^{3/2}}}\ .}$

In case of ${\displaystyle n=1}$ (Archimedean spiral) ${\displaystyle \kappa ={\tfrac {\varphi ^{2}+2}{a(\varphi ^{2}+1)^{3/2}}}}$.
Only for ${\displaystyle -1<n<0}$ the spiral has an inflection point.

The curvature of a logarithmic spiral ${\displaystyle \;r=ae^{k\varphi }\;}$ is ${\displaystyle \;\kappa ={\tfrac {1}{r{\sqrt {1+k^{2}}}}}\;.}$

Sector area

The area of a sector of a curve (see diagram) with polar equation ${\displaystyle r=r(\varphi )}$ is

${\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}r(\varphi )^{2}\;d\varphi \ .}$

For a spiral with equation ${\displaystyle r=a\varphi ^{n}\;}$ one gets

• ${\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}a^{2}\varphi ^{2n}\;d\varphi ={\frac {a^{2}}{2(2n+1)}}{\big (}\varphi _{2}^{2n+1}-\varphi _{1}^{2n+1}{\big )}\ ,\quad {\text{if}}\quad n\neq -{\frac {1}{2}},}$

${\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}{\frac {a^{2}}{\varphi }}\;d\varphi ={\frac {a^{2}}{2}}(\ln \varphi _{2}-\ln \varphi _{1})\ ,\quad {\text{if}}\quad n=-{\frac {1}{2}}\ .}$

The formula for a logarithmic spiral ${\displaystyle \;r=ae^{k\varphi }\;}$ is ${\displaystyle \ A={\tfrac {r(\varphi _{2})^{2}-r(\varphi _{1})^{2})}{4k}}\ .}$

Arc length

The length of an arc of a curve with polar equation ${\displaystyle r=r(\varphi )}$ is

${\displaystyle L=\int \limits _{\varphi _{1}}^{\varphi _{2}}{\sqrt {\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}}\,\mathrm {d} \varphi \ .}$

For the spiral ${\displaystyle r=a\varphi ^{n}\;}$ the length is

• ${\displaystyle L=\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {{\frac {n^{2}r^{2}}{\varphi ^{2}}}+r^{2}}}\;d\varphi =a\int \limits _{\varphi _{1}}^{\varphi _{2}}\varphi ^{n-1}{\sqrt {n^{2}+\varphi ^{2}}}d\varphi \ .}$

Not all these integrals can be solved by a suitable table. In case of a Fermat's spiral, the integral can be expressed by elliptic integrals only.

The arc length of a logarithmic spiral ${\displaystyle \;r=ae^{k\varphi }\;}$ is ${\displaystyle \ L={\tfrac {\sqrt {k^{2}+1}}{k}}{\big (}r(\varphi _{2})-r(\varphi _{1}){\big )}\ .}$

Circle inversion

The

${\displaystyle \ (r,\varphi )\mapsto ({\tfrac {1}{r}},\varphi )\ }$

• The image of a spiral ${\displaystyle \ r=a\varphi ^{n}\ }$ under the inversion at the unit circle is the spiral with polar equation ${\displaystyle \ r={\tfrac {1}{a}}\varphi ^{-n}\ }$. For example: The inverse of an Archimedean spiral is a hyperbolic spiral.

A logarithmic spiral ${\displaystyle \;r=ae^{k\varphi }\;}$ is mapped onto the logarithmic spiral ${\displaystyle \;r={\tfrac {1}{a}}e^{-k\varphi }\;.}$

Bounded spirals

The function ${\displaystyle r(\varphi )}$ of a spiral is usually strictly monotonic, continuous and unbounded. For the standard spirals ${\displaystyle r(\varphi )}$ is either a power function or an exponential function. If one chooses for ${\displaystyle r(\varphi )}$ a bounded function, the spiral is bounded, too. A suitable bounded function is the

Example 1

Setting ${\displaystyle \;r=a\arctan(k\varphi )\;}$ and the choice ${\displaystyle \;k=0.1,a=4,\;\varphi \geq 0\;}$ gives a spiral, that starts at the origin (like an Archimedean spiral) and approaches the circle with radius ${\displaystyle \;r=a\pi /2\;}$ (diagram, left).

Example 2

For ${\displaystyle \;r=a(\arctan(k\varphi )+\pi /2)\;}$ and ${\displaystyle \;k=0.2,a=2,\;-\infty <\varphi <\infty \;}$ one gets a spiral, that approaches the origin (like a hyperbolic spiral) and approaches the circle with radius ${\displaystyle \;r=a\pi \;}$ (diagram, right).

Three-dimensional

Two well-known spiral

coiled coil filaments

.

Conical spirals

Main article: conical spiral

If in the ${\displaystyle x}$-${\displaystyle y}$-plane a spiral with parametric representation

${\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi }$

is given, then there can be added a third coordinate ${\displaystyle z(\varphi )}$, such that the now space curve lies on the cone with equation ${\displaystyle \;m(x^{2}+y^{2})=(z-z_{0})^{2}\ ,\ m>0\;}$:

• ${\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi \ ,\qquad \color {red}{z=z_{0}+mr(\varphi )}\ .}$

Spirals based on this procedure are called conical spirals.

Example

Starting with an archimedean spiral ${\displaystyle \;r(\varphi )=a\varphi \;}$ one gets the conical spiral (see diagram)

${\displaystyle x=a\varphi \cos \varphi \ ,\qquad y=a\varphi \sin \varphi \ ,\qquad z=z_{0}+ma\varphi \ ,\quad \varphi \geq 0\ .}$

Spherical spirals

If one represents a sphere of radius ${\displaystyle r}$ by:

${\displaystyle {\begin{array}{cll}x&=&r\cdot \sin \theta \cdot \cos \varphi \\y&=&r\cdot \sin \theta \cdot \sin \varphi \\z&=&r\cdot \cos \theta \end{array}}}$

and sets the linear dependency ${\displaystyle \;\varphi =c\theta ,\;c>2\;,}$ for the angle coordinates, one gets a

spherical curve called spherical spiral^[11]

with the parametric representation (with ${\displaystyle c}$ equal to twice the number of turns)

• ${\displaystyle {\begin{array}{cll}x&=&r\cdot \sin \theta \cdot \cos {\color {red}c\theta }\\y&=&r\cdot \sin \theta \cdot \sin {\color {red}c\theta }\\z&=&r\cdot \cos \theta \qquad \qquad 0\leq \theta \leq \pi \ .\end{array}}}$

Spherical spirals were known to Pappus, too.

Remark: a rhumb line is not a spherical spiral in this sense.

• 
  Spherical spiral
• 
  Loxodrome

A

where n is the index number of the floret and c is a constant scaling factor, and is a form of Fermat's spiral. The angle 137.5° is the golden angle which is related to the golden ratio and gives a close packing of florets.^[14]

Spirals in plants and animals are frequently described as whorls. This is also the name given to spiral shaped fingerprints.

• 
  An artist's rendering of a spiral galaxy.
• 
  Sunflower head displaying florets in spirals of 34 and 55 around the outside.

As a symbol

A spiral like form has been found in Mezine, Ukraine, as part of a decorative object dated to 10,000 BCE.^{[citation needed]} The spiral and

Spirals are also a symbol of

• 
  Cucuteni Culture spirals on a bowl on stand, a vessel on stand, and an amphora, 4300-4000 BCE, ceramic, Palace of Culture, Iași, Romania
• 
  Neolithic spirals on the Newgrange entrance slab, unknown sculptor or architect, 3rd millennium BC
• 
  Mycenaean spirals on a burial stela, Grave Circle A, c.1550 BC, stone, National Archaeological Museum, Athens, Greece
• 
  Meroitic spirals on a ram of the alley of the Amun Temple of Naqa, unknown sculptor, 1st century AD, stone, in situ
• 
  Islamic spiral design of the Great Mosque of Samarra, Samarra, Iraq, unknown architect, c.851
• 
  Gothic Revival spiralling bell tower of the Maison des compagnons du tour de France, Nantes

  , unknown architect, c.1910

• Celtic maze (straight-line spiral)
• Concentric circles
• DNA
• Fibonacci number
• Hypogeum of Ħal-Saflieni
• Megalithic Temples of Malta
• Patterns in nature
• Seashell surface
• Spirangle
• Spiral vegetable slicer
• Spiral stairs
• Triskelion

References

Related publications

• Cook, T., 1903. Spirals in nature and art. Nature 68 (1761), 296.
• Cook, T., 1979. The curves of life. Dover, New York.
• Habib, Z., Sakai, M., 2005. Spiral transition curves and their applications. Scientiae Mathematicae Japonicae 61 (2), 195 – 206.
• Dimulyo, Sarpono; Habib, Zulfiqar; Sakai, Manabu (2009). "Fair cubic transition between two circles with one circle inside or tangent to the other". Numerical Algorithms. 51 (4): 461–476.
  S2CID 22532724
  .
• Harary, G., Tal, A., 2011. The natural 3D spiral. Computer Graphics Forum 30 (2), 237 – 246 [1] Archived 2015-11-22 at the Wayback Machine.
• Xu, L., Mould, D., 2009. Magnetic curves: curvature-controlled aesthetic curves using magnetic fields. In: Deussen, O., Hall, P. (Eds.), Computational Aesthetics in Graphics, Visualization, and Imaging. The Eurographics Association [2].
• Wang, Yulin; Zhao, Bingyan; Zhang, Luzou; Xu, Jiachuan; Wang, Kanchang; Wang, Shuchun (2004). "Designing fair curves using monotone curvature pieces". Computer Aided Geometric Design. 21 (5): 515–527.
  doi:10.1016/j.cagd.2004.04.001
  .
• Kurnosenko, A. (2010). "Applying inversion to construct planar, rational spirals that satisfy two-point G2 Hermite data". Computer Aided Geometric Design. 27 (3): 262–280.
  S2CID 14476206
  .
• A. Kurnosenko. Two-point G2 Hermite interpolation with spirals by inversion of hyperbola. Computer Aided Geometric Design, 27(6), 474–481, 2010.
• Miura, K.T., 2006. A general equation of aesthetic curves and its self-affinity. Computer-Aided Design and Applications 3 (1–4), 457–464 [3] Archived 2013-06-28 at the Wayback Machine.
• Miura, K., Sone, J., Yamashita, A., Kaneko, T., 2005. Derivation of a general formula of aesthetic curves. In: 8th International Conference on Humans and Computers (HC2005). Aizu-Wakamutsu, Japan, pp. 166 – 171 [4] Archived 2013-06-28 at the Wayback Machine.
• Meek, D.S.; Walton, D.J. (1989). "The use of Cornu spirals in drawing planar curves of controlled curvature". Journal of Computational and Applied Mathematics. 25: 69–78.
  doi:10.1016/0377-0427(89)90076-9
  .
• Thomas, Sunil (2017). "Potassium sulfate forms a spiral structure when dissolved in solution". Russian Journal of Physical Chemistry B. 11 (1): 195–198.
  S2CID 99162341
  .
• Farin, Gerald (2006). "Class a Bézier curves". Computer Aided Geometric Design. 23 (7): 573–581.
  doi:10.1016/j.cagd.2006.03.004
  .
• Farouki, R.T., 1997. Pythagorean-hodograph quintic transition curves of monotone curvature. Computer-Aided Design 29 (9), 601–606.
• Yoshida, N., Saito, T., 2006. Interactive aesthetic curve segments. The Visual Computer 22 (9), 896–905 [5] Archived 2016-03-04 at the Wayback Machine.
• Yoshida, N., Saito, T., 2007. Quasi-aesthetic curves in rational cubic Bézier forms. Computer-Aided Design and Applications 4 (9–10), 477–486 [6] Archived 2016-03-03 at the Wayback Machine.
• Ziatdinov, R., Yoshida, N., Kim, T., 2012. Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions. Computer Aided Geometric Design 29 (2), 129—140 [7].
• Ziatdinov, R., Yoshida, N., Kim, T., 2012. Fitting G2 multispiral transition curve joining two straight lines, Computer-Aided Design 44(6), 591—596 [8].
• Ziatdinov, R., 2012. Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function. Computer Aided Geometric Design 29(7): 510–518, 2012 [9].
• Ziatdinov, R., Miura K.T., 2012. On the Variety of Planar Spirals and Their Applications in Computer Aided Design. European Researcher 27(8–2), 1227—1232 [10].

External links

Wikimedia Commons has media related to Spiral.

• [11] Archived 2021-07-02 at the Wayback Machine
• Archimedes' spiral transforms into Galileo's spiral. Mikhail Gaichenkov, OEIS