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Spiral

From Wikipedia, the free encyclopedia
(Redirected fromSpherical spiral)
Curve that winds around a central point
For other uses, seeSpiral (disambiguation).
Cutaway of anautilus shell showing the chambers arranged in an approximatelylogarithmic spiral

Inmathematics, aspiral is acurve which emanates from a point, moving farther away as it revolves around the point.[1][2][3][4] It is a subtype ofwhorled patterns, a broad group that also includesconcentric objects.

Two-dimensional

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Main article:List of spirals
Spirals generated by six mathematical relationships between radius and angle

Atwo-dimensional, or plane, spiral may be easily described usingpolar coordinates, where theradiusr{\displaystyle r} is amonotoniccontinuous function of angleφ{\displaystyle \varphi }:

The circle would be regarded as adegenerate case (thefunction not being strictly monotonic, but ratherconstant).

Inx{\displaystyle x}-y{\displaystyle y}-coordinates the curve has the parametric representation:

Examples

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Some of the most important sorts of two-dimensional spirals include:

  • Archimedean spiral
    Archimedean spiral
  • Hyperbolic spiral
    Hyperbolic spiral
  • Fermat's spiral
    Fermat's spiral
  • The lituus
    The lituus
  • Logarithmic spiral
    Logarithmic spiral
  • Cornu spiral
    Cornu spiral
  • Spiral of Theodorus
    Spiral of Theodorus
  • Fibonacci Spiral (golden spiral)
    Fibonacci Spiral (golden spiral)
  • The involute of a circle (black) is not identical to the Archimedean spiral (red).
    The involute of a circle (black) is not identical to the Archimedean spiral (red).
Hyperbolic spiral as central projection of a helix

AnArchimedean spiral is, for example, generated while coiling a carpet.[5]

Ahyperbolic spiral appears as image of a helix with a special central projection (see diagram). A hyperbolic spiral is some times calledreciproke spiral, because it is the image of an Archimedean spiral with a circle-inversion (see below).[6]

The namelogarithmic spiral is due to the equationφ=1k⋅ln⁡ra{\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:

ACornu spiral has two asymptotic points.
Thespiral of Theodorus is a polygon.
TheFibonacci Spiral consists of a sequence of circle arcs.
Theinvolute of a circle looks like an Archimedean, but is not: seeInvolute#Examples.

Geometric properties

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

Definition of sector (light blue) and polar slope angleα{\displaystyle \alpha }
Polar slope angle

The angleα{\displaystyle \alpha } between the spiral tangent and the corresponding polar circle (see diagram) is calledangle of the polar slope andtan⁡α{\displaystyle \tan \alpha } thepolar slope.

Fromvector calculus in polar coordinates one gets the formula

tan⁡α=r′r .{\displaystyle \tan \alpha ={\frac {r'}{r}}\ .}

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

In case of anArchimedean spiral (n=1{\displaystyle n=1}) the polar slope istan⁡α=1φ .{\displaystyle \tan \alpha ={\tfrac {1}{\varphi }}\ .}

In alogarithmic spiral,tan⁡α=k {\displaystyle \tan \alpha =k\ } is constant.

Curvature

The curvatureκ{\displaystyle \kappa } of a curve with polar equationr=r(φ){\displaystyle r=r(\varphi )} is

κ=r2+2(r′)2−rr″(r2+(r′)2)3/2 .{\displaystyle \kappa ={\frac {r^{2}+2(r')^{2}-r\;r''}{(r^{2}+(r')^{2})^{3/2}}}\ .}

For a spiral withr=aφn{\displaystyle r=a\varphi ^{n}} one gets

In case ofn=1{\displaystyle n=1}(Archimedean spiral)κ=φ2+2a(φ2+1)3/2{\displaystyle \kappa ={\tfrac {\varphi ^{2}+2}{a(\varphi ^{2}+1)^{3/2}}}}.
Only for−1<n<0{\displaystyle -1<n<0} the spiral has aninflection point.

The curvature of alogarithmic spiralr=aekφ{\displaystyle \;r=ae^{k\varphi }\;} isκ=1r1+k2.{\displaystyle \;\kappa ={\tfrac {1}{r{\sqrt {1+k^{2}}}}}\;.}

Sector area

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

A=12∫φ1φ2r(φ)2dφ .{\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}r(\varphi )^{2}\;d\varphi \ .}

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

A=12∫φ1φ2a2φdφ=a22(ln⁡φ2−ln⁡φ1) ,ifn=−12 .{\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 alogarithmic spiralr=aekφ{\displaystyle \;r=ae^{k\varphi }\;} is A=r(φ2)2−r(φ1)2)4k .{\displaystyle \ A={\tfrac {r(\varphi _{2})^{2}-r(\varphi _{1})^{2})}{4k}}\ .}

Arc length

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

L=∫φ1φ2(r′(φ))2+r2(φ)dφ .{\displaystyle L=\int \limits _{\varphi _{1}}^{\varphi _{2}}{\sqrt {\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}}\,\mathrm {d} \varphi \ .}

For the spiralr=aφn{\displaystyle r=a\varphi ^{n}\;} the length is

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

The arc length of alogarithmic spiralr=aekφ{\displaystyle \;r=ae^{k\varphi }\;} is L=k2+1k(r(φ2)−r(φ1)) .{\displaystyle \ L={\tfrac {\sqrt {k^{2}+1}}{k}}{\big (}r(\varphi _{2})-r(\varphi _{1}){\big )}\ .}

Circle inversion

Theinversion at the unit circle has in polar coordinates the simple description: (r,φ)↦(1r,φ) {\displaystyle \ (r,\varphi )\mapsto ({\tfrac {1}{r}},\varphi )\ }.

A logarithmic spiralr=aekφ{\displaystyle \;r=ae^{k\varphi }\;} is mapped onto the logarithmic spiralr=1ae−kφ.{\displaystyle \;r={\tfrac {1}{a}}e^{-k\varphi }\;.}

Bounded spirals

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Bounded spirals:
r=aarctan⁡(kφ){\displaystyle r=a\arctan(k\varphi )} (left),
r=a(arctan⁡(kφ)+π/2){\displaystyle r=a(\arctan(k\varphi )+\pi /2)} (right)

The functionr(φ){\displaystyle r(\varphi )} of a spiral is usually strictly monotonic, continuousand unbounded. For the standard spiralsr(φ){\displaystyle r(\varphi )} is either a power function or an exponential function. If one chooses forr(φ){\displaystyle r(\varphi )} abounded function, the spiral is bounded, too. A suitable bounded function is thearctan function:

Example 1

Settingr=aarctan⁡(kφ){\displaystyle \;r=a\arctan(k\varphi )\;} and the choicek=0.1,a=4,φ≥0{\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 radiusr=aπ/2{\displaystyle \;r=a\pi /2\;} (diagram, left).

Example 2

Forr=a(arctan⁡(kφ)+π/2){\displaystyle \;r=a(\arctan(k\varphi )+\pi /2)\;} andk=0.2,a=2,−∞<φ<∞{\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 radiusr=aπ{\displaystyle \;r=a\pi \;} (diagram, right).

Three-dimensional

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"Space spiral" redirects here. For the gyro tower, seeSpace Spiral.

Helices

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An Archimedean spiral (black), a helix (green), and a conical spiral (red)

Two major definitions of "spiral" in theAmerican Heritage Dictionary are:[7]

  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; ahelix.

The first definition describes aplanar curve, that extends in both of the perpendicular directions within its plane; the groove on one side of agramophone record closely approximates a plane spiral (and it is by the finite width and depth of the groove, butnot by the wider spacing between than within tracks, that it falls short of being a perfect example); note that successive loopsdiffer in diameter. In another example, the "center lines" of the arms of aspiral galaxy tracelogarithmic spirals.

The second definition includes two kinds of 3-dimensional relatives of spirals:

  • A conical orvolute spring (including the spring used to hold and make contact with the negative terminals of AA or AAA batteries in abattery box), and thevortex that is created when water is draining in a sink is often described as a spiral, or as aconical helix.
  • Quite explicitly, definition 2 also includes a cylindrical coil spring and a strand ofDNA, both of which are fairly helical, so that "helix" is a moreuseful description than "spiral" for each of them. In general, "spiral" is seldom applied if successive "loops" of a curve have the same diameter.[7]

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

"Space spiral" redirects here. For the building, seeSpace Spiral.

Two well-known spiralspace curves areconical spirals andspherical spirals, defined below.Another instance of space spirals is thetoroidal spiral.[8] A spiral wound around a helix,[9] also known asdouble-twisted helix,[10] represents objects such ascoiled coil filaments.

Conical spirals

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Conical spiral with Archimedean spiral as floor plan
Main article:Conical spiral

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

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

is given, then there can be added a third coordinatez(φ){\displaystyle z(\varphi )}, such that the now space curve lies on thecone with equationm(x2+y2)=(z−z0)2 , m>0{\displaystyle \;m(x^{2}+y^{2})=(z-z_{0})^{2}\ ,\ m>0\;}:

Spirals based on this procedure are calledconical spirals.

Example

Starting with anarchimedean spiralr(φ)=aφ{\displaystyle \;r(\varphi )=a\varphi \;} one gets the conical spiral (see diagram)

x=aφcos⁡φ ,y=aφsin⁡φ ,z=z0+maφ ,φ≥0 .{\displaystyle x=a\varphi \cos \varphi \ ,\qquad y=a\varphi \sin \varphi \ ,\qquad z=z_{0}+ma\varphi \ ,\quad \varphi \geq 0\ .}

Spherical spirals

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Clelia curve withc=8{\displaystyle c=8}

Anycylindrical map projection can be used as the basis for aspherical spiral: draw a straight line on the map and find its inverse projection on the sphere, a kind ofspherical curve.

One of the most basic families of spherical spirals is theClelia curves, which project to straight lines on anequirectangular projection. These are curves for whichlongitude andcolatitude are in a linear relationship, analogous to Archimedean spirals in the plane; under theazimuthal equidistant projection a Clelia curve projects to a planar Archimedean spiral.

If one represents a unit sphere byspherical coordinates

x=sin⁡θcos⁡φ,y=sin⁡θsin⁡φ,z=cos⁡θ,{\displaystyle x=\sin \theta \,\cos \varphi ,\quad y=\sin \theta \,\sin \varphi ,\quad z=\cos \theta ,}

then setting the linear dependencyφ=cθ{\displaystyle \varphi =c\theta } for the angle coordinates gives aparametric curve in terms of parameter⁠θ{\displaystyle \theta }⁠,[11]

(sin⁡θcos⁡cθ,sin⁡θsin⁡cθ,cos⁡θ).{\displaystyle {\bigl (}\sin \theta \,\cos c\theta ,\,\sin \theta \,\sin c\theta ,\,\cos \theta \,{\bigr )}.}
  • Clelia curve
    Clelia curve
  • Loxodrome
    Loxodrome

Another family of spherical spirals is therhumb lines or loxodromes, that project to straight lines on theMercator projection. These are the trajectories traced by a ship traveling with constantbearing. Any loxodrome (except for the meridians and parallels) spirals infinitely around either pole, closer and closer each time, unlike a Clelia curve which maintains uniform spacing in colatitude. Understereographic projection, a loxodrome projects to a logarithmic spiral in the plane.

In nature

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The study of spirals innature has a long history.Christopher Wren observed that manyshells form alogarithmic spiral;Jan Swammerdam observed the common mathematical characteristics of a wide range of shells fromHelix toSpirula; andHenry Nottidge Moseley described the mathematics ofunivalve shells.D’Arcy Wentworth Thompson'sOn Growth and Form gives extensive treatment to these spirals. He describes how shells are formed by rotating a closed curve around a fixed axis: theshape of the curve remains fixed, but its size grows in ageometric progression. In some shells, such asNautilus andammonites, the generating curve revolves in a plane perpendicular to the axis and the shell will form a planar discoid shape. In others it follows a skew path forming ahelico-spiral pattern. Thompson also studied spirals occurring inhorns,teeth,claws andplants.[12]

A model for the pattern offlorets in the head of asunflower[13] was proposed by H. Vogel. This has the form

θ=n×137.5∘, r=cn{\displaystyle \theta =n\times 137.5^{\circ },\ r=c{\sqrt {n}}}

wheren is the index number of the floret andc is a constant scaling factor, and is a form ofFermat's spiral. The angle 137.5° is thegolden angle which is related to thegolden ratio and gives a close packing of florets.[14]

Spirals in plants and animals are frequently described aswhorls. This is also the name given to spiral shapedfingerprints.

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

As a symbol

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TheCeltic triple-spiral is in fact a pre-Celtic symbol.[15] It is carved into the rock of a stone lozenge near the main entrance of the prehistoricNewgrange monument inCounty Meath, Ireland. Newgrange was built around 3200BCE, predating the Celts; triple spirals were carved at least 2,500 years before the Celts reached Ireland, but have long since become part of Celtic culture.[16] Thetriskelion symbol, consisting of three interlocked spirals or three bent human legs, appears in many early cultures: examples includeMycenaean vessels, coinage fromLycia,staters ofPamphylia (atAspendos, 370–333 BC) andPisidia, as well as theheraldic emblem on warriors' shields depicted on Greek pottery.[17]

Spirals occur commonly in pre-Columbian art in Latin and Central America. The more than 1,400petroglyphs (rock engravings) inLas Plazuelas,GuanajuatoMexico, dating 750-1200 AD, predominantly depict spirals, dot figures and scale models.[18] In Colombia, monkeys, frog and lizard-like figures depicted in petroglyphs or as gold offering-figures frequently include spirals, for example on the palms of hands.[19] In Lower Central America, spirals along with circles, wavy lines, crosses and points are universal petroglyph characters.[20] Spirals also appear among theNazca Lines in the coastal desert of Peru, dating from 200 BC to 500 AD. Thegeoglyphs number in the thousands and depict animals, plants and geometric motifs, including spirals.[21]

Spirals are also a symbol ofhypnosis, stemming from thecliché of people and cartoon characters being hypnotized by staring into a spinning spiral (one example beingKaa in Disney'sThe Jungle Book). They are also used as a symbol ofdizziness, where the eyes of a cartoon character, especially inanime andmanga, will turn into spirals to suggest that they are dizzy or dazed. The spiral is also found in structures as small as thedouble helix ofDNA and as large as agalaxy. Due to this frequent natural occurrence, the spiral is the official symbol of theWorld Pantheist Movement.[22]The spiral is also a symbol of thedialectic process and ofDialectical monism.

The spiral is a frequent symbol for spiritual purification, both withinChristianity and beyond (one thinks of the spiral as the neo-Platonist symbol for prayer and contemplation, circling around a subject and ascending at the same time, and as aBuddhist symbol for the gradual process on the Path to Enlightenment). [...] while a helix is repetitive, a spiral expands and thus epitomizes growth – conceptuallyad infinitum.[23]

In art

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The spiral has inspired artists throughout the ages. Among the most famous of spiral-inspired art isRobert Smithson'searthwork, "Spiral Jetty", at theGreat Salt Lake in Utah.[24] The spiral theme is also present in David Wood's Spiral Resonance Field at theBalloon Museum in Albuquerque, as well as in the critically acclaimedNine Inch Nails 1994 concept albumThe Downward Spiral. The Spiral is also a prominent theme in the animeGurren Lagann, where it represents a philosophy and way of life. It also central in Mario Merz and Andy Goldsworthy's work. The spiral is the central theme of the horror mangaUzumaki byJunji Ito, where a small coastal town is afflicted by a curse involving spirals.

See also

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References

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  1. ^"Spiral | mathematics".Encyclopedia Britannica. Retrieved2020-10-08.
  2. ^"Spiral Definition (Illustrated Mathematics Dictionary)".www.mathsisfun.com. Retrieved2020-10-08.
  3. ^"spiral.htm".www.math.tamu.edu. Archived fromthe original on 2022-01-23. Retrieved2020-10-08.
  4. ^"Math Patterns in Nature".The Franklin Institute. 2017-06-01. Archived fromthe original on 2021-03-12. Retrieved2020-10-08.
  5. ^Weisstein, Eric W."Archimedean Spiral".mathworld.wolfram.com. Retrieved2020-10-08.
  6. ^Weisstein, Eric W."Hyperbolic Spiral".mathworld.wolfram.com. Retrieved2020-10-08.
  7. ^ab"Spiral,American Heritage Dictionary of the English Language, Houghton Mifflin Company, Fourth Edition, 2009.
  8. ^von Seggern, D.H. (1994).Practical Handbook of Curve Design and Generation. Taylor & Francis. p. 241.ISBN 978-0-8493-8916-0. Retrieved2022-03-03.
  9. ^"Slinky -- from Wolfram MathWorld".Wolfram MathWorld. 2002-09-13. Retrieved2022-03-03.
  10. ^Ugajin, R.; Ishimoto, C.; Kuroki, Y.; Hirata, S.; Watanabe, S. (2001). "Statistical analysis of a multiply-twisted helix".Physica A: Statistical Mechanics and Its Applications.292 (1–4). Elsevier BV:437–451.Bibcode:2001PhyA..292..437U.doi:10.1016/s0378-4371(00)00572-0.ISSN 0378-4371.
  11. ^Kuno Fladt:Analytische Geometrie spezieller Flächen und Raumkurven, Springer-Verlag, 2013,ISBN 3322853659, 9783322853653, S. 132
  12. ^Thompson, D'Arcy (1942) [1917].On Growth and Form. Cambridge : University Press ; New York : Macmillan. pp. 748–933.
  13. ^Ben Sparks."Geogebra: Sunflowers are Irrationally Pretty".
  14. ^Prusinkiewicz, Przemyslaw;Lindenmayer, Aristid (1990).The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107.ISBN 978-0-387-97297-8.
  15. ^Anthony Murphy and Richard Moore,Island of the Setting Sun: In Search of Ireland's Ancient Astronomers, 2nd ed., Dublin: The Liffey Press, 2008, pp. 168-169
  16. ^"Newgrange Ireland - Megalithic Passage Tomb - World Heritage Site". Knowth.com. 2007-12-21.Archived from the original on 2013-07-26. Retrieved2013-08-16.
  17. ^For example, the trislele onAchilles' round shield on an Attic late sixth-centuryhydria at theBoston Museum of Fine Arts, illustrated in John Boardman, Jasper Griffin and Oswyn Murray,Greece and the Hellenistic World (Oxford History of the Classical World) vol. I (1988), p. 50.
  18. ^"Rock Art Of Latin America & The Caribbean"(PDF). International Council on Monuments & Sites. June 2006. p. 5.Archived(PDF) from the original on 5 January 2014. Retrieved4 January 2014.
  19. ^"Rock Art Of Latin America & The Caribbean"(PDF). International Council on Monuments & Sites. June 2006. p. 99.Archived(PDF) from the original on 5 January 2014. Retrieved4 January 2014.
  20. ^"Rock Art Of Latin America & The Caribbean"(PDF). International Council on Monuments & Sites. June 2006. p. 17.Archived(PDF) from the original on 5 January 2014. Retrieved4 January 2014.
  21. ^Jarus, Owen (14 August 2012)."Nazca Lines: Mysterious Geoglyphs in Peru". LiveScience.Archived from the original on 4 January 2014. Retrieved4 January 2014.
  22. ^Harrison, Paul."Pantheist Art"(PDF). World Pantheist Movement. Retrieved7 June 2012.
  23. ^Bruhn, Siglind (1997). "The Exchange of Natures and the Nature(s) of Time and Silence".Images and Ideas in Modern French Piano Music: The Extra-musical Subtext in Piano Works by Ravel, Debussy, and Messiaen. Aesthetics in music, ISSN 1062-404X, number 6. Stuyvesant, New York: Pendragon Press. p. 353.ISBN 978-0-945193-95-1. Retrieved30 June 2024.
  24. ^Israel, Nico (2015).Spirals : the whirled image in twentieth-century literature and art. New York Columbia University Press. pp. 161–186.ISBN 978-0-231-15302-7.

Related publications

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  • 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.Bibcode:2009NuAlg..51..461D.doi:10.1007/s11075-008-9252-1.S2CID 22532724.
  • Harary, G., Tal, A., 2011.The natural 3D spiral. Computer Graphics Forum 30 (2), 237 – 246[1]Archived 2015-11-22 at theWayback 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.arXiv:0902.4834.doi:10.1016/j.cagd.2009.12.004.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 theWayback 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 theWayback 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.Bibcode:2017RJPCB..11..195T.doi:10.1134/S1990793117010328.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 theWayback 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 theWayback 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].

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