Editing Hyperbolic spiral (section)

==References==
{{reflist|refs=

<ref name=bernoulli>{{citation
 | last = Bernoulli | first = Johann | author-link = Johann Bernoulli
 | journal = Mémoires de l'Académie des Sciences
 | pages = 521–33
 | title = Extrait de la Réponse de M. Bernoulli à M. Herman, datée de Basle le 7 Octobre 1710
 | year = 1710}}. As cited by {{harvtxt|Guicciardini|1995}}, footnote 47, p. 554.</ref>

<ref name=bowser>{{citation
 | last = Bowser | first = Edward Albert
 | contribution = The reciprocal or hyperbolic spiral
 | contribution-url = https://archive.org/details/anelementarytre09bowsgoog/page/n250
 | edition = 4th
 | page = 232
 | publisher = D. Van Nostrand
 | title = An Elementary Treatise on Analytic Geometry: Embracing Plane Geometry and an Introduction to Geometry of Three Dimensions
 | year = 1882}}</ref>

<ref name=cotes>{{citation|title=Harmonia Mensurarum, Sive Analysis & Synthesis per Rationum & Angulorum Mensuras|location=Cambridge|year=1722|first=Rogerum|last=Cotesium|author-link=Roger Cotes|editor-last=Smith|editor-first=Robertus|language=la}}. For the Cotes spirals, see pp. 30–35; the hyperbolic spiral is case 4, [https://books.google.com/books?id=J6BGAAAAcAAJ&pg=PA34 p. 34]. Hammer dates this material to 1714, but it was not published until after Cotes's death.</ref>

<ref name=curvature>{{citation
 | last = Rutter | first = J.W.
 | contribution = Theorem 7.11
 | contribution-url = https://books.google.com/books?id=yUlZDwAAQBAJ&pg=PA143
 | isbn = 9781482285673
 | page = 143
 | publisher = CRC Press
 | title = Geometry of Curves
 | year = 2018}}</ref>

<ref name=drabek>{{citation
 | last = Drábek | first = Karel
 | editor-last = Rektorys | editor-first = Karel
 | contribution = Plane curves and constructions
 | doi = 10.1007/978-94-015-8308-4_4
 | isbn = 9789401583084
 | pages = 112–166
 | publisher = Springer Netherlands
 | series = Mathematics and Its Applications
 | title = Survey of Applicable Mathematics
 | volume = 280–281
 | year = 1994}}; see p. 138</ref>

<ref name=dunham>{{citation
 | last = Dunham | first = Douglas
 | editor1-last = Barrallo | editor1-first = Javier
 | editor2-last = Friedman | editor2-first = Nathaniel
 | editor3-last = Maldonado | editor3-first = Juan Antonio
 | editor4-last = Martínez-Aroza | editor4-first = José
 | editor5-last = Sarhangi | editor5-first = Reza
 | editor6-last = Séquin | editor6-first = Carlo | editor6-link = Carlo Séquin
 | contribution = Hyperbolic spirals and spiral patterns
 | contribution-url = https://archive.bridgesmathart.org/2003/bridges2003-521.html
 | isbn = 84-930669-1-5
 | location = Granada, Spain
 | pages = 521–528
 | publisher = University of Granada
 | title = Meeting Alhambra, ISAMA-BRIDGES Conference Proceedings
 | year = 2003}}</ref>

<ref name=galaxy1>{{citation
 | last = Kennicutt | first = R. C. Jr.
 | bibcode = 1981AJ.....86.1847K
 | date = December 1981
 | doi = 10.1086/113064
 | journal = [[The Astronomical Journal]]
 | page = 1847
 | publisher = American Astronomical Society
 | title = The shapes of spiral arms along the Hubble sequence
 | volume = 86}}</ref>

<ref name=galaxy2>{{citation
 | last1 = Savchenko | first1 = S. S.
 | last2 = Reshetnikov | first2 = V. P.
 | date = September 2013
 | doi = 10.1093/mnras/stt1627 | doi-access = free
 | issue = 2
 | journal = [[Monthly Notices of the Royal Astronomical Society]]
 | pages = 1074–1083
 | title = Pitch angle variations in spiral galaxies
 | volume = 436| arxiv = 1309.4308
 }}</ref>

<ref name=ganguli>{{citation
 | last = Ganguli | first = Surendramohan
 | contribution = 289: The hyperbolic spiral
 | contribution-url = https://archive.org/details/theoryofplanecur033158mbp/page/n389
 | edition = 2nd
 | pages = 364–365
 | publisher = University of Calcutta
 | title = The Theory of Plane Curves
 | volume = II
 | year = 1926}}</ref>

<ref name=guiccardini>{{citation
 | last = Guicciardini | first = Niccolò
 | doi = 10.1080/00033799500200401
 | issue = 6
 | journal = Annals of Science
 | pages = 537–575
 | title = Johann Bernoulli, John Keill and the inverse problem of central forces
 | volume = 52
 | year = 1995}}</ref>

<ref name=haines>{{citation
 | last = Haines | first = C. R.
 | date = December 1977
 | doi = 10.2307/3617399
 | issue = 418
 | journal = [[The Mathematical Gazette]]
 | jstor = 3617399
 | pages = 262–266
 | title = Old curves in a new setting
 | volume = 61| s2cid = 189050097
 }}</ref>

<ref name=hammer>{{citation
 | last = Hammer | first = Øyvind
 | contribution = 15: The case of the staircase
 | doi = 10.1007/978-3-319-47373-4_15
 | pages = 65–68
 | publisher = Springer International Publishing
 | title = The Perfect Shape: Spiral Stories
 | year = 2016}}</ref>

<ref name=indra>{{citation
 | last1 = Mumford | first1 = David | author1-link = David Mumford
 | last2 = Series | first2 = Caroline | author2-link = Caroline Series
 | last3 = Wright | first3 = David
 | contribution = Inversions and the Riemann sphere
 | contribution-url = https://books.google.com/books?id=OzTzAgAAQBAJ&pg=PA54
 | isbn = 9781107717190
 | mr = 3558870
 | page = 54
 | publisher = Cambridge University Press
 | title = Indra's Pearls: The Vision of Felix Klein
 | title-link = Indra's Pearls (book)
 | year = 2002}}</ref>

<ref name=kepr>{{citation
 | last = Kepr | first = Bořivoj
 | editor-last = Rektorys | editor-first = Karel
 | contribution = Differential geometry
 | doi = 10.1007/978-94-015-8308-4_9
 | isbn = 9789401583084
 | pages = 260–335
 | publisher = Springer Netherlands
 | series = Mathematics and Its Applications
 | title = Survey of Applicable Mathematics
 | volume = 280–281
 | year = 1994}}. For an equivalent formula for the direction angle (the [[complementary angle]] to the pitch angle) see Section 9.9, Theorem 1, p. 300</ref>

<ref name=loria-roever>{{citation
 | last1 = Loria | first1 = Gino
 | last2 = Roever | first2 = W. H.
 | date = February 1919
 | doi = 10.1080/00029890.1919.11998485
 | issue = 2
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2973138
 | pages = 45–53
 | title = On certain constructions of descriptive geometry
 | volume = 26}}; for the central projection of a helix, see p. 51</ref>

<ref name=mactutor>{{citation|url=https://mathshistory.st-andrews.ac.uk/Curves/Hyperbolic/|work=MacTutor History of Mathematics Archive|title=
Curves: Hyperbolic Spiral|publisher=University of St Andrews}}</ref>

<ref name=maxwell>{{citation
 | last = Maxwell | first = James Clerk | author-link = James Clerk Maxwell
 | doi = 10.1017/s008045680002247x
 | issue = 5
 | journal = [[Transactions of the Royal Society of Edinburgh]]
 | pages = 519–540
 | title = XXXV.—On the theory of rolling curves
 | volume = 16
 | year = 1849| id = {{zenodo|2250749}} | url = https://archive.org/details/membershipdirect16asli/page/519/ }}</ref>

<ref name=morris>{{citation
 | editor-last = Morris | editor-first = Christopher G.
 | contribution = Hyperbolic spiral
 | contribution-url = https://books.google.com/books?id=nauWlPTBcjIC&pg=PA1068
 | page = 1068
 | publisher = Academic Press
 | title = Academic Press Dictionary of Science and Technology
 | year = 1992}}</ref>

<ref name=ngc4622>{{citation
 | last1 = Ringermacher | first1 = Harry I.
 | last2 = Mead | first2 = Lawrence R.
 | arxiv = 0908.0892
 | bibcode = 2009MNRAS.397..164R
 | date = July 2009
 | doi = 10.1111/j.1365-2966.2009.14950.x
 | issue = 1
 | journal = [[Monthly Notices of the Royal Astronomical Society]]
 | pages = 164–171
 | title = A new formula describing the scaffold structure of spiral galaxies
 | volume = 397| doi-access = free
 }}</ref>

<ref name=polezhaev>{{citation
 | last = Polezhaev | first = Andrey
 | editor1-last = Tsuji | editor1-first = Kinko
 | editor2-last = Müller | editor2-first = Stefan C.
 | contribution = Spirals, their types and peculiarities
 | doi = 10.1007/978-3-030-05798-5_4
 | isbn = 9783030057985
 | pages = 91–112
 | publisher = Springer International Publishing
 | series = The Frontiers Collection
 | title = Spirals and Vortices: In Culture, Nature, and Science
 | year = 2019| s2cid = 150149152
 }}; see especially Section 2.2, Hyperbolic spiral, p. 96</ref>

<ref name=scott-noland>{{citation
 | last1 = Scott | first1 = Thomas R.
 | last2 = Noland | first2 = J. H.
 | doi = 10.1037/h0022204
 | id = {{ProQuest|614277135}}
 | issue = 5
 | journal = [[Psychological Review]]
 | pages = 344–357
 | title = Some stimulus dimensions of rotating spirals
 | volume = 72
 | year = 1965| pmid = 5318086
 }}</ref>

<ref name=shikin>{{citation
 | last = Shikin | first = Eugene V.
 | contribution = Hyperbolic spiral (reciprocal spiral)
 | contribution-url = https://books.google.com/books?id=H05ZDwAAQBAJ&pg=PA222
 | isbn = 9781498710671
 | pages = 222–223
 | publisher = CRC Press
 | title = Handbook and Atlas of Curves
 | year = 2014}}</ref>

<ref name=varignon>{{citation
 | last = Varignon | first = Pierre | author-link = Pierre Varignon
 | journal = Mémoires de l'Académie des sciences de l'Institut de France
 | pages = 94–103
 | title = Nouvelle formation de Spirales – exemple II
 | url = https://books.google.com/books?id=o73OAAAAMAAJ&pg=RA1-PA94
 | year = 1704}}</ref>

<ref name=volute>{{citation
 | last = Nicholson | first = Peter
 | page = 436
 | publisher = G. B. Whittaker
 | title = A Popular Course of Pure and Mixed Mathematics for the Use of Schools and Students
 | url = https://books.google.com/books?id=Wtc2AAAAMAAJ&pg=PA436
 | year = 1825}}</ref>

<ref name=waud>{{citation
 | last = Waud | first = Samuel Wilkes
 | page = 194
 | publisher = Baldwin and Cradock
 | title = A Treatise on Algebraical Geometry
 | url = https://archive.org/details/treatiseonalgebr00wauduoft/page/194
 | year = 1835}}</ref>

}}