View source for Engineering:Flat spline

{{short description|Long flexible batten used to produce a fair curve through a set of points}}
[[File:Spline (PSF).png|thumb|right|A spline]]
A '''spline''' consists of a long strip fixed in position at a number of points whose tension creates a smooth curve passing through those points, for the purpose of transferring that curve to another material.<ref name=":0">{{Cite book|last=Stephens|first=William Picard|title=Canoe and Boat Building: A Complete Manual for Amateurs.|url=https://archive.org/details/canoeboatbuildin00stepiala|publisher=Forest and Stream Publishing Company|year=1889|isbn=1360838279}}</ref>

Before computers were used for [[Engineering:Computer-aided design|creating engineering designs]], [[Engineering:Technical drawing|drafting tools]] were employed by designers drawing by hand.<ref>{{cite web
|url = http://www.cs.wisc.edu/~deboor/draftspline.html
|title = A {{sic|nolink=y|draftman's|expected=draftsman's}} spline
|first = Carl
|last = de Boor
|publisher = [[Organization:University of Wisconsin–Madison|University of Wisconsin–Madison]]
|access-date = 2012-02-24}}</ref> To draw curves, especially for [[Engineering:Shipbuilding|shipbuilding]], draftsmen often used long, thin, flexible strips of wood, plastic, or metal called splines (or laths, not to be confused with [[Engineering:Lathe|lathe]]s).<ref name=":0" /> The splines were held in place with lead weights (called ducks because of their [[Biology:Duck|duck]]-like shape). The [[Physics:Elasticity|elasticity]] of the spline material combined with the constraint of the control points, or [[Spline (mathematics)#Definition|knots]], would cause the strip to take the shape that minimized the energy required for bending it between the fixed points, this being the smoothest possible shape.<ref name=":1">{{Cite journal|last=Newsam|first=G. N.|date=1991|title=Some topical variational geometry problems in computer graphics|url=https://projecteuclid.org/euclid.pcma/1416323561|journal=Proceedings of the Centre for Mathematics and Its Applications|language=EN|publisher=Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University|volume=26|page=181}}</ref>

One can recreate an original draftsman's spline device with weights and a length of thin plastic or wood, flexible to bend enough without breaking. Crosses are marked on the paper to designate the knots or control points. The spline is placed on the drafting paper, and weights are attached to the shaft near each knot so that the spline passes through each one. Once adjusted to the satisfaction of the drafter, a line may be traced along the shaft, creating a template for a smooth curve.<ref name=":0" /><ref name=":1" />

== Etymology and history ==
The Oxford English Dictionary finds the first recorded usage in the 18th century in East Anglia, England, and suggests the term spline may be related to ''splinter.''<ref>{{Cite book|last=Fowler, H. W. (Henry Watson), 1858-1933.|title=The concise Oxford dictionary of current English : 1911 first edition|date=2011|publisher=Oxford University Press|others=Fowler, F. G. (Francis George), 1870-1918.|isbn=978-0-19-969612-3|edition=100th Anniversary|location=Oxford|oclc=706025127}}</ref>

Spline devices have been used to designs shapes for pianos, violins, and other wooden instruments. The [[Biography:Wright brothers|Wright brothers]] used one to shape the wings of their aircraft.<ref>{{Cite web|title=Solving geoscience problems with math {{!}} UCAR Center for Science Education|url=https://scied.ucar.edu/students/careers/profiles/doug-nychka|website=scied.ucar.edu|access-date=2020-05-09}}</ref>

== Mathematical splines ==
By 1946, mathematicians had begun to devise mathematical formulae to serve a similar purpose,<ref>{{Cite journal|last=Schoenberg|first=I. J.|date=1946|title=Contributions to the problem of approximation of equidistant data by analytic functions. Part A. On the problem of smoothing or graduation. A first class of analytic approximation formulae|url=https://www.ams.org/qam/1946-04-01/S0033-569X-1946-15914-5/|journal=Quarterly of Applied Mathematics|language=en|volume=4|issue=1|pages=45–99|doi=10.1090/qam/15914|issn=0033-569X|doi-access=free}}</ref> and ultimately created efficient algorithms to find piecewise [[Polynomial|polynomial]] curves, also known as [[Spline (mathematics)|splines]], that go smoothly through designated points. This has led to the widespread use of such functions in [[Engineering:Computer-aided design|computer-aided design]], especially in the surface designs of vehicles, replacing the draftsman's spline.<ref>{{Cite news|last=Grandine|first=Thomas|date=May 2005|title=The Extensive Use of Splines at Boeing|volume=38|work=SIAM News|publisher=Society for Industrial and Applied Mathematics|issue=4|url=https://archive.siam.org/pdf/news/53.pdf|access-date=May 9, 2020}}</ref> [[Biography:Isaac Jacob Schoenberg|I. J. Schoenberg]] gave the spline function its name after its resemblance to the mechanical spline used by draftsmen.<ref>{{cite journal
|first = I. J.
|last = Schoenberg
|title = Spline Functions and the Problem of Graduation
|journal = Proceedings of the National Academy of Sciences of the United States of America
|publisher = National Academy of Sciences
|date = August 19, 1964
|volume = 52
|issue = 4
|pages = 947–950
|doi=10.1073/pnas.52.4.947|pmc = 300377
|pmid=16591233
|bibcode = 1964PNAS...52..947S
|doi-access = free
}}</ref>

== Other curve drawing tools ==
[[File:Krzywik.jpg|thumb|A modern flexible curve|alt=]]A related but distinct device is the "flexible curve", which can be molded by hand and used to design or copy a complex curve. Unlike a spline, the flexible curve does not have significant tension, so it maintains a given shape, instead of minimizing its curvature between point. The equivalent device was known in antiquity as a [[Engineering:Lesbian rule|lesbian rule]].<ref>{{OED|lesbian rule}}</ref> The ancient form was made of lead (sourced on the island of Lesbos; hence the name); while the modern form consists of a lead core enclosed in vinyl or rubber.<ref>{{Cite journal|last1=Rheault|first1=W.|last2=Ferris|first2=S.|last3=Foley|first3=J. A.|last4=Schaffhauser|first4=D.|last5=Smith|first5=R.|date=1989|title=Intertester reliability of the flexible ruler for the cervical spine|journal=The Journal of Orthopaedic and Sports Physical Therapy|volume=10|issue=7|pages=254–256|doi=10.2519/jospt.1989.10.7.254|issn=0190-6011|pmid=18791322}}</ref>

==See also==
* {{annotated link|French curve}}
* {{annotated link|Engineering:Lesbian rule}}
* {{annotated link|Engineering:Technical drawing tool}}
*[[Spline (mathematics)]] – piecewise polynomial curves that smoothly interpolate points

==References==
{{Reflist}}

{{Measuring and alignment tools}}

[[Category:Tools]]
[[Category:Shipbuilding]]
[[Category:Technical drawing tools]]

{{Sourceattribution|Flat spline}}