# The Euler spiral: a mathematical history

• Published in 2008
In the collections
The beautiful Euler spiral, deﬁned by the linear relationship between curvature and arclength, was ﬁrst proposed as a problem of elasticity by James Bernoulli, then solved accurately by Leonhard Euler. Since then, it has been independently reinvented twice, ﬁrst by Augustin Fresnel to compute diffraction of light through a slit, and again by Arthur Talbot to produce an ideal shape for a railway transition curve connecting a straight section with a section of given curvature. Though it has gathered many names throughout its history, the curve retains its aesthetic and mathematical beauty as Euler had clearly visualized. Its equation is related to the Gamma function, the Gauss error function (erf), and is a special case of the conﬂuent hypergeometric function.

## Comment

I love the Euler spiral. I don't know why. Maybe it's because I first learnt of it as the "clothoid", which is an excellent name, or maybe it's because it gives me something to think about when I'm driving.This shortish essay by Raph Levien gives a readable potted history of the spiral's multiple discoveries and applications, illustrated with some lovely sparse diagrams of the sort that maths-illiterate Etsy craftspeople love.

## Other information

journal
Opera
pages
1--14

### BibTeX entry

@article{Levien2008,
title = {The Euler spiral: a mathematical history},
author = {Levien, Raph},
url = {http://raph.levien.com/phd/euler{\_}hist.pdf},
urldate = {2012-04-07},
abstract = {The beautiful Euler spiral, deﬁned by the linear relationship between curvature and arclength, was ﬁrst proposed as a problem of elasticity by James Bernoulli, then solved accurately by Leonhard Euler. Since then, it has been independently reinvented twice, ﬁrst by Augustin Fresnel to compute diffraction of light through a slit, and again by Arthur Talbot to produce an ideal shape for a railway transition curve connecting a straight section with a section of given curvature. Though it has gathered many names throughout its history, the curve retains its aesthetic and mathematical beauty as Euler had clearly visualized. Its equation is related to the Gamma function, the Gauss error function (erf), and is a special case of the conﬂuent hypergeometric function.},
comment = {I love the Euler spiral. I don't know why. Maybe it's because I first learnt of it as the "clothoid", which is an excellent name, or maybe it's because it gives me something to think about when I'm driving.This shortish essay by Raph Levien gives a readable potted history of the spiral's multiple discoveries and applications, illustrated with some lovely sparse diagrams of the sort that maths-illiterate Etsy craftspeople love.},
journal = {Opera},
pages = {1--14},
year = 2008,
collections = {History,Geometry}
}