# The Euler spiral: a mathematical history

### Raph Levien

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2008-111

September 2, 2008

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-111.pdf

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. This report is adapted from a chapter of a Ph.D. thesis done under the direction of Prof. C. H. Sequin.

BibTeX citation:

@techreport{Levien:EECS-2008-111, Author = {Levien, Raph}, Title = {The Euler spiral: a mathematical history}, Institution = {EECS Department, University of California, Berkeley}, Year = {2008}, Month = {Sep}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-111.html}, Number = {UCB/EECS-2008-111}, 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. This report is adapted from a chapter of a Ph.D. thesis done under the direction of Prof. C. H. Sequin.} }

EndNote citation:

%0 Report %A Levien, Raph %T The Euler spiral: a mathematical history %I EECS Department, University of California, Berkeley %D 2008 %8 September 2 %@ UCB/EECS-2008-111 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-111.html %F Levien:EECS-2008-111