\chapter{History of the Euler spiral}
\label{hist-euler-chapter}
This chapter traces the history of the Euler spiral, a beautiful and
useful curve known by several other names, including ``clothoid,'' and
``Cornu spiral.'' The underlying mathematical equation is also most
commonly known as the Fresnel integral. The profusion of names
reflects the fact that the curve has been discovered several different
times, each for a completely different application: first, as a
particular problem in the theory of elastic springs; second, as a
graphical computation technique for light diffraction patterns; and
third, as a railway transition spiral. Through all the names, 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.
The Euler spiral is defined as the curve in which the curvature
increases linearly with arc length. Changing the constant of
proportionality merely scales the entire curve. Considering curvature
as a signed quantity, it forms a double spiral with odd symmetry, a
single inflection point at the center, as shown in Figure
\ref{euler-spiral-fig}. According to Alfred Gray, it is ``one of the
most elegant of all plane curves.'' \cite{Gray}
\section{James Bernoulli poses a problem of elasticity -- 1694}
The first appearance of the Euler spiral is as a problem of
elasticity, posed by James Bernoulli in the same 1694 publication as his
solution to a related problem, that of the elastica.
The elastica is the shape defined by an initially straight band of
thin elastic material (such as spring metal) when placed under load at
its endpoints. The Euler spiral can be defined as something of the
inverse problem; the shape of a pre-curved spring, so that when placed
under load at one endpoint, it assumes a straight line.
\begin{figure*}[ht]
\begin{center}
\includegraphics[scale=0.8]{figs/euler_elastic.pdf}
\caption{\label{euler-elastic-fig}Euler's spiral as an elasticity problem.}
\end{center}
\end{figure*}
The problem is shown graphically in Figure
\ref{euler-elastic-fig}. When the curve is straightened out, the
moment at any point is equal to the force $F$ times the distance $s$ from
the force. The curvature at the point in the original curve is
proportional to the moment (according to elementary elasticity
theory). Because the elastic band is assumed not to stretch, the
distance from the force is equal to the arc length. Thus, curvature is
proportional to arc length, the definition of the Euler spiral.
James Bernoulli, at the end of his monumental 1694 \emph{Curvatura
Laminae Elasticae} (see Section \ref{bernoulli-curvatura-sec}),
presenting the solution to the problem of the elastica, sets out a
number of other problems he feels may be addressed by the techniques
set forth in that paper, for example, cases where the elastica isn't
of uniform density or thickness.
%In excess now and retain common extension hypothesis
%speculating our move forward further, search out what kind
%of curve comes out if elastic lamina individual weight hang load
%bend: if bend by both directions at the same time: if not be
%uniform density or thickness
%...
%nor not what kind lamina curved have own, to by hang load, or
%individual weight by both sides at the same time right extend
A single sentence among a list of many, poses the mechanics problem
whose solution is the Euler spiral: To find the curvature a lamina
should have in order to be straightened out horizontally by a weight
at one end\footnote{``Nec non qualem Lamina curvaturam habere debeat, ut
ab appenso onere, vel proprio pondere, vel ab utroque simul in
rectam extedatur.'' The translation here is due to Raymond Clare
Archibald \cite{amm1917}.}.
\begin{figure*}
\begin{center}
\includegraphics[scale=1]{figs/fig28.png}
\caption{\label{bernoulli-fig28}Bernoulli's construction.}
\end{center}
\end{figure*}
The same year, Bernoulli wrote a note containing the integral\footnote{The
1694 original (Latin title ``Invenire curvam, quae ab appenso
pondere flectitur in rectam; h.e. construere curvam aa = sz'' is
No. CCXVI of Jacob Bernoulli's \emph{Thoughts, notes, and
remarks}. An expanded version was published in slightly expanded
form as No. XX of his ``Varia Posthuma,'' which were collected in
volume 2 of his \emph{Opera}, published in 1744 (and available
online through Google
Books) \cite[pp. 1084--1086]{BernoulliOpera2}. Both works are also
scheduled to be be published in Volume 6 of \emph{Die Werke von
Jacob Bernoulli.}}
entitled ``To find the curve which an attached weight bends into a
straight line; that is, to construct the curve $a^2 = sR$''.
\begin{quote}
Quia nominatis abscissa $= x$, applicata $= y$, arcu curv\ae\ $s$, \&
posita $ds$ constante, radius circuli oscularis, curvedini reciproce
proportionalis, est $dxds:-ddy$; habebitur, ex hypothesi, h\ae c \ae
quatio $-aaddy = sdsdx$. ...
\end{quote}
Translated into English (and with slightly modernized notation):
\begin{quote}
Let us call the abscissa $x$, the ordinate $y$, the arc length $s$, and
hold $ds$ constant. Then, the radius of the osculating circle, which
is proportional to the reciprocal of the moment, is
$dx\,ds/d^2y$. Thus, we have, by hypothesis, the equation $-a^2\,d^2y =
sds\, dx$.
\end{quote}
The remainder of the note is a geometric construction of the
curve. According to Truesdell \cite[p. 109]{Truesdell60}, ``it is not
enlightening, as it does not reveal that the curve is a spiral, nor is
this indicated by his figure.'' Nonetheless, it is most definitely the
equation for the Euler spiral. This construction is illustrated in
Figure \ref{bernoulli-fig28}. The curve of interest is $ET$, and the
others are simply scaffolding from the construction. However, it is
not clear from the figure that Bernoulli truly grasped the shape of
the curve. Perhaps it is simply the fault the draftsman, but the curve
$ET$ is barely distinguishable from a circular arc.
In summary, Bernoulli had written the equation for the curve, but did
not draw its true shape, did not compute any values numerically, and
did not publish his reasoning for \emph{why} the equation was
correct. His central insight was that curvature is additive; more
specifically, the curvature of an elastic band under a moment force is
its curvature in an unstressed state plus the product of the moment
and a coefficient of elasticity. But he never properly published this
insight. In editing his work for publication in 1744, his nephew
Nicholas I Bernoulli wrote about the equation $s = -a^2/R$, ``I have
not found this identity established'' \cite[p. 108]{Truesdell60}.
%Original title: Invenire curvam, quae ab appenso pondere
% flectitur in rectam; h.e. construere curvam aa = sz. No. CCXVI of
% \emph{Thoughts, notes, and remarks}, published in slightly expanded
% form as No. XX, pp. 1084--1086, of the ``Varia Posthuma,'' Opera
% \textbf{2}.\cite[pp. 1084--1086]{BernoulliOpera2}}
\section{Euler characterizes the curve -- 1744}
\begin{figure*}
\begin{center}
\includegraphics[width=2.5in]{figs/fig17.png}
\caption{\label{euler-fig17}Euler's drawing of his spiral, from Tabula V
of the Additamentum.}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics{figs/watchspring.pdf}
\caption{\label{euler-fig17-recon}Reconstruction of Euler's Fig. 17, with complete spiral superimposed.}
\end{center}
\end{figure*}
The passage introducing the Euler spiral appears in section 51 of the
Additamentum\footnote{Additamentum 1 to \emph{Methodus inveniendi
lineas curvas maximi minimive proprietate gaudentes, sive solutio
problematis isoperimetrici lattissimo sensu accepti}
\cite{Euler1744}. The quotes here are based on Oldfather's 1933
translation \cite{Oldfather33}, with additional monkeying by the
author.}, referring to his Fig. 17, which is reproduced here as
Figure \ref{euler-fig17}:
\begin{quote}
51. Hence the figure $amB$, which the lamina must have in its
natural state, can be determined, so that by the force $P$, acting in
the direction $AP$, it can be unfolded into the straight line $AMB$. For
letting $AM = s$, the moment of the force acting at the point $M$ will
equal $Ps$, and the radius of curvature at $M$ will be infinite by
hypothesis, or $1/R = 0$. Now the arc $am$ in its natural state being
equal to $s$, and the radius of curvature at $m$ being taken as $r$,
because this curve is convex to the axis $AB$, the quantity $r$ must
be made negative. Hence $Ps = Ekk/r$, or $rs = aa$, which is the
equation of the curve $amB$.
\hfill \cite[\S 51]{Euler1744}
% TODO: this language seems nice, shouldn't it be included?
%Consider an elastic spring freely coiled up in the form of a
%spiral. Let us suppose that the interior extremity is fixed and that
%the spring can be developed into a horizontal position by a weight $p$
%suspended at the other extremity. Under these conditions the action of
%the weight on an element of the spring placed at a distance $s$ from the
%extremity is $ps$; and the elasticity of the element preserves it in
%equilibrium. This elasticity is ``the reciprocal of the osculating
%radius of the spring in its unrestricted state.''
%\hfill (\cite{amm1917}, loosely translating the passage from
%\cite[p. 276]{Euler1744})
\end{quote}
In modern terms (and illustrated by the modern reconstruction of
Euler's drawing, Figure \ref{euler-fig17-recon}), the lamina in this
case is not straight in its natural (unstressed) state, as is the case
in his main investigation of the elastica, but begins with the shape
$amB$. At point $B$, the curve is held so the tangent is horizontal
(i.e. point $B$ is fixed into the wall), and a weight $P$ is suspended
from the other end of the lamina, pulling that endpoint down from
point $A$ to point $a$, and overall flattening out the curve of the
lamina.
The problem posed is this: what shape must the lamina $amB$ take so
that it is flattened into an exactly straight line when the free end
is pulled down by weight $P$? The answer derived by Euler appeals to
the simple theory of moments: the moment at any point $M$ along the
(straightened) lamina is the force $P$ times the distance $s$ from $A$
to $M$.
The curvature of the curve resulting from the original shape stressed
by force $P$ is equal to the original curvature plus the moment $Ps$
divided by the lamina's stiffness $Ek^2$. Since this resulting curve
must be a straight line with curvature zero, the solution for the
curvature of the original curve is $\kappa = -Ps(Ek^2)^{-1}$. Euler
flips the sign for the curvature and groups all the force and
elasticity constants into one constant $a$ for convenience, yielding
$1/r = \kappa = s/aa$.
From this intrinsic equation, Euler derives the curve's quadrature
($x$ and $y$ as a function of the arc length parameter $s$), giving
equations for the modern expression of the curve
(formatting in this case preserved from the original):
\begin{equation}
\mbox{$x=\int ds\;\mbox{sin.}\;\frac{ss}{2aa},\ \&\ y = \int
ds\;\mbox{cos.}\;\frac{ss}{2aa}$}
\label{euler-spiral-eq}
\end{equation}
Euler then goes on to describe several properties of the curve,
particularly, ``Now from the fact that the radius of curvature
continuously decreases the greater the arc $am = s$ is taken, it is
manifest that the curve cannot become infinite, even if the arc $s$ is
taken infinite. Therefore the curve will belong to the class of
spirals, in such a way that after an infinite number of windings it
will roll up a certain definite point as a center, which point
seems very difficult to find from this construction.'' \cite[\S
52]{Euler1744}
Euler did give a series expansion for the above integral, but in the
1744 publication is not able to analytically determine the coordinates
of this limit point, saying
%``Therefore we must admit that
%analysis will make no small gain should anyone find a method whereby,
%approximately at least, the value of this integral would be determined
%in the case that $s$ is taken to be infinite. This problem does not seem
%to be unworthy the best strength of geometers.'' (again, translation
%from \cite{amm1917}).
``Therefore analysis should gain no small advance, if someone
were to discover a method to assign a value, even if only
approximate, to the integrals [of Equation \ref{euler-spiral-eq}], in
the case where $s$ is infinite; this problem does not seem unworthy
for geometers to exercise their strength.''
Euler also derived a series expansion for the integrals \cite[\S
53]{Euler1744}, which still remains a viable method for computing
them for reasonably small $s$ (here $b = \sqrt{2}a$, chosen for
notational convenience).
\begin{equation}
\begin{gathered}
\label{euler-series}
x = \frac{s^3}{1\cdot 3\;b^2} - \frac{s^7}{1\cdot 2 \cdot 3 \cdot 7\; b^6}
+ \frac{s^{11}}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 11\; b^{10}} -
\frac{s^{15}}{1\cdot 2\cdot ... \cdot 7\cdot 15\; b^{14}} + \&c. \\
y = s - \frac{s^5}{1\cdot 2 \cdot 5\; b^4}
+ \frac{s^9}{1\cdot 2\cdot 3\cdot 4\cdot 9\; b^8} -
\frac{s^{13}}{1\cdot 2\cdot ... \cdot 6\cdot 13\; b^{12}} + \&c. \\
\end{gathered}
\end{equation}
\section{Euler finds the limits -- 1781}
It took him about thirty-eight years to solve the problem of the
integral's limits. In his 1781 ``On the values of integrals extended
from the variable term $x = 0$ up to $x = \infty$''\footnote{``De
valoribus integralium a terminus variabilis $x = 0$ usque ad $x =
\infty$ extensorum'', presented to the Academy at Petrograd on April
30, 1781, E675 in the Ennestr\"om index. Jordan Bell has recently
translated this paper into English \cite{euler-2007}.}, he finally
gave the solution, which he had ``recently found by a happy chance and
in an exceedingly peculiar manner'', of $x = y = \frac{a}{\sqrt
2}\sqrt{\frac{\pi}{2}}$. This limit is marked with a cross in
Figure~\ref{euler-spiral-fig}, as is its mirror-symmetric twin.
\begin{figure*}
\begin{center}
\includegraphics[width=2.5in]{figs/e675_works_fig1.pdf}
\raisebox{0.32in}{\includegraphics[width=2.1in]{figs/e675_spiral.pdf}}
\caption{\label{euler-fig2}The figure from
Euler's ``De valoribus integralium'', with modern reconstruction.}
\end{center}
\end{figure*}
The paper references a ``Fig. 2''. Unfortunately, the original figure
is not easy to track down. The version appearing in Figure
\ref{euler-fig2} is from the 1933 edition of Euler's collected works.
For reference, an accurately plotted reconstruction is shown
alongside.
Euler's technique in finding the limits is to substitute $s^2/2a^2 = v$,
resulting in these equivalences (this part of the derivation had
already been done in his 1744 Additamentum \cite[\S 54]{Euler1744}):
\begin{equation}
\begin{gathered}
\int_0^\infty \sin \frac{x^2}{2a^2}dx = \frac{a}{\sqrt{2}}\int_0^\infty \frac{\sin
v}{\sqrt{v}} dv\:, \\
\int_0^\infty \cos \frac{x^2}{2a^2}dx = \frac{a}{\sqrt{2}}\int_0^\infty \frac{\cos
v}{\sqrt{v}} dv\:.
\end{gathered}
\end{equation}
To solve these integrals, Euler considers the Gamma
function\footnote{Euler published his discovery of the Gamma function
in 1729, as \emph{De progressionibus transcendentibus seu
quarum termini generales algebraice dari nequeunt}, Ennestr\"om
index E19.} (which he calls $\Delta$), defined as:
\begin{equation}
\label{gamma-def}
\Gamma(z) = \int_0^\infty t^{z-1}e^{-t} dt
\end{equation}
Through some manipulation, Euler solves a pair of fairly general
integrals, of which the limit point of the Euler spiral will be a
special case. Assuming $p = r\cos \alpha,\ q = r\sin\alpha$, he
derives
\begin{equation}
\begin{gathered}
\int_0^\infty t^{z-1}e^{-px}\cdot\cos qt\;dt = \frac{\Gamma(z) \cos
z\alpha}{r^z}\:, \\
\int_0^\infty t^{z-1}e^{-px}\cdot\sin qt\;dt = \frac{\Gamma(z) \sin
z\alpha}{r^z}\:.
\end{gathered}
\end{equation}
Euler sets $q=1,\ p = 0,\ z = \tfrac{1}{2}$ and derives the limit of $\frac{a}{\sqrt
2}\sqrt{\frac{\pi}{2}}$, which follows straightforwardly from
the well-known value $\Gamma(\tfrac{1}{2}) = \sqrt{\pi}$.
\section{Relation to the elastica}
\begin{figure*}[ht]
\begin{center}
\includegraphics[scale=0.8]{figs/threecurves.pdf}
\caption{\label{threecurves-fig}Euler's spiral, rectangular elastica,
and cubic parabola.}
\end{center}
\end{figure*}
The Euler spiral can be considered something of a cousin to the
elastica. Both curves were initially described in terms of elasticity
problems. In fact, James Bernoulli was responsible for posing both
problems, and Leonhard Euler described both curves in detail about
fifty years later, in his 1744 Additamentum \cite[p. 276]{Euler1744}.
Both curves also appear together frequently in the literature on
spline curves, starting from Birkhoff and de Boor's 1965 survey of
nonlinear splines \cite{Birkhoff65}, through Mehlum's work on the
Autokon system \cite{Mehlum74}, and Horn's independent derivation of
the rectangular elastica (the ``curve of least energy'') \cite{Horn}.
The close relationship between the curves is also apparent in their
mathematical formulations. The simplest equation of the elastica is
$\kappa = x$, while that of the Euler spiral is $\kappa = s$ (here,
$\kappa$ represents curvature, $x$ is a Cartesian coordinate, and $s$
is the arc length of the curve). This similarity of equation is
reflected in the similarity of shape, especially in the region where
the arc is roughly parallel to the $x$ axis, as can be seen in Figure
\ref{threecurves-fig}, which shows both the rectangular elastica and Euler
spiral, as well as the ``cubic parabola,'' the curve that
results under the very small angle approximation $\kappa \approx y''$.
Similarly, both curves can be expressed in terms of minimizing a
functional. The elastica is the curve that minimizes
\begin{equation}
E[\kappa(s)] = \int \kappa^2\, ds\:.
\end{equation}
The Euler spiral is one of many solutions that minimizes the $L^2$-norm
of the \emph{variation} of curvature (known as the MVC, or minimum
variation curve) \cite{Moreton93}. It is, in fact, the optimal solution
when the curvatures (but not the endpoint angles) are constrained.
\begin{equation}
E[\kappa(s)] = \int \Big (\frac{d\kappa}{ds}\Big )^2\, ds\:.
\end{equation}
It is also the curve that minimizes the $L^\infty$ norm of the
curvature variation, subject to endpoint constraints.
\section{Fresnel on diffraction problems -- 1818}
Around 1818, Augustin Fresnel considered a problem of light
diffracting through a slit, and independently derived integrals
equivalent to those defining the Euler spiral\footnote{A. J. Fresnel, \emph{M\'emoire
sur la diffraction de la lumi\`ere}, Annales de Chimie et de
Physique, t.x., 288 1819. Henry Crew
published a translation into English in 1900 \cite{Fresnel}}. At the time, he seemed
to be unaware of the fact that Euler (and Bernoulli) had already
considered these integrals, or that they were related to a problem of
elastic springs. Later, this correspondence was recognized, as well as
the fact that the curves could be used as a graphical computation
method for diffraction patterns.
\begin{figure*}[ht]
\begin{center}
\includegraphics[scale=0.8]{figs/diffrac.pdf}
\caption{\label{diffrac-fig}Diffraction through a slit.}
\end{center}
\end{figure*}
The following presentation of Fresnel's results loosely follows
Preston's 1901 \emph{The Theory of Light} \cite{Preston1901}, which is
among the earliest English-language accounts of the theory. Another
readable account is Houstoun's 1915 Treatise on Light \cite{Houstoun1915}.
Consider a monochromatic light source diffracted through a slit. Based
on fundamental principles of wave optics, the wavefront emerging from
the slit is the integral of point sources at each point along the
slit, shown in Figure \ref{diffrac-fig} as $s_0$ through
$s_1$. Assuming the wavelength is $\lambda$, the phase $\phi$ of the
light emanating from point $s$ reaching the target on the right is
\begin{equation}
\phi = \frac{2\pi}{\lambda}\sqrt{x^2 + s^2}\:.
\end{equation}
Assuming that $s \ll x$, apply the simplifying approximation
\begin{equation}
\phi \approx \frac{2\pi}{\lambda}\big(x + \tfrac{1}{2}s^2\big)\:.
\end{equation}
Again assuming $s \ll x$, the intensity of the wave can be considered
constant for all $s_0 < s < s_1$. Dropping the term including $x$ (it
represents the phase of the light incident on the target, but doesn't
affect total intensity), and choosing units arbitrarily to
simplify constants, assume $\lambda = \tfrac{1}{2}$, and then the intensity
incident on the target is
\begin{equation}
\label{intensity-eq}
%I = \Big [ \int_{s_0}^{s_1} e^{i\phi(s)} ds \Big ] ^2 =
% \Big [ \int_{s_0}^{s_1} e^{i\frac{\pi}{2}s^2} ds \Big ] ^2
I = \Big [ \int_{s_0}^{s_1} \cos \phi\; ds \Big ] ^2 +
\Big [ \int_{s_0}^{s_1} \sin \phi(s)\; ds \Big ] ^2 =
\Big [ \int_{s_0}^{s_1} \cos{\frac{\pi}{2}s^2} ds \Big ] ^2 +
\Big [ \int_{s_0}^{s_1} \sin{\frac{\pi}{2}s^2} ds \Big ] ^2\:.
\end{equation}
The indefinite integrals needed to compute this intensity are best
known as the \emph{Fresnel integrals}:
\begin{equation}
\begin{gathered}
\label{fresnel-int-def}
S(z) = \int_0^z \sin \Big(\frac{\pi t^2}{2} \Big)dt\:, \\
C(z) = \int_0^z \cos \Big(\frac{\pi t^2}{2} \Big)dt\:.
\end{gathered}
\end{equation}
Choosing $a = 1/\sqrt{\pi}$, these integrals are obviously equivalent
to the formula for the Cartesian coordinates of the Euler spiral,
Equation \ref{euler-spiral-eq}. The choice of scale factor gives a
simpler limit: $S(z) = C(z) = 0.5$ as $z \rightarrow \infty$. Fresnel
gives these limits, but does not justify the result \cite[p. 124]{Fresnel}.
Given these integrals, the formula for intensity, Equation
\ref{intensity-eq}, can be rewritten simply as
\begin{equation}
I = \big(S(s_1) - S(s_0)\big)^2 + \big(C(s_1) - C(s_0)\big)^2\:.
\end{equation}
Fresnel included in his 1818 publication a table of fifty values (with
equally spaced $s$) to four decimal places.
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=4in]{figs/cornu_plot.pdf}
\caption{\label{cornu-plot-fig}Cornu's plot of the Fresnel integrals.}
\end{center}
\end{figure*}
Alfred Marie Cornu plotted the spiral accurately in 1874
\cite{Cornu1874} and proposed its use as a graphical computation
technique for diffraction problems. His main insight is that the
intensity $I$ is simply the square of the Euclidean distance between
the two points on the Euler spiral at arc length $s_0$ and $s_1$. Cornu
observes the same principle as Bernoulli's proposal of the formula for
the integrals: \emph{Le rayon de courbure est en raison inverse de
l'arc} (the radius of curvature is inversely proportional to
arc length), but he, like Fresnel, also seems unaware of Euler's prior
investigation of the integral, or of the curve.
Today, it is common to use complex numbers to obtain a more
concise formulation of the Fresnel integrals, reflecting the intuitive
understanding of the propagation of light as a complex-valued wave
\begin{equation}
\label{fresnel-complex-def}
C(z) + iS(z) = \int_0^z e^{i\frac{\pi}{2} t^2}dt\:. \\
\end{equation}
Even though Euler anticipated the important mathematical results, the
phrase ``spiral of Cornu'' became popular. At the funeral of Alfred
Cornu on April 16, 1902, Henri Poincar\'e had these glowing words:
``Also, when addressing the study of diffraction, he had quickly
replaced an unpleasant multitude of hairy integral formulas with a
single harmonious figure, that the eye follows with pleasure and where
the spirit moves without effort.'' Elaborating further in his sketch
of Cornu in his 1910 \textit{Savants et \'ecrivains}, `` Today,
everyone, to predict the effect of an arbitrary screen on a beam of
light, makes use of the spiral of Cornu.'' (Translations from
the original French are mine.)
% Aussi, quand il aborda l'\etude de la diffraction, il eut bient\^ot fait
% de remplacer cette multitude rebarbative de formule heris\'ees
% d'int\'egrales par une figure unique et harmonieuse, que l'oeil suit
% avec plaisir et o\`u l'esprit se dirige sans effort.
Since apparently two names were not adequate, Ernesto Ces{\`a}ro
around 1886 dubbed the curve
``clothoide'', after Clotho ($K\lambda\omega\theta\acute\omega$), the
youngest of the three Fates of Greek mythology, who spun the threads
of life, winding them around her distaff -- since the curve
spins or twists about its asymptotic points. Today, judging from the
number of documents retrieved by keyword from an Internet search
engine, the term ``clothoid'' is by far the most popular\footnote{As of
25 Aug 2008, Google search reports 17,700 results for ``clothoid'',
5,660 hits for ``Cornu spiral'', 935 hits for ``Euler spiral'', and
17,800 for ``Fresnel integral''.}. However, as Archibald wrote in
1917 \cite{amm1917}, by modern standards of attribution, it is clear
that the proper name for this beautiful curve is the Euler spiral, and
that is the name used here throughout.
\section{Talbot's railway transition spiral -- 1890}
The third completely independent discovery of the Euler spiral is in
the context of designing railway tracks to provide a smooth riding
experience. Over the course of the 19th century, the need for a track
shape with gradually varying curvature became clear. William Rankine,
in his \emph{Manual of Civil Engineering} \cite[p. 651]{Rankine1889},
gives Mr. William Gravatt credit for the first such curve, about 1828 or
1829, based on a sine curve. The elastica makes another appearance, in
a proposal about 1842 by William Froude to use circles for most of the
curve, but ''a curve approximating the elastic curve, for the purpose
of making the change of curvature by degrees.''
Charles Crandall \cite[p. 1]{Crandall1893} gives priority for the
``true transition curve'' to Ellis Holbrook, in the Railroad Gazette,
Dec. 3, 1880. Arthur Talbot was also among the first to approach the
problem mathematically, and derived exactly the same integrals as
Bernoulli and Fresnel before him. His introduction to ``The Railway
Transition Spiral'' \cite{Talbot1899} describes the problem and his
solution articulately:
\begin{quote}
A transition curve, or easement curve, as it is sometimes called, is a
curve of varying radius used to connect circular curves with tangents
for the purpose of avoiding the shock and disagreeable lurch of
trains, due to the instant change of direction and also to the sudden
change from level to inclined track. The primary object of the
transition curve, then, is to effect smooth riding when the train is
entering or leaving a curve.
The generally accepted requirement for a proper transition curve is
that the degree-of-curve shall increase gradually and uniformly from
the point of tangent until the degree of the main curve is reached,
and that the super-elevation\footnote{Super-elevation is the
difference in elevation between the outer and inner rails, banking a
moving train to reduce the lateral acceleration felt by passengers.}
shall increase uniformly from zero at the tangent to the full amount
at the connection with the main curve and yet have at any point the
appropriate super-elevation for the curvature. In addition to this, an
acceptable transition curve must be so simple that the field work may
be easily and rapidly done, and should be so flexible that it may be
adjusted to meet the varied requirements of problems in location and
construction.
Without attempting to show the necessity or the utility of transition
curves, this paper will consider the principles and some of the
applications of one of the best of these curves, the railway
transition spiral.
\emph{The Transition Spiral is a curve whose degree-of-curve increases
directly as the distance along the curve from the point of spiral.}
\end{quote}
Thus, we have yet another concise statement of what Bernoulli and
Euler wrote as $rs = aa$.
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=4in]{figs/transition_spiral.pdf}
\caption{\label{transition-spiral-fig}Talbot's Railway Transition Spiral.}
\end{center}
\end{figure*}
In his introductory figure (here reproduced as Figure
\ref{transition-spiral-fig}), Talbot shows the spiral connecting a
straight section tangent to point $A$ to a circular arc $LH$ (of which
$DLH$ continues the arc, and $C$ is the center of the circle). The
remainder of the paper consists of many tables of values for this
spiral, as well as examples of its application to specific problems.
Talbot derives the basic equations in terms of the angle of direction
(he uses $\Delta$, apparently standing for ``degrees'', having already
used up $\theta$ to represent the angle $BAL$; also, $L$ is arc length,
not to be confused with point $L$ in the figure): $\Delta =
\tfrac{1}{2}a L^2$ \cite[p. 9]{Talbot1899}. He then expresses the $x$
and $y$ coordinates in terms of the simple equations $dy = ds\,\sin
\Delta$ and $dx = ds\,\cos \Delta$, quickly moving to a series
expansion for easy numerical evaluation, then integrating. In
particular, Talbot writes \cite[p.10]{Talbot1899},
\begin{equation}
\begin{gathered}
y = .291\:a\:L^3 - .00000158\: a^3\: L^7 ...\\
x = 100\:L - .000762\:a^2\:L^5 + .0000000027\: a^4\: L^9 ...
\end{gathered}
\end{equation}
Aside from a switch of $x$ and $y$, and a change of constants
representing actual physical units used in railroad engineering, these
equations are effectively identical to those derived in Euler's
1744 Additamentum, see Equation \ref{euler-series}.
Obviously, Talbot was unaware of Euler's original work, or of the
identity of his spiral to the original problem in
elasticity. Similarly, Archibald did not cite any of the railroad work
in his otherwise extremely complete 1917 survey \cite{amm1917}. The
earliest published connection between the railway transition spiral
and the clothoid I could find is in a 1922 book by Arthur Higgins
\cite{Higgins22}, where it is also referred to as the ``Glover's
spiral'', a reference to a 1900 derivation by James Glover
\cite{Glover1900} of results similar to Talbot's, but with
considerably more mathematical notation.
\section{Mathematical properties}
Euler's spiral and the Fresnel integrals that generate it have a
number of interesting mathematical properties. This section presents a
selection.
The Fresnel integrals are closely related to the error function
\cite[p.301]{abramowitz+stegun}, \S 7.3.22:
\begin{equation}
C(z) + iS(z) = \frac{1+i}{2}\mbox{erf}\Big(\frac{\sqrt{\pi}}{2}(1-i)z\Big)
\end{equation}
The Fresnel integrals can also be considered a special case of the
confluent hypergeometric function \cite[p. 301]{abramowitz+stegun}, \S
7.3.25:
\begin{equation}
C(z) + iS(z) = z\;_1F_1\Big(\tfrac{1}{2}; \tfrac{3}{2}; i\frac{\pi}{2}z^2\Big)\:.
\end{equation}
The confluent hypergeometric function is
\begin{equation}
_1F_1(a; b; z) = 1 + \frac{a}{b}z + \frac{a(a+1)}{b(b+1)}\frac{z^2}{2!} + \ldots =
\sum_{k=0}^\infty \frac{(a)_k}{(b)_k}\frac{z^k}{k!}\:,
\end{equation}
\noindent where $(a)_k$ and $(b)_k$ are \emph{Pochhammer symbols},
also known as the ``rising factorial,''
\begin{equation}
(a)_k = \frac{(a + k - 1)!}{(a - 1)!} = \frac{\Gamma(x +
k)}{\Gamma(x)}\:.
\end{equation}
Due to these equivalences, the Fresnel integrals are often considered
part of a larger family of related ``special functions'' containing
the Gamma function, erf, Bessel functions, and others.
\section{Use as an interpolating spline}
\label{euler-interp-sec}
Several systems and publications have proposed the use of the Euler
spiral as an interpolating spline primitive, but for some reason it
has not become popular. Perhaps one reason is that many authors seem
to consider the Euler spiral nothing but an approximation to a
``true'' spline based on the elastica.
\newcommand{\kurgla}{{\sc kurgla}}
An early reference to the Euler spiral for use as an interpolating
spline is Birkhoff and de Boor's 1965 survey of both linear and
nonlinear splines \cite{Birkhoff65}. There, they present the
rectangular elastica as the curve simulating the mechanical spline,
but also point out that an exact simulation of this curve is not
``particularly desirable.'' As an alternative, they suggest that
``they approximate equally well to Hermite interpolation by segments
of Euler's spirals'' \cite[p. 172]{Birkhoff65}, and cite Archibald
\cite{amm1917} as their source.
Probably the earliest actual application of Euler's spiral for splines
was in the Autokon system, developed in the beginning of the
1960's. According to Even Mehlum \cite{Meh94}, early versions of
Autokon used the \kurgla\ 1 algorithm, which was a numerical
approximation to the elastica. Later versions used Euler's spiral,
having derived it as a small curvature approximation to the elastica
equation. (Note that \kurgla\ 1 is based on the elastica, but not
quite the same as a true minimal energy curve (MEC), as the curve
segments can take elastica forms other than the rectangular
elastica). Mehlum writes: ``There is hardly any visible difference
between the curves the two algorithms produce, but \kurgla\ 2 is
preferred because of its better computational economy. There is also a
question of stability in the \kurgla\ 1 version if the total arc
length is not kept under control. This question disappears in the case
of \kurgla\ 2.''
Mehlum published both \kurgla\ algorithms in 1974
\cite{Mehlum74}. It is fairly clear that he considered the Euler
spiral an approximation to true splines based on the elastica. He
writes, ``In Section 5 we make a `mathematical approximation' in
addition to the numerical, which makes the resulting curves of Section
5 slightly different from those of Section 4 [which is based on the
elastica as a primitive]. The difference is, however, not visible in
practical applications.''
The 1974 publication notes that the spline solution with linear
variation of curvature and $G^2$-continuity is a piecewise Cornu spiral,
and that the Fresnel integrals represent the Cartesian coordinates for
this curve, but does not cite sources for these facts.
Mehlum's numerical techniques are based on approximating the Euler
spiral using a sequence of stepped circular arcs of linearly
increasing curvature.
%Around the same time, Nutbourne proposed defining curves as given
%piecewise linear intrinsic functions, and integrating those to produce
%the curve. Nutbourne used the term ``linarc'' to refer to Euler spiral
%segments. ***
Stoer \cite{Stoer82} writes that the Euler spiral spline can be
considered an approximation to the ``true'' problem of finding a
minimal energy spline. His results are broadly similar to Mehlum's, but
he presents his algorithms in considerably more detail, including a
detailed construction of a band-diagonal Jacobian matrix. He also
presents an application of Euler spirals as a smoothing
(approximating), rather than interpolating spline.
Stoer also brings up the point that there may be many discrete
solutions to the interpolating spline problem, each $G^2$ continuous
and piecewise Euler spiral, based on higher winding numbers. From
these, he chooses the solution minimizing the total bending energy as
the ``best'' one.
Later, Coope \cite{Coope93} again presents the ``spiral spline'' as a
good approximation to the minimum energy curve, gives a good Newton
approximation method with band-diagonal matrices for globally solving
the splines, and notes that ``for sensible end conditions and
appropriately calculated chord angles convergence always occurs.'' A
stronger convergence result is posed as an open problem.
\section{Efficient computation}
Throughout the history of the Euler spiral, a major focus of
mathematical investigation is to compute values of the integral
efficiently. Fresnel, in particular, devotes many pages to approximate
formulas for the definite integral when the limits of integration are
close together \cite{Fresnel}, and used these techniques to produce
his table. Many subsequent researchers throughout the 19th century
refined these techniques and the resulting tables, including
Knochenhauer, Cauchy, Gilbert, Peters, Ignatowsky, Lommel and Peters
(see \cite{amm1917} for more detail and citations on these results).
Today, the problem can be considered solved. At least two published
algorithms provide
accurate results in time comparable to that needed to evaluate
ordinary trigonometric functions.
One technique is that of the Cephes library \cite{Moshier1989}, which
uses a highly numerical technique, splitting the function into two
ranges. For values of $s < 1.6$, a simple rational Chebyshev
polynomial gives the answer with high precision. For the section
turning around the limit point, a similar polynomial perturbs this
approximation (valid for $z \gg 1$) into an accurate value,
\begin{equation}
\label{asymptote}
C(z) + iS(z) \approx \frac{1 + i}{2} - \frac{i}{\pi z}e^{i\frac{\pi}{2}z^2}\:.
\end{equation}
Numerical Recipes \cite{NR} uses a similar technique to split the
range. For the section $s < 1.5$ near the inflection point, the series
given by Euler in 1744 (Equation \ref{euler-series}) accurately
computes the function. For the section spiraling around the limit point, the
recipe calls for a continued fraction based on the erf
function. Thanks to equivalences involving $e^{iz^2}$, this solution
also converges to Equation \ref{asymptote} as $z \rightarrow \infty$
\cite[p. 255]{NR}.
Thus, the Fresnel integrals can be considered an ordinary ``special
function'', and the coordinates of the Euler spiral can be efficiently
computed without fear of requiring significant resources.