.. role:: raw-latex(raw)
   :format: latex


.. contents::
   :depth: 3

================
The Euler spiral
================


The first example in the beautiful book
:raw-latex:`\cite[pp.~1~ff.]{havil}` is the *Euler spiral*, which is a
*clothoid*, i. e. a curve with a curvature depending linearly on the arc
length. Clothoids play a significant role in road and railway track
construction. Imagine driving along a straight road, approaching a
curve. If the straight part of the road was directly connected to an arc
of a circle forming the curve, one would have to switch from a steering
angle of zero to a non-zero steering angle instantaneously, which is
impracticable even at rather low speeds. Therefore, a safe curve in a
road has a curvature that starts at zero, rises to a maximum and then
falls back to zero again when the next straight segment of the road is
entered. This can be done by using clothoids and many curves encountered
on highways and railway tracks are based on this type of curve.

Generally, a curve parameterized by some functions :math:`x(t)`,
:math:`y(t)` has a slope

.. math:: m=\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\dot{y}}{\dot{x}},\label{equ_slope_1}
   :label: euler01

an arc length

.. math:: l=\int\limits_{t_0}^{t_1}\sqrt{\dot{x}^2+\dot{y}^2}\,\mathrm{d}t,\label{equ_arc_length_1}
   :label: euler02
   
and a curvature

.. math:: \kappa=\frac{\dot{x}\ddot{y}-\dot{y}\ddot{x}}{\sqrt{\left(\dot{x}^2+\dot{y}^2\right)^3}}.\label{equ_curvature_1}
   :label: euler03
   
Using the general parameterization

.. math::

   \begin{aligned}
     x(t)&=\int\limits_0^T\cos\left(f(t)\right)\,\mathrm{d}t\text{~and}\\
     y(t)&=\int\limits_0^T\sin\left(f(t)\right)\,\mathrm{d}t
    \end{aligned}

yields the following time derivatives:

.. math::

   \begin{aligned}
     \dot{x}&=\cos\left(f(t)\right)&\dot{y}&=\sin\left(f(t)\right)\\
     \ddot{x}&=-\dot{f}\sin\left(f(t)\right)&\ddot{y}&=\dot{f}\cos\left(f(t)\right)
    \end{aligned}

Inserting these into **(1)**, **(2)** and **(3)** yields

.. math::

   \begin{aligned}
     m&=\frac{\sin\left(f(t)\right)}{\cos\left(f(t)\right)}=\tan\left(f(t)\right),\\
     l&=\int\limits_0^T\sqrt{\cos^2\left(f(t)\right)+\sin^2\left(f(t)\right)}\,\mathrm{d}t=\int\limits_0^T\mathrm{d}t=T,\text{~and}\\
     \kappa&=\frac{\dot{f}\left(\cos^2\left(f(t)\right)+\sin^2\left(f(t)\right) \right)}{\sqrt{\left(\cos^2\left(f(t)\right)+\sin^2\left(f(t)\right)\right)^3}}=\dot{f}.
    \end{aligned}

The Euler spiral satisfies :math:`\kappa=t`, i. e. :math:`\dot{f}=t`
yielding :math:`f(t)=\frac{t^2}{2}` from which its parameterization

.. math::

   \begin{aligned}
     x(t)&=\int\limits_0^T\cos\left(\frac{t^2}{2}\right)\,\mathrm{d}t\text{~and}\\
     y(t)&=\int\limits_0^T\sin\left(\frac{t^2}{2}\right)\,\mathrm{d}t
    \end{aligned}

follows. This looks pretty straightforward to implement on an analog
computer but classic function generators for :math:`\sin(\varphi)` and
:math:`\cos(\varphi)` have a limited interval for :math:`\varphi`,
typically :math:`[-\frac{\pi}{2},\frac{\pi}{2}]`. If more than a very
short segment of the Euler spiral is to be generated by an analog
computer, a trick must be employed. The idea is to use a quadrature
generator [1]_ which gets :math:`\dot{\varphi}` instead of
:math:`\varphi` as its input. Such a generator, basically consisting of
two integrators, two multipliers (to introduce :math:`\dot{\varphi}`
into the underlying DEQ) and an inverter.

Figure 1 shows the analog computer program for generating an Euler spiral. The quadrature output
output signals :math:`\cos\left(\frac{t^2}{2}\right)` and
:math:`\sin\left(\frac{t^2}{2}\right)` are fed to integrators yielding
:math:`x(t)` and :math:`y(t)`, while :math:`\dot{\varphi}` is obtained
by another integrator generating a linear ramp from :math:`-1` to
:math:`+1`.

.. list-table::
   :widths: 75 
   :header-rows: 0
   
   * - .. figure:: circuit01.png
   
       
   * - Figure 1: Analog computer program for generating an Euler spiral
   
=========== ================= =============
Coefficient Description       Typical value
=========== ================= =============
P1          :math:`x` scaling :math:`0.6`
P2          :math:`x` shift   :math:`0.87`
P3          :math:`y` scaling :math:`0.6`
P4          :math:`y` shift   :math:`0.75`
P5          ramp slope        :math:`0.1`
=========== ================= =============

Figures 3 and 4 show the actual implementation of the program on
*THE ANALOG THING* and a typical curve
display as seen on an oscilloscope screen. The analog computer was set
to repetitive operation and the operation time was set so that the
output of the integrator yielding :math:`\dot{\varphi}` is a ramp from
:math:`-1` to :math:`+1` which corresponds to roughly :math:`20` ms.

.. list-table::
   :widths: 75 75
   :header-rows: 0

   * - .. figure:: euler_setup.jpg
   		:width: 400
   		
     - .. figure:: euler_result.jpg
   		:width: 400
		
   * - Figure 2: Setup for generating an Euler spiral Figure 3: Typical output of the Euler spiral pro-
       on THE ANALOG THING
     - Figure 3: Typical output of the Euler spiral program

Increasing the operating time slowly allows to show the growth of the
Euler spiral on the oscilloscope screen. Ideally, all integrators are
run with the highest time scale factor :math:`k_0` selected. [2]_


References
==========

[Havil(2019)] Julian Havil, *Curves for the Mathematically Curious – an Anthology of
the Unpredictable, Historical, Beautiful, and Romantic*, Princeton
University Press, 2019

[Ulmann(2020)] Bernd Ulmann, *Analog and Hybrid Computer
Programming*, DeGruyter, 2020

.. [1]
   Cf. :raw-latex:`\cite[pp.~60~ff]{ap2}`.

.. [2]
   On THE ANALOG THING this corresponds to :math:`k_0=10^3`.