# The Clothoid

#### INTRODUCTION

In transportation infrastructure design the route is defined based on its axis (centreline) – the alignment model. This simplified abstract model is designed in such a way to clearly define the principle course of the infrastructure project.

For most of the transportation means the infrastructure alignment design is split into two main two-dimensional complex strings: the horizontal alignment and the vertical alignment.

The horizontal alignment consist of three types of elements:

• Line segments – straights
• Circular arcs
• Transition curves

The movement of the transportation vehicle over circular elements is generating lateral accelerations, dependant on the speed of the vehicle and on the radius of the curve. In order to avoid the sudden change of these accelerations when passing from a curve to a different curve or a straight, the horizontal alignment includes transition curves. These curves have a continuous variation of curvature, ensuring the smooth transition between the horizontal alignment elements of constant curvature.

The most well-known transition curve is the Clothoid. In its real world application the clothoid enables a car driver to ride smoothly by turning the steering wheel with a constant speed, defining a clothoidal spiral, a continuous and linear curvature variation.

A curve with so many names …

The clothoid equations were first defined by Leonhard Euler; this is why, in general Physics the curve is often called Euler spiral. The French physicists Augustin-Jean Fresnel and, later, Alfred Cornu, rediscovered the curve and defined its parametric equations – hence the curve is sometimes called Fresnel or Cornu spiral. (source of the image: Levien, R. (2008) The Euler Spiral: A mathematical history.)

In 1890, Arthur N. Talbot, Professor of Municipal and Sanitary Engineering at the University of Illinois, defined for civil engineering the “railway transition curve” (Talbot – 1912), with similar equations as Euler did for elasticity and Fresnel and Cornu for optical applications.

The name clothoid was suggested by the Italian mathematician Ernesto Cesàro. The word clothoid comes from klothos, the Greek word for spin (wool) the shape of the curve thread that wraps around the spindle. The same root appears in the name of Clotho (The Spinner), one of the three Fates who holds the thread of human destiny.

#### Clothoid geometry and (some) math

The clothoid equations can be defined starting from the condition of linear relation between radius and length: This defines an infinite spiral, starting from the origin (x=0, y=0, R=∞, L=0) and spinning in two infinite loops to two points where R=0 and L=∞: The constant A is called flatness or homothetic parameter of the clothoid. The clothoid coordinates and the majority of the other characteristic elements of the spiral can be defined based on this essential parameter. For example, the clothoid curvature gradient is 1 in A².

In road design, an alignment developed to have all the transitions with practically the same A is seen as an optimum design, providing similar comfort conditions from the point of view of the variation of the lateral acceleration across all the transitions of that route. Some of the national road design standards are considering this criteria together with superelevation and aesthetic conditions (TAC – 2009).

In railway design an alignment developed with constant A is providing constant rate of change of the Equilibrium Cant (the sum of the rates of change of cant and cant deficiency). If this is done together with a constant E/D ration across all the curves, that alignment will provide practically the same comfort conditions over all the transitions and also similar wear conditions for top of railhead.

The angle α, measured between the tangent in the current point i of the clothoid and the initial direction (where R = ∞),  is called direction angle and is computed dependant on the length of the clothoid arc to the current point L , and the current radius R: where L is the total length of the clothoid and R its final radius.

The Cartesian coordinates of the clothoids can be defined starting from the above equation; using Euler (Fresnel) integrals for sine and cosine, these coordinates are:  Both equations have infinite terms. For railway track and road design, only a part of the clothoid can be practically used – the arc starting from origin to the area where the deflection angle α ≈ 90°.

For this section of the clothoid arc the first 4 terms of the equations provide sufficient accuracy to allow the site tracing of the transition curve.

Before the introduction of the Computerised Aided Design (CAD) in the alignment design for road and railway, the clothoid coordinates were computed based on a set of tables defined for a reference clothoid (usually the one defined by Aₒ = 100) and using the following relation:  A snapshot from Talbot, A. N. (1912) The Railway Transition Curve.

Any other clothoid can be defined based on this property (called homothety relative to the origin O) : Another option largely used was to consider only the first term of the x and y equations, defining in this way the cubic parabola, a close approximation of the clothoid.

#### Clothoid shift

As the clothoid spiral develops, the circle defined by the radius R is gradually shifted away from the x axis. This shift is defined as: The first term of this equation defines the shift for the cubic parabola.
(to be continued)

Later edit: (09/10/2016) The direction angle equation was added based on the blog comments.

##### References:

Cope, G. (1993) British Railway Track – Design, Construction and Maintenance. Permanent Way Institution, Echo Press, Loughborough.

Racanel, I. (1987) Drumuri moderne. Racordari cu clotoida. (Modern roads. Clothoidal transitions). Editura Tehnica, Bucharest. Romania.

Levien, R. (2008) The Euler Spiral: A mathematical historyUniversity of California at Berkeley. (report available online on digitalassets.lib.berkeley.edu)

Radu, C. (2001) Parabola cubica imbunatatita (Improved cubic parabola). Technical University of Civil Engineering Bucharest.

Radu, C., Ciobanu, C. (2004) Elemente referitoare la utilizarea parabolei cubice imbunatatite (Elements related to the use of the improved cubic parabola). Third Romanian National Railway Symposium, Technical University of Civil Engineering Bucharest. Romania.

Talbot, A. N. (1912) The Railway Transition Curve. Fifth Edition. Engineering News Publishing Co. New York. (available online on archive.org)

## 20 thoughts on “The Clothoid”

1. Noel says:

This is the formula We use before in Africa for x & y at any point in the spiral.
R = A ^ 2 / L
T = L / (2 * R) ‘ spiral angle at SC.
X = (1 – T ^ 2 / 10 + T ^ 4 / 216 – T ^ 6 / 9360) * A * Sqr(2 * T)
Y = (1 – T ^ 2 / 14 + T ^ 4 / 440 – T ^ 6 / 25200) * T / 3 * A * Sqr(2 * T)

I created a program before to plot the whole spiral curve with its property and planning to create a new one to plot in screen given two tangent line , Ls1,Ls2 and Rc, and plan to plot in AutoCAD.

We are the same up to 6 to 8 digit for x and y values.

Thanks,

Noel Tecson, PE
Phoenix, AZ

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2. Constantin says:

Hi Noel. Thank you for your comment and for checking this. Actually, the formula you quote is the same as ‘mine’ , for l = L, giving the coordinates x and y at the end of the clothoidal arc. So no surprise the result are the same.
Keep in mind, however, you can’t use the x and y formulas you gave for the rest of the points within the clothoid.

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3. Noel says:

I compare your formula to our formula and the result is totally the same up to the last 14 digit
x = L * (1 – L ^ 2 / (40 * R ^ 2) + L ^ 4 / (3456 * R ^ 4) – L ^ 6 / (599040 * R ^ 6))
y = L ^ 2 / (6 * R) * (1 – L ^ 2 / (56 * R ^ 2) + L ^ 4 / (7040 * R ^ 4) – L ^ 6 / (1612800 * R ^ 6))
so when L=280 and R= 2864.934
x=279.933144545306
y=4.56011833936189

my Ts1=Ts2= 545.049985805895 (Spiral Tangent Length)

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4. John Lacker says:

First, thank you very much. I tried this and it seems to be correct

Thanks
John

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5. Constantin says:

If your entire alignment is offset you have the same final direction angle (alfa) for the old and new clothoid. You will get from the equation for alfa Lnew =Lold x (R+0.02)/R.

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6. John Lacker says:

Hello Again
My question (on the same topic) is:
If I have design values for a clothoid (Radius and Length of the spiral), but I am working on an offset alignment (0.20m), how can I calculate the new value for L when R=R=+/- 0.20m.
Thank You
John Lacker

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7. Constantin says:

I haven’t tried this but I presume the best way to do this is the following (see the image attached/embedded):
– compute the coordinates of the points Si along the spiral – sufficiently dense to allow the following steps.
– compute the distance from the “measured point” M to the spiral points S and find the shortest distance M-S. This will allow you to find Si, the closest spiral point to the “measured point” M.
– compute the perpendicular distance MN between the measured point and the spiral segments defined by the closest spiral point.
– which side of the clothoid is M? … tricky … Try computing the azimuth angle of the clothoid segment Si-Si+1 and compare it to the azimuth angle of Si-M (or MN). That should answer the question. Like

8. John Lacker says:

Hello again. I have another question.

I want to determine the distance between a point and a nearby spiral curve. I also need to know which side of the spiral the point is on. Finally I want to calculate the chainage to the point on the spiral where the distance to the point would be zero. Typically (and hopefully) the distance will be very small, a few centimeters at most. I have attached a drawing to illustrate the problem.

Thank you
John

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9. Constantin says:

Hi John. Thank you for your question. The direction angle – αi in the drawings – can be computed dependant on the length of the clothoid arc to the current point Li, and the current radius Ri. αi =Li/(2Ri). It is also αi=Li²/(2RL) – all these are radians and measured referenced to the straight tangent of the clothoid. I presume you are looking for the radial angle which is perpendicular to the direction angle. That would be αi +/- π/2. Please contact me directly if you need more details about this issue.
I presume you have the length Li and not only the (x,y) coordinates. This Li would be the difference between the chainage of current point i and the the chaiange of the current point of the clothoid.
Please consider this just a theoretical information – check it before applying it on a project.

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10. John Lacker says:

I am a survey engineer working on a tunneling project. How would I find the direction that is perpendicular to the spiral at any point (x,y) on the spiral.
Thank you
John

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