Cycloids, Clothoids and Other Fun Curves


What happens if you strap an LED to a bike tyre and ride it along at night? Well, something like the picture below.

You might say that this shape looks like just another semi-circle, but there’s more to it than meets the eye. The actual curve that is traced by a point on a rolling circle is called a cycloid, and it has some pretty unique properties. In fact, the cycloid has been of such interest to mathematicians that it is sometimes called the “Helen of Geometers” after the frequent quarrels it sparked in the 17th century.

A cycloid being traced by the locus of a single point on a rolling circle

One of the neat things about a cycloid is that if you were to turn it upside-down and start two marbles rolling from any two points, they will reach the centre at the exact same time. This means the cycloid is called a tautochrone (tauto – equal, chrono – time). In fact, this property was discovered from clocks. The circular arc of an ordinary pendulum is not quite isochronous, an unfortunate property which means that a clock will keep a different time depending on the size of the pendulum’s swing. In 1659, Christiaan Huygens managed to geometrically solve this problem, showing what the solution to the long-standing tautochrone problem was – the cycloid. A mass on a string would wrap around a cycloid on either side as it swung, producing a cycloidical arc thus having equal timings.

A tautochrone pendulum tracing the path of a cycloid

Unfortunately in real life, friction exists (and ruins everything), so the cycloidical solution turned out to be pretty rubbish. But at least we found a cool property of the cycloid, right?

In 1696, Johann Bernoulli posed a challenging problem to his fellow mathematicians:

Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time? 

In other words, what shaped hill lets me roll down it the fastest. In the end, 5 mathematicians managed to solve the problem –  Isaac Newton, Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus, Jakob Bernoulli (Johann’s brother) and Guillaume de l’Hôpital (note that 3 of these people are well-known for their early ventures into calculus). The solution was, of course, the cycloid.



Who likes roller coasters? Have you ever looked at a loop-the-loop and realised that it’s not exactly circular? No? Just me? Well, here’s a picture to prove it:

Could be a circle as drawn by a 5-year-old

This balloon-like shape is used because if a carriage went from straight track to circular track, it would literally be the ride of your life, because you’d be dead afterwards. You can read more about the physics involved here, but basically the sudden jolt experienced at the junction of straight and circular track sections has enough G-force to kill you. The solution to this problem is using a track that produces a constant acceleration as you move throught the field of gravity. Such a track is shaped like a section of the clothoid, a curve whose curvature increases with length.

A clothoid curve

Clothoids are also used in railroads. Back when railroads were first introduced, the engineers of the time decided to keep things simple by designing tracks composed of straight sections and circular arcs. While this got them where they wanted to go, passengers would frequently complain of neck pain and spilt tea (as is the British way) from all of the jolts experienced when transitioning from one track section to another. The solution was to design tracks as sections of the clothoid, so that the track’s curvature changed continuously. Clothoids are still used in highway design, and is the recommended driving line of many racing drivers.



If you’ve ever tried building an archway out of bricks, you probably went with the traditional Roman straight-lines-topped-with-a-semicircle. This is fairly convenient to construct, but you’ll need a bunch of mortar to keep the whole thing from collapsing. In true infomercial style: there has to be a better way!

Well, the solution comes from string. While I wouldn’t suggest building an archway out of string, you’ll find that the shape a string makes when it is fixed at two ends and allowed to hang free in the middle is of interest. This shape is called a catenary, and can be plotted using  y = cosh(x). This shape is generated by an equal force of gravity acting on each point of the string. If you were to freeze the string in place and then flip it over, the force of gravity exerted on each point is just negated, so no forces are unbalanced and the arch is completely self-supporting. In practice, this arch design has been used extensively in architecture, such as the Gateway Arch and the dome of St Paul’s Cathedral. Antoni Gaudi even designed the whole Casa Milà upside-down by suspending chains from a board.

The Gateway Arch


…And More!

Now, this is by no way an exhaustive list of all of the cool curves out there. If you’ve found these interesting, be sure to check out cardioids, lituus curves and Lissajous curves and do some exploring of your own!

1 thought on “Cycloids, Clothoids and Other Fun Curves”

  1. The cycloid is very special, the fact that its own evolute curve is itself a cycloid, a linear transformation of its involute (the original cycloid).


    To be more specific in supplementing this short article, the advantage of the cycloidal pendulum is not merely that its time period is independent of its initial amplitude, but that in the evaluation of its time period by means of the approach of establishing the time period as an integral it does not resolve itself into an elliptic integral.

    A general solution for the time period of the motion of a point mass traveling along a curve from a point A to a point B when applied to that of an ordinary pendulum resolves itself into an elliptic integral, we are unable to resolve it into elementary forms and precursory approximation approaches would be the continual attempt.

    An elementary approximation attempt was, for example, as given by the great Richard Courant in “Introduction to Calculus and Analysis, Vol. 1” on page 411, finds that (with the given method) with an initial amplitude that is less than 10 degrees, there thus yields a relative error that is less than 0.5%.

    For larger initial amplitudes, the approximation method will thus yield a greater relative error.


    A quick look on the Wikipedia page of “Pendulum” will you realize initially given approximations for the time period of the ordinary pendulum, under the subtitle “Period of oscillation”, you will notice that all the solutions are approximations.

    The main point of this comment is, the cycloidal pendulum does not require such approximations.

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