Properties of the Tautochrone

A Tautochrone is a curve described parametrically by the equations $x=a(\theta+\mathrm{sin}\theta)$ and $y=a(1-\mathrm{cos}\theta)$. Here is a graph of the curve for $\theta \in (-2\pi, 2\pi)$

The graph of a tautochrone drawn in SAGE math

This curve is not commonly encountered in mathematics, indeed, I only came across this curve for the first time about two months ago – but it has some very interesting properties. The curve itself is quite simple; however, if a particle were to be released from rest on one of the slopes of the tautochrone, then the time taken for the particle to reach the bottom of the slope is independent of its starting position assuming that the only force acting on it is the gravitational force. In other words, if you were to simultaneously set a ball rolling down the slope from the top of the slope and another ball from halfway down the slope then they would both arrive at the bottom of the slope at exactly the same time.

This property can be proved using energy considerations and some basic trigonometric identities to form the differential equation

$$a\dot{\theta}^{2}\mathrm{cos}^{2}\frac{\theta}{2}=g(\mathrm{sin}^{2}\frac{\theta_{0}}{2}-\mathrm{sin}^{2}\frac{\theta}{2})$$

where $g$ is the gravitational force – and then using integration by substitution we find that the time taken for the particle to reach the bottom is $T=\pi\sqrt{\frac{a}{g}}$ which is independent of starting position – you can download my full, detailed proof of this property here – Basic Properties of the Tautochrone. The derivation of the differential equation and then the integration that follows can all be done using techniques from the A-Level maths and further maths courses.

Since we are assuming that the only force acting on the particle is the gravitational force we can assume that all gravitational potential energy lost (remember that the particle will move down the slope and therefore lose gravitational potential energy) will be converted to kinetic energy. This is the starting point of the whole derivation of the above differential equation and although the resulting differential equation is non-linear, we are fortunate that it is nice enough to be able to solve – non-linear differential equations are notoriously difficult to solve and often impossible to solve analytically.

Parametric Curves

A function in mathematics is a very precise thing – one of the conditions is that a function can take a number in the domain to only one other number in the range at a time. For example $y=x^{2}$ will take 2 to 4 but no other number and $y=x^{3}$ will take $-3$ to $-27$ and no other number. This doesn’t mean that a number in the range can only have come from one number, for example $y=x^{2}$ will also take $-2$ to $4$.

We often see graphs of functions such as $y=x^{2}$ and $y=\mathrm{e}^{x}$ and it is a common mistake to think that every graph is the graph of a function – this is not true. For example take the graph of $y=\sqrt{x}$ as shown below

Graph of a curve that is not a function (?)

this graph is not the graph of a function because $x=4$ is taken to both $-2$ and $2$; this is not allowed by the definition of a function.

To resolve this we have to introduce a new independent variable (often $t$) called a parameter and make $x$ and $y$ functions of the parameter $t$. This is like adding a third axis to the usual $x$-$y$ plane – this new axis is the $t$-axis.

Here is a graph of the parametric equations $x=\mathrm{cos}t$ and $y=\mathrm{sin}^{3}t$

2-dimensional graph of a parametric equation created in SAGE Math

This, however, appears not to satisfy the definition of a function since there are several values of $x$ that are taken to two different $y$-values. Lets look a little closer at what is going on. If we add a further axis – the $t$-axis – to the diagram then this is what we see,

A 3-dimensional parametric plot created in SAGE_Math

As you can see from the plot each value of $t$ corresponds to exactly one point in three-dimensional space.

If we look at this plot from a certain angle – in particular, straight down the $t$-axis we see the following

3-dimensional parametric plot created in SAGE Math

Now if we ignore the effects of perspective we see exactly the original graph that we started with – so even though we originally thought that this did not satisfy the conditions to be a function, it actually does. By using the trigonometric identity $\mathrm{sin}^{2}t+\mathrm{cos}^{2}t \equiv 1$ we can show that the Cartesian form of this curve is given by $y^{2}=(1-x^{2})^{3}$ which is much more difficult to deal with than the parametric form of $x=\mathrm{cos}t$ and $y=\mathrm{sin}^{3}t$

Similarly $y=\sqrt{x}$ can be represented by the parametric equations $x=t^{2}$ and $y=t$ and again looking at the three dimensional plot straight down the $t$-axis gives a graph like the one earlier in the post. So sometimes things aren’t always what they first seem. With the help of parametric equations we can produce graphs of some very interesting curves that we can now treat as functions – not only that but parameterisations of curves and paths are critical to being able to do higher level calculus.