Challenge Level

Suppose that the distance between the two wheels on a bike is $1$ unit (note that this is a modelling assumption: see the foot of this problem for more details). The bike moves forward and steers from the front. The rear wheel of the bike traces a curve $y = f(x)$ in the plane for some function $f(x)$.

Find an algebraic expression for the path travelled by the front wheel in terms of $x$ and $f(x)$.

**Further numerical exploration**

Use a spreadsheet to plot the path of the front and back wheels when the back wheel follows the paths:

- $\tan(x)$ between $-1.2$ and $1.2$ radians (like switching from one side of the road to another)
- $\frac{1}{x}$ between $0.1$ and $4$ (like turning a corner)
- the arc of a circle (like spiralling down the exit of a multi story carpark)
- $\sin(x)$ between $0$ and $14$ radians (like weaving through bollards)

Examine the form of these curves. Can you identify any common themes? Can you make any conjectures about the curves? Can you prove any of these conjectures?

NOTES AND BACKGROUND

In reality there is not a fixed distance between the points of contact with the ground of the front and back wheels. In a real bicycle the front stem is not vertical and in many bicycles the front wheel is offset from the stem axis so that the point at which the front wheel touches the ground is behind the stem axis extended.

As a result the distance between the points of contact of the two wheels varies as the front wheel is turned and the bike tilts.

You may wish to experiment with different bicycles to see these effects in action and to consider the effects of different cycling tracks on this.

In reality there is not a fixed distance between the points of contact with the ground of the front and back wheels. In a real bicycle the front stem is not vertical and in many bicycles the front wheel is offset from the stem axis so that the point at which the front wheel touches the ground is behind the stem axis extended.

As a result the distance between the points of contact of the two wheels varies as the front wheel is turned and the bike tilts.

You may wish to experiment with different bicycles to see these effects in action and to consider the effects of different cycling tracks on this.