From experience from riding a bicycle, one has direct control over the speed of the back tire and the turning angle, the angle between the unit tangent of the front tire track and the frame of the bicycle (or the tangent of the back tire track), which we denote by . For our purposes, the turning angle needs to be signed and satisfy the inequality . This sign requirement is so that we can distinguish between a right hand turn and a left hand turn. We use positive angles to denote a left-hand turn and negative angles to denote a right-hand turn. The restriction that the angles must be less than a right angle implies that

and

have the same sign, and will hold until a singularity develops in the tire tracks.

We can now write the differential equations relating

and

in terms of the these natural controls for a bicycle. This will allow us to create a bicycle track in a natural fashion by specifying how the bicycle is turned and how fast the bicycle is travelling. First, we note that the angle relates the Frenet frames of the front tire and the back tire,

(1)

This gives us

and thus

(2)

We can now write differential equations as

(3)

where we interpret as . We can rewrite (3) as

(4)

These differential equations allow us to create a pair of tire tracks knowing how the bicycle is turned and the speed of the back tire by solving the relevant Frenet frame equations. In particular, these differential equation allow us to construct the back-tire track knowing the front-tire track, as knowing the curvature of the front-tire track we can solve for the angle , which then determines the curvature of the back tire track. The tracks can then be recovered by solving the relevant Frenet frame equations.