Derivations of Equations
Recall that the question that we are studying is:
Do there exist any curves 
parameterized with respect to arclength such
that the positions of the front tire and back tire of a bicycle at time 
,
respectively satisfy 
and 
for some positions 
and 
of the curve 
?
We will assume that the bicycle is ridden on a flat surface,
and that the bicycle is not banked into the ground, that is the plane of each
tire meet the ground in a right angle. We suspect that one can create a
unicycle track without these assumptions, but the equations become much more
complicated. In particular, using these
we have that
where L is the length of the bicycle and 
is the unit tangent vector to the back tire 
at time t. Equation (1) arises from two simple observations concerning the
manner in which a bicycle is built. The
first observation is that the back tire is fixed in the frame, meaning that the
back tire and the frame are aligned.
This means that the tangent line to the back tire track at time t
is equal to the secant line through the front and back tire positions at time t,
that is
The second observation is that the frame is rigid, meaning
that the distance between 
the points of contact between the tires and
the ground is a constant independent if
the position of the tires.
Rather than express all the equations for describing the
tracks of a bicycle in terms of the tracks 
and 
,
we describe the motion of the bicycle in terms of the angle 
between the unit tangent vectors of the
tracks, the angle between the planes of the tires. We use this angle because when riding a bicycle, we have direct
control of this angle since by controlling the angle 
and the speed of the bicycle the position of
the bicycle. We require the angle 
be signed and satisfy 
to distinguish between a right hand turn and
a left hand turn. A left-hand turn will
have a positive sign and right-hand turn will have a negative sign, see diagram
below. This convention allows us to
quantify left and right by using the principal unit normal vector N(t)
for the curve, which is given by
If 
is positive then 
lies on the same side of 
as 
,
while if 
is negative then 
lies on the same side of 
as 
,
which from standard convention respectively corresponds to a left hand and a
right hand turn.
A principal use of the angle 
is to convert between the two sets of
orthogonal basis vectors 
and 
. Using trigonometry and basic vector
arithmetic, we have
By differentiating equations (1) to (3) with respect to
time, and using the Frenet frame equations, we derive the main equations needed
to construct a unicycle track with a bicycle.
Differentiating equation (1) respect to time, we get
where 
is the signed curvature of the back tire
track, and 
and 
are the arclength parameters of the curves 
and 
respectively. There are a couple of important conclusions that we can make from
equation (4). Namely, that
and therefore
Equating the expression for 
in (3) and (6), we find that
from which we get
Differentiating the equation for 
in (3) with respect to t, we have
which yields
Using that 
,
we have
and thus implies that
The equations (8), (10), (11) allow us to construct a
bicycle track knowing the angle 
as a function either arclength parameter 
or 
since with this information we can solve the
Frenet frame equations to construct either the back tire track or the front
tire track. We could also solve the
Frenet frame equations knowing the angle 
and the speed 
of the back tire, which are of course the
physical controls of a bicycle.
It is also useful to have the curvature of the front tire in
terms of the curvature for the pact tire.
Differentiating (6) we have
thus
Equations (1), (6), (12) are useful for determining the
conditions needed on a bicycle track for it to be possible to create a unicycle
track with a bicycle. We need every
point on the front tire track to be a point on the back tire track. The equations (1), (6), (12) provide useful
relations between two points on the unicycle track, say 
and 
,
that correspond to 
and 
. These equations say that for a unicycle
track to be created by a bicycle we need
where s is the arclength parameter on 
and 
is the parameter for 
. Repeatedly differentiating (12) with respect
to time or arclength we get conditions specifying the derivatives of the
curvature of 
when 
in terms of the curvature and the derivatives
of the curvature when 
. A cursory examination of these conditions
reveals that they are satisfied by having the curvature and the derivatives of
the curvature identically equal to zero when 
and 
.
To create a unicycle track with a bicycle, we first create a
track segment 
with 
,
,
and the curvature and the derivatives of the curvature identically equal to
zero when 
and 
.
Pushing the bicycle forward on the initial track segment, we can extend the
segment to the interval 
by using the formula
The conditions on the curvature imply that the curvature and
the derivatives of the curvature when 
are identically equal to zero, and therefore
we can extend the unicycle track forward indefinitely. We can also push the bicycle backwards on
the initial unicycle track to extend the track for 
. However, we can only guarantee that the
bicycle can be pushed for a short amount of time and arclength in the backwards
direction at present time, because we generate the backward direction for the
unicycle track by solving a differential difference equation. The differential
difference equation comes from noticing that for a unicycle track that can be
created with a bicycle there exists for every time 
a time 
with 
,
and therefore at every point we have two methods for calculating the
curvature. Thus by equating the
curvature of the back tire track at time 
to the curvature of the front tire track at
time 
,
we have that the curvature of the unicycle track must satisfy
In terms of the angle 
which controls the construction of the
bicycle, this is equivalent to the equation
Given an initial segment, 
,
we can solve these differential difference equations for 
only.
If we attempt to solve for 
we would overwrite the initial segment and
the extension of the initial segment by 
.
