A Numerical Algorithm

for Creating a Unicycle Track with a Bicycle

 

To create a unicycle track with a bicycle, we first create an initial track segment  as described in Creating the Initial Track Segment.  Then, we sample the initial track segment and information that describes the initial track segment.  Specifically, we need at each point on the curve the unit tangent vector,  the principal unit normal vectors, and the signed curvature.  From the signed curvature, we can then compute the angle  by  where  is the signed curvature.  We then use the process of pushing the bicycle forward and pushing the bicycle backward to extend the initial track segment to

a unicycle track that can be created by a bicycle.

 

Sampling Curve: Let  be the initial track segment.  To sample this segment, we choose M+1 points, for instance set  with i=0,1,2, ... ,M then define .  We then symbolically (or numerically) calculate the unit tangent vector  and the curvature  for each sampled point . From this information, we can determine the principal unit normal vector  and the angle  at each sample point, by using the equation  and the trick for computing the principal unit normal from the unit tangent vector  if

 

Forward Direction: The data points from the initial segment can be used to generate new data points on the unicycle track  following the method for pushing the bicycle forward.  Following the method for pushing the bicycle forward (for details see Pushing the Bicycle Forward), we get the iterative formulas

(6)

which will produce an extension of the initial track segment in the forward direction,   that is for i>M.

 

Backward Direction The data points from the initial segment can be used to generate new data points on the unicycle track  following the method for pushing the bicycle backward.  Following the method for pushing the bicycle backward (for details see Pushing the Bicycle Backward), we get the iterative formulas

(7)

which will produce an extension of the initial track segment in the backward direction, that is for i<0

 

For more details, on the equations used see the page Deriving the Equations.