Odometry with one vertical tracking wheel

After reading the ingenious position tracking system on the Pilons document, I would like to implement a similar position tracking system where the robot’s position is tracked over an infinite arc. However, I was wondering how the math would work out if only one vertical tracking wheel was used. Does anyone know how to do the arc tracking if only one vertical tracking wheel is used instead of 2?

Only calculating the orientation of the bot requires 2 parallel tracking wheels. The arcs are calculated with only one. The third wheel perpendicular to the other two is to detect when the robot is pushed off course.