  This paper revisits the fundamental mathematics of Taylor series to
approximate curves with function representation and arc-length-based parametric
representation. Parametric representation is shown to preserve its form in
coordinate transformation and parameter shifting. These preservations can
significantly facilitate lane estimation in vehicle control since lanes
perceived by cameras are typically represented in vehicle body-fixed frames
which are translating and rotating. Then we derived the transformation from
function representation to arc-length-based parametric representation and its
inverse. We applied the transformation to lane estimation in vehicle control
problem, and derived the evolution of coefficients for parametric
representation that can be used for prediction. We come up with a procedure to
simulate the whole process with perception, lane estimation and control for the
path-following problem. Simulations are performed to demonstrate the efficacy
of the proposed lane estimation algorithm using parametric representation. The
results indicate that the proposed technique ensures that vehicle control can
achieve reasonably good performance at very low perception updating rate.
