What is arc length parameterization?

Arc length parameterization refers to the description of the path of a particle that travels at a constant rate of one unit per second along a curve. It is the simplest representation of a curve with regard to the dynamics of the path of a particle.

Parameterization of a curve by arch length involves finding its arch length s(t), then calculating the inverse of the arch length t(s) and finally setting the new parameters, u = t(s). Both the arch length and its inverse are monotonically increasing functions if the curve is regular. A parameterized curve typically increases in length uniformly.