Existence and uniqueness for Legendre curves

We give a moving frame of a Legendre curve (or, a frontal) in the unite tangent bundle and define a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. The existence and uniqueness for Legendre curves are holded like as regular plane curves. It is quite useful to analyse the Legendre curves. As applications, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unite tangent bundle.