Abstract: Based on the improved criteria for parameterization, a new optimal parameterization with the
help of the piecewise linear transformation is obtained. By taking the nodes as the free variables and the
variation of the parameter speed as the objective function, the simple analytical solution of a system of
equations is presented. The new expressions of the original curve are a series of Bézier curves, which
are different from the expression by the Möbius transformation. The new optimal parameterization is
close to the arc-length parameterization, which is displayed by the numerical example.

Key words: optimal parameterization, arc-length parameter, piecewise linear transformation