Offset-rational parametric plane curves Original Research Article

This paper identifies all polynomial and rational parametric plane curves whose offsets are rational. By using complex polynomials to represent parametric curves, we prove that a necessary and sufficient condition for properly parameterized plane curves to be offset-rational is that their hodographs have at most one complex root (with nonzero imaginary part) of odd multiplicity. With this we present explicit formulae for all the offset-rational polynomial curves and rational curves with proper parameterizations. The formulae given by Farouki and Sakkalis, and by Pottmann are shown to be special cases.