Nonexistence of rational rotation-minimizing frames on cubic curves Original Research Article

We prove there is no rational rotation-minimizing frame (RMF) along any non-planar regular cubic polynomial curve. Although several schemes have been proposed to generate rational frames that approximate RMFʹs, exact rational RMFʹs have been only observed on certain Pythagorean-hodograph curves of degree seven. Using the Euler–Rodrigues frames naturally defined on Pythagorean-hodograph curves, we characterize the condition for the given curve to allow a rational RMF and rigorously prove its nonexistence in the case of cubic curves.