Nonexistence of rational rotation-minimizing frames on cubic curves

Authors:
Chang Yong Han
Affiliations:
Department of Applied Mathematics, Kyung Hee University, Yongin-si, Gyeonggi-do 446-701, South Korea
Venue:
Computer Aided Geometric Design
Year:
2008

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Abstract

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 RMF's, exact rational RMF's 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.