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Nonexistence of rational rotation-minimizing frames on cubic curves. (English) Zbl 1172.65327

Summary: 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.

MSC:

65D17 Computer-aided design (modeling of curves and surfaces)
53A04 Curves in Euclidean and related spaces
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