Nonexistence of Rational Rotation Minimizing Frames on Quintic Helices

Fatma Şengüler Çiftçi and Gert Vegter


Quintic helices, rotation minimizing frames


Rotation minimizing frames (RMFs) have the property of minimum twist which makes them useful for computer graphics, swept surface or generalised cylinder constructions, motion design and control in computer animation and robotics, streamline visualization, and tool path planning in CAD/CAM. Recent advances in two main research topics, polynomial helices and rational rotation minimizing frame (RRMF) curves, in the theory of Pythagorean-hodograph (PH) curves let us study RRMFs on quintic helices. We first give a condition on a polynomial helix of any degree to have an RRMF, which leads a simplification of rational approximation to RMFs on monotone-helical PH quintics. Then we prove the nonexistence of RRMFs on both general and monotone quintic helices.

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