On the existence of Bertrand pairs
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In this note we introduce the Bertrand Group for a curve. While there is no mention of this group in it, Saban (Simon Stevin 57:37–45, 1983) contains a statement which would mean that for a simple, closed, twisted curve that group would be trivial. However there appears to be a flaw in the proof given there. Here we are able to show that for a simple, twisted curve that group is either trivial or Z 2. If this latter group can occur we do not know. We also show that for some classes of curves the non-triviality of that group forces the curve to be plane.
KeywordsCurves in Euclidean space Frenet-Serret frame Bertrand mates
Mathematics Subject Classification (2000)53A04 51H30 53A55
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