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NOTES ON BERTRAND CURVES

Hiroo Matsuda, Shinsuke Yorozu

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Abstract

Every circular helix in ＄E^{3}＄ is a typical example of Bertrand curve. The circular helix is one in a family of special Frenet curves. We prove that no special Frenet curve in ＄E＄￥ ＄(n>4)＄ is a Bertrand curve. Thus the notion of Bertrand curve stands only on ＄E^{￥overline{2}}＄ and ＄E^{3}＄ . In ＄E^{4}＄ , we can show an idea of a generalization of Bertrand curve.

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Every circular helix in ＄E^{3}＄ is a typical example of Bertrand curve. The circular helix is one in a family of special Frenet curves. We prove that no special Frenet curve in ＄E＄￥ ＄(n>4)＄ is a Bertrand curve. Thus the notion of Bertrand curve stands only on ＄E^{￥overline{2}}＄ and ＄E^{3}＄ . In ＄E^{4}＄ , we can show an idea of a generalization of Bertrand curve.

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Available abstract

Every circular helix in ＄E^{3}＄ is a typical example of Bertrand curve. The circular helix is one in a family of special Frenet curves. We prove that no special Frenet curve in ＄E＄￥ ＄(n>4)＄ is a Bertrand curve. Thus the notion of Bertrand curve stands only on ＄E^{￥overline{2}}＄ and ＄E^{3}＄ . In ＄E^{4}＄ , we can show an idea of a generalization of Bertrand curve.

Key concepts: Bertrand competition, Mathematics, Economics, Mathematical economics, Oligopoly, Cournot competition

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