On Harmonicity and Differential Equations of a Bertrand Curve
S\"uleyman \c{S}enyurt, Osman \c{C}ak{\i}r
TL;DR
This paper derives new Frenet formulas for Bertrand partner curves, establishes differential equations and conditions for harmonicity, and demonstrates these results using the example of a helix.
Contribution
It introduces novel Frenet formulas and harmonicity conditions for Bertrand partner curves based on curvature relations.
Findings
New Frenet formulas for Bertrand partner curves
Differential equations characterizing harmonicity
Application to the helix example
Abstract
In the present paper, we give new Frenet formulas for the Bertrand partner curve by taking the advantage of relations between curvatures and a curve itself. Then making use of these formulas we write the differential equations and sufficient conditions of harmonicity of the Bertrand partner curve in terms of the main curve. Finally, we exemplify our assertions on the curve helix to see how the formulas we developed work.
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Polynomial and algebraic computation · Algebraic Geometry and Number Theory