Determination of the position vectors of general helices from intrinsic equations in $\e^3$

Details Determination of the position vectors of general helices from intrinsic equations in $\e^3$ Determination of the position vectors of general helices from intrinsic equations in $\e^3$

2009-04-02

arxiv.org/abs/0904.0301v1

Mathematics

Differential Geometry

10 pages only

Scientific paper

In this paper, we prove that the position vector of every space curve satisfies a vector differential equation of fourth order. Also, we determine the parametric representation of the position vector $\psi=\Big(\psi_1,\psi_2,\psi_3\Big)$ of general helices from the intrinsic equations $\kappa=\kappa(s)$ and $\tau=\tau(s)$ where $\kappa$ and $\tau$ are the curvature and torsion of the space curve $\psi$, respectively. Our result extends some knwown results. Moreover, we give four examples to illustrate how to find the position vector from the intrinsic equations of general helices.

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