Combining Principal Normal Indicatrix Curves and Direction Curves with an Alternative Frame

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Abstract
This work examined the direction curves of principal normal indicatrix curves regarding a regular curve in Euclidean space of three dimension. The {N,C,W} frame was assigned as a new alternative frame for the mentioned curve. Investigating relationships among the main curve and direction curves using such novel alternative frame is an innovative approach. The Frenet members, curvature, torsion, harmonic curvature and geodesic curvature of direction curves were determined. Direction curve characterizations in the forms of the C-slant helix , slant helix and general helix were provided. In the end, the relevant figures for these curves were displayed.

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