On Vectorial Moment of the Darboux Vector

Abstract

In this paper we define a new curve denoted by (c*). It is well known that any regular curve can be written by means of Frenet vectors and also via the vectorial moments. In a space we know a regular curve moves around an instantaneous rotation vector called as the Darboux vector. In this study we are interested in a curve plotted by the vectorial moment of the unit Darboux vector. The curve on which we worked generated by the vectorial moment of the unit Darboux vector satisfying the following condition that the curve is created by the vectorial moment of the unit Darboux vector whose components are of the Frenet vectors of a regular curve in Euclidean 3-space. We use c* to denote the vectorial moment vector of the unit Darboux vector and also c to denote the unit Darboux vector. We show that the new curve (c*) doesn't form a constant width curve pairs with the main curve. Then we calculate the Frenet apparatus of the regular curve (c*), drawn by the vectorial moment vector of c*. Also we point out that this new curve (c*) can be expressed as a linear combination of Frenet vectors. Further we assert that the principle normal and binormal of the curve (c*) doesn't form a constant width curve pairs with the main curve. Finally we draw a conclusion and compute the Frenet apparatus of the curve (c*) when the main curve is supposed to be an helix.

Keywords

• Darboux vector; vectorial moment; constant width curve pair; regular curve

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