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A new version of Bishop frame and an application to spherical images. (English) Zbl 1207.53003

A classical Bishop frame is a moving frame of a curve in Euclidean three space [R. L. Bishop, Am. Math. Mon. 82, 246–251 (1975; Zbl 0298.53001)] with the following properties. Considering a curve parametrized by arclength $s$, the first vector is the unit tangent vector $T\left(s\right)$, the second and third vector ${M}_{1}\left(s\right),{M}_{2}\left(s\right)$ are selected arbitrarily so that $\left(T\left(s\right),{M}_{1}\left(s\right),{M}_{2}\left(s\right)\right)$ is an orthonormal frame and the (skew symmetric) frame equations read ${T}^{\text{'}}={k}_{1}{M}_{1}+{k}_{2}{M}_{2},{M}_{1}^{\text{'}}=-{k}_{1}T,{M}_{2}^{\text{'}}=-{k}_{2}T$.

In the present paper the authors introduce an other version of a Bishop frame using the binormal vector shared with the Frenet frame. The relations to the Frenet frame and the classical Bishop frame are discussed as well as the spherical images. As applications, general helices and slant helices (regular curves the principal normals of which form a constant angle with a fixed direction) are discussed.

##### MSC:
 53A04 Curves in Euclidean space
##### Keywords:
Bishop frame; general helix; slant helix
##### References:
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