A helix is a curve that in every point makes a constant angle with a fixed direction, ...

A helix is a curve that in every point makes a constant angle with a fixed direction,

and a slant helix is a curve with a principal normal that has this property.

In this paper, the natural equations of the slant helix are developed and an explicit

parametrization of its Frenet apparatus is given. To this goal, a general method is

presented for transforming a Frenet space curve into a successor curve, which has the

predecessor’s tangent as principal normal. Helices are exactly the successor curves of

plane curves and slant helices are the successor curves of helices. By the same method,

an infinite series of successor curves can be constructed. The new method is not restricted

to curves with positive curvature. To achieve maximal generality, the paper

first gives a generalization of conventional Frenet theory. Bishop frames are examined

along with Frenet frames as a useful tool in the theory of space curves.

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