The Successor Transformation of Frenet Curves and its Application to Obtain an Explicit Parametrization of the Frenet Apparatus of the Slant Helix
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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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