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Space curves and their duals. (English) Zbl 0882.53004
Let \(f: \mathbb{R}\to E^3\) be a smooth immersion. The dual \(f_*\) of \(f\) is given by \(f_*(t)= f(t) \wedge T(t)\), \(T(t)\) denoting the unit tangent of \(f\) and \(\wedge\) the vector cross product. The authors consider this construction for curves on unit spheres (the origin being at the center of the sphere) and circular cones of apex angle \(\pi/2\) (the origin being at the apex of the cone). [Reviewer’s remark: These are the only surfaces where the dual remains in the surface in general.] Results on the involutionary character of this construction are obtained. Furthermore, the invariance of Little’s regular homotopy classes is investigated for this duality.
53A04 Curves in Euclidean and related spaces
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