Helix of a curve

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Definition

Let \gamma be a parametrized curve on a manifold M. The helix or spiral of \gamma is a parametrized curve in \R \times M that sends t in the parameter domain to (t,f(t)). In other words, the helix of a curve is the Cartesian product of the curve with the identity-parametrized curve.

Note that the above definition requires a parametrized curve. However, for a Riemannian manifold, any smooth curve in the manifold acquires a natural parametrization by arc-length, and we can thus give a natural choice of helix, or spiral, associated with the curve.

For a closed curve, viz a map S^1 \to M, we construct the helix using the map \R \to M that composes the modulo map with the curve parametrization.