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 Incoming links: Logarithmic conical helix

For a regular space curve parametrized by arc length

s

, its curvature is

κ = ∥ d T / d s ∥

, where

T = d x / d s

is the unit tangent. Differentiating the given parametrization gives

x^{'} (t) = \frac{e ^{t}}{2} (cos t - sin t, sin t + cos t, 2), ∥ x^{'} (t) ∥ = e^{t} .

The curve approaches the origin as

t \to - \infty

; the origin is a limiting endpoint. Its arc length measured from that endpoint is

s = \int_{- \infty}^{t} e^{u} d u = e^{t}

, so

x (s) = (\frac{s}{2} cos lo g s, \frac{s}{2} sin lo g s, \frac{s}{2}), s > 0.

T (s) = \frac{1}{2} (cos lo g s - sin lo g s, sin lo g s + cos lo g s, 2), T^{'} (s) = \frac{1}{2 s} (- sin lo g s - cos lo g s, cos lo g s - sin lo g s, 0) .

Consequently,

κ (s) = \frac{1}{2 s} .

The infinitely many turns near the limiting origin have finite total arc length, while the curvature diverges there.