Curves in $\mathbb{R} ^3 $

Asked Dec 06 '21 at 07:49

Active Dec 06 '21 at 07:49

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Let $\gamma :I \subset \mathbb{R} \rightarrow \mathbb{R} ^ 3 $an arc length parameterization of a smooth curve C.

If $\tau (s)= 0$ for all $s \in I $, C is contained in a plane?

And if also, the curvature is constant and none zero for all $s \in I $. Is C (or is contained in) a circle?

I don´t know how to prove it, Do I have to use the Frenet-Serret equations?

asked Dec 06 '21 at 07:49

What is $\tau$? – 5xum Dec 06 '21 at 07:49
@5xum Is the unit tangent vector of $\gamma$, and $\tau = \frac{\gamma'}{ | \gamma'' |}$ – Jessica Flores Robles Dec 06 '21 at 08:04
@jessica No, $\tau$ is the torsion. – Milten Dec 06 '21 at 08:17
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Duplicate of A space curve is planar if and only if its torsion is everywhere 0 and https://math.stackexchange.com/q/1505638/620957 – Milten Dec 06 '21 at 08:20

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