Questions tagged [frenet-frame]

Use this tag for questions on Frenet frames and the Frenet-Serret formulae. Related tags include (differential-geometry), (multivariable-calculus), and (curves).

84 questions
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Using Frenet coordinates as a metric

Having two trajectories, one that is my target and one which is the result of a system trying to follow the target trajectory, I get two slightly different trajectories as a result. The first is my ...
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in 2D dimensional plane, is it problematic to have Frenet-Serret frame with zero curvature?

I have a Frenet-Serret frame moving on a 2-D plane. As of now, I do not care about the binormal vector. So my equations are given by, \begin{align} \dot{T} = v\kappa N \\ \dot{N} = -v\kappa T \end{...
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Frenet- Serret and Darboux frame

I am aware the two frames are different, aside from sharing the same tangent unit vector in their basis. But I was wondering, why or when would one choose to use/work with one other frame and not the ...
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Using local canonical form to approximate a curve (not necessarily unit speed)

Let $\alpha: I \mapsto \mathbb{R^3}$ be a regular curve with non vanishing curvature. Check that if $t_0 \in I$ is fixed, it's possible to choose a cartesian coordinate system and reparameterize the ...
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Proving $W(s) = \alpha(s) + pN(s) - p' \sigma B(s)$ is constant, given that $||W(s) - \alpha(s)|| = R^2$

Let $\alpha: I \mapsto \mathbb{R^3}$ be a regular unit speed curve such that $k(s) > 0$ and $\tau(s) \neq 0, \forall s \in I$. Prove that: a) If $\alpha(I)$ is contained in a sphere $S$ ...
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