# 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).

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### a regular curve with $\kappa(t)>0$ is helix if and only if $\frac{\kappa}{\tau}$ is constant

A curve is said to be helix if its tangent line have a constant angle with a fixed direction. i.e. $\langle T(t),u\rangle$ is constant for some unit vector $u$. I am trying to prove: a regular curve ...
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### Find unit normal vector given speed-, acceleration- and jerk vectors, Calculus III

So we have been given the following: $$\frac{dr}{dt} = (-3,2,0)$$ $$\frac{d^2r}{dt^2} = (0,3,-3)$$ $$\frac{d^3r}{dt^3} = (0,0,1)$$ With the information above, I have found the unit tangent vector by ...
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### Why is this proof of the Frent-Serret formulae wrong?

I'm aware that similar questions have been asked, but none of them appear relevant to my own question. By definition and by simple geometric observations, $N,B,T$ are all mutually perpendicular unit ...
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### Showing that the angle between $\alpha’(s)$ and $\gamma’(s)$ is constant $\forall s\in I$ when $\gamma = \alpha + \lambda \vec{n_\alpha}$ and (...)

Let $\alpha: I\rightarrow \mathbb{R}^3$ be a curve parameterized by arc-length such that $\tau_\alpha(s) \neq 0$, $k_\alpha(s) \neq 0$. We also know that $$\lambda k_\alpha(s) + \mu \tau_\alpha(s) =1$$...
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A curve $\gamma : I \rightarrow \mathbb{R}^n$ is called a curve of general type in $\mathbb{R}^n$ if the first $n-1$ derivatives are linearly independent $\forall t \in I$ A moving (orthonormal) ...
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### Projection of moving point onto static curve and respective velocities / Frenet Coordinates

Consider a curve in 2D $\vec{p}(s)$ parameterized by arclength $s$ and the usual local coordinate system on the curve (Frenet Frame with unit vectors $\vec{n}(s)\perp\vec{t}(s)$, no torsion, curvature ...
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### The curvature of a line defined in terms of s

I have a curve defined in terms of arclength $s$ : $$x=\arctan(s)$$ $$y=\frac{\sqrt{2}}{2}(s^2+1)$$ $$z=s-\arctan(s)$$ So to compute its curvature, I started by writing its first and second ...
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### Can N and B be graphed when undefined?

I was asked to graph T, N, and B on a 3 dimensional graph. In this case, N(0) and B(0) are both undefined. Is there a way to graph them? I will include a picture of my work on the problem so you can ...
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### Prove that the projection of loxodrome helical curves of cone projected on the base is a logarithmic spiral

Show that loxodrome helices on a cone of revolution project on a plane perpendicular to their axes (the base) as logarithmic spirals and then show that the intrinsic equations of these conical ...
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