# 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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1 vote
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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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I am interested in the following result: "If all the normal planes of a curve pass through a particular point, then the curve is contained in a sphere". My approach: Let $\alpha: I \to \... 1 vote 0 answers 13 views ### What is$W$when$W\times X = X'$for every$X\in \{T, N, B\}$? ($\times$denotes the usual cross product) Let$\alpha: I \to R^3$be a smooth regular curve with non-zero curvature, parametrized by arc length. Given there exists$W: I \to R^3$such that$W\times X ...
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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) ...
1 vote
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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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1 vote
Given the following curve $$\alpha(t)=(t,t^{2},t^{3})$$ I gotta find its Frenet vectors. I know, for example $$N(t)=\frac{\alpha'(t)\times(\alpha''(t)\times\alpha'(t))}{||\alpha'(t)||\cdot||\alpha''(t)... 0 votes 1 answer 26 views ### Computer Algebra System for TNB Frames (How to write in 4 Dimensions)? So I am working through a problem and we are allowed to use a computer algebra system to check our answers. I am trying to input a 4 dimensional curve and get mathematica to display the relevant ... 1 vote 1 answer 127 views ### How did we get from this to this? (TNB Frames and Calculus Question) Edit Fixed a typo: I am looking at a teachers notes and I see the following:$$N = x'' – (x''\cdot T)T = x'' – \frac12 \frac{\mathrm{d}}{\mathrm{d}t} (T•T)T = x’'$$I don't see how (x'' \cdot T)T ... 1 vote 1 answer 99 views ### Verifying Serret-Frenet equations I need to verify the Serret-Frenet equations for  \gamma(t) = (4/5 \cos t, 1-\ \sin t, -3/5 \cos t) That is I need to verify \dot t = \kappa n, \dot n = -\kappa t+ \tau b, \dot b = -\tau n Here ... 4 votes 2 answers 235 views ### Find all functions f(t) such that x = (cost, sint, f(t)) is a plane curve Okay so I have a question How do we find all function f(t) such that x = (cost, sint, f(t)) is a plane curve I know this means the torsion is 0. So I know that we can find the pieces of the TNB needed ... 1 vote 0 answers 78 views ### Let x lie on the surface of a sphere centered at the origin. Prove that (\tau /\kappa) + ( (1/\tau )(1/\kappa)' )' = 0 So this problem is really confusing me. Here is a hint we had: (Hint: since the Frenet frame is a frame, write x = aT + bN + cB and work from there) I had an initial idea based on the hint, but my ... 4 votes 1 answer 99 views ### If acceleration is decomposed into the T and N directions, why can an object leave the plane? I'm reading Thomas' Calculus, and had a question similar to this question, why-is-there-no-b-component-of-acceleration-in-my-multivariable-calculus-class I understand the math part, but cannot quite ... 0 votes 2 answers 53 views ### Can I find b from a = b \times c? So I just started studying the TNB (or Frenet-Serret) frame, where B = T × N. Then my book also goes on to say that T = N × B and N = B × T. Basically, we can find a new valid cross-product equation ... 1 vote 0 answers 26 views ### Sum of reciprocal of curvatures of a curve of constant width and its opposite is constant The question I am trying to answer states that: Given a curve \alpha (s) of constant width R, then \beta (s) = \alpha (s) + R \mathbf{n}(s) is its opposite. Show that,$$\frac{1}{\kappa_\alpha}+\...
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 ...