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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3
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1answer
43 views

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 ...
0
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1answer
31 views

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 ...
5
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1answer
50 views

The Frenet frame is orthogonal

I have proved $P'=AP$ where $$P= \begin{pmatrix} T \\ N \\B \end{pmatrix}$$ $$A= \begin{pmatrix} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \\ \end{...
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0answers
20 views

Area of closed curve $\gamma $ : $ \frac{1}{2} \oint_{\gamma} (\gamma \times T \cdot B ds)$

I want to extend the area formula of the closed curve as follows : First for some curve $C(t) = (x(t), y(t))$. I know by Green's theorem, then the area have the following form : \begin{align}A = \...
0
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1answer
50 views

Can we use the matrix exponential trick to solve the Frenet-Serret frame?

The Frenet-Serret frame has the following property: $$\begin{bmatrix}T'\\N'\\ B'\end{bmatrix}=\begin{bmatrix}0&\kappa&0\\-\kappa&0&\tau\\0&-\tau&0\end{bmatrix}\begin{bmatrix}T\\...
1
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1answer
96 views

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 ...
4
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1answer
47 views

Normal planes and spherical curves

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 \...
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0answers
12 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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0answers
28 views

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 $$\lambda k_\alpha(s) + \mu \tau_\alpha(s) =1$$...
2
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1answer
53 views

Questions about Frenet Frames

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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0answers
48 views

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 ...
0
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1answer
13 views

$\|\mathbf{r}'\|$ in terms of Frenet-Serret Frame variables

On Wikipedia I find the equation: $$\frac{d}{dt} \begin{bmatrix} \mathbf{T}\\ \mathbf{N}\\ \mathbf{B} \end{bmatrix} = \|\mathbf{r}'(t)\| \begin{bmatrix} 0&\kappa&0\\ -\kappa&0&\tau\\ 0&...
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0answers
14 views

Parametrize a curve $\vec{y(t)}= r\cos(wt)i+r\sin(wt)j+hwtk$

I have a curve $\vec{y(t)} = r\cos(wt)i+r\sin(wt)j+hwtk$, $-\infty < t < \infty$ and $w = (r^2+h^2)^{\frac{-1}{2}}$ and $r,h,w$ are all positive... I am trying to work out the Frenet Frame field,...
0
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1answer
31 views

Conditions for Bishop frame along closed curve to be closed

Given a smooth, unit-speed curve $\gamma \colon [0,L] \to \mathbb{R}^{3}$ and a unit vector $v$ in the normal plane $\gamma'(0)^{\perp}$, it is well-known that there is a unique normal parallel vector ...
0
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1answer
46 views

α is a plane curve iff the binormal vector is constant

I've just started studing Differencial Geometry at college and I came across the following exercise "α is a plane curve iff the binormal vector is constant" Would you have any hints for this ...
2
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1answer
74 views

Parametric equations for a curved helix in 3D

I am using a helix to parameterise biological molecules in 3D. Currently I have a script that first fits a straight cylinder to my molecule and then plots a helix. I refine this helix using a least ...
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0answers
28 views

Curvature formula for general regular curve

If we have a curve $\gamma :(a,b)\rightarrow \mathbb{R}^n $. Then we have the same curve but with an arc length parametrisation $\hat{\gamma} $ such that $\gamma =\hat{\gamma } \circ s .$ (So $\hat{\...
1
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1answer
61 views

Proving that two curves are orthogonal

Let $α(s)=(x(s)),y(s))$ be a regular plane curve that is parameterized by arc length, and let $n(s)$ be its normal vector. Consider the family of curves: $β(s,r)=α(s)+rn(s),−ϵ≤r≤ϵ$ I need to prove ...
0
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1answer
74 views

Finding the curvature & torsion of the derivative of a smooth regular curve in $\mathbb{R}^3$

Let $\beta(s)$ be a smooth regular curve in $\mathbb{R}^3$ parameterized by arclength with nowhere vanishing curvature. Let $\gamma(s) = \beta'(s)$. Find $\kappa_\gamma(s)$ and $\tau_\gamma(s)$ in ...
1
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1answer
62 views

Constant orthonormal frame

I'm reading a geometry article, and some doubts arose, and I hope someone can help me. At a certain point in the work, the author says: Choosing a point wise constant local orthonormal frame $\{e_1, .....
1
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2answers
81 views

Why is $d(T \times N)/ds$ orthogonal to the unit tangent vector $T$?

Consider a curve in $\mathbb{R}^3$ and its Frenet-Serret frame $(T,N,B)$. A calculation gives $$\frac d{ds} (T \times N)= T \times \frac {dN}{ds}.$$ Question: I read that this implies that $\frac d{...
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2answers
28 views

What is $\left( \frac{t^2}{t^2+2}i+\frac{2}{t^2+2}j+\frac{2t}{t^2+2}k \right) \times \frac{2ti-2tj-(t^2+2)k}{t^2+2}$?

I did \begin{align} & \left( \frac{t^2}{t^2+2}i+\frac{2}{t^2+2}j+\frac{2t}{t^2+2}k \right) \times \frac{2ti-2tj-(t^2+2)k}{t^2+2} \\[8pt] = {} & \frac{2t^2i-4tj-2t(t^2+2)k}{(t^2+2)^2} \\[8pt] = ...
1
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1answer
57 views

Finding the Frenet frame field of a curve $\vec{C}(t)$ = $(\frac{t^3}{3}, 2t-1,t^2+2)$

Working out the unit tangent $T$ $\vec{r} = \frac{t^3}{3}i + (2t-1)j + (t^2+2)k$ $\frac{dr}{dt} = t^2 +2j+2tk$ $\frac{ds}{dt} = |\frac{dr}{dt}| = \sqrt{t^4+4+2t^2} = t^2+2$ $T = \frac{dr}{ds} = \frac{...
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0answers
20 views

Working out the The Unit Tangent $T$ and the Principal Normal $N$ of a curve $\vec{C}(t)$ = $(\frac{t^3}{3}, 2t-1,t^2+2)$

I am able to work this out. But in the end I get really complicated numbers to work with. This creates an issue when wanting to work out the Principal Normal N. Maybe I'm making a mistake. $\vec{r} = \...
1
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1answer
43 views

Jacobian Determinant of frenet transformation

If anybody can help with this. Given a point $\boldsymbol x = (x,y)$, it can be represented as $\boldsymbol x=\boldsymbol p(s)+r \boldsymbol u(s) $. p is a curve parametrized with arc length s. $r$ is ...
0
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0answers
35 views

Finding curavature and torsion of a curve in terms of its Bishop frame

Suppose we are given a regular, naturally parametrized curve $\gamma :I \rightarrow \mathbb{R}^3$, how does one then compute the curvature and torsion of $\gamma$ in terms of the entries of the Bishop ...
0
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0answers
81 views

Plotting Frenet Frame on Python(Jupyter Notebook)

Can someone help me plot a self explanatory version of the TNB frame on jupyter notebook. It need not be an animation but just has to have a surface with the three orthogonal vectors and how it ...
3
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0answers
173 views

Prove curvature and torsion of these two curves are equal using Frenet frame

I have the following problem. Let $\alpha,\beta:I\rightarrow\mathbb{R}^3$ 2-regular curves (curvature $k\not=0$ for all $t\in I$), arc-length parametrized. Let $(t_\alpha, n_\alpha, b_\alpha)$ and $(...
0
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0answers
113 views

Why is the distance between Bertrand mates constant?

I was reading this question about Bertrand curves and I got a bit confused about some steps of the solution given by the author. In section a) of the problem it's stated that if $\alpha$ and $\beta$ ...
0
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0answers
38 views

Frenet equation applicability

The Frenet - orthogonal basis $(t, n, b)$ at $t0$.But $t,n$ are orthogonal only for unit speed curve. So is there any method for general curves without unit speed reparametrization. I found that in ...
0
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1answer
56 views

Frenet-Serret formula: why is $T$'s magnitude unitary?

Why is $T$'s tangent vector magnitude unitary? $$T=\frac{dr}{ds}$$
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1answer
181 views

Torsion of tangent indicatrix curve

Disclaimer: there are many posts about this problem here, but none with exactly what I'm looking for. I'll state the problem and then link some references. This is ...
1
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1answer
58 views

Showing a curve which lays on a sphere of radius 1 is plane

Assuming $\alpha$ is a unit speed curve, I'm trying to prove that $\alpha$ is plane. By hypothesis, I know its curvature is such that $$\kappa=1$$ I'm trying to use the torsion's formula: $$\tau=\frac{...
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0answers
40 views

Frenet formulas of $\alpha(t)=(t,t^{2},t^{3})$

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)...
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1answer
20 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
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1answer
118 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
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1answer
68 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
2answers
146 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
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0answers
46 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
1answer
76 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
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2answers
46 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 ...
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0answers
22 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}+\...
2
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0answers
44 views

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 ...
0
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0answers
11 views

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 ...
2
votes
1answer
189 views

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 ...
4
votes
1answer
557 views

Frenet-Serret and Vector Fields

The well-known Frenet-Serret equations, $\dot T(s) = \kappa N(s), \tag 1$ $\dot N(s) = -\kappa(s) T(s) + \tau(s) B(s), \tag 2$ $\dot B(s) = -\tau(s) N(s), \tag 3$ where $T = \dot \alpha(s), \tag ...
1
vote
1answer
690 views

Do Carmo 1.5.18 - Bertrand curves

I am trying to solve the excercise 1.5.18 from Do Carmo's curves and surfaces. Although hints are included in the book I have difficulties with the details. Part of the Exercise is: Let $\alpha: I \...
1
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1answer
110 views

Frenet serret formulas vector analysis question

If unit tangent vector $\bar{t}$ and binormal vector $\bar{b}$ make angle $\theta, \phi$ with a constant unit vector $\hat{a}$. So prove that $$\frac{\sin\theta}{\sin\phi}\cdot\frac{d\theta}{d\phi}=-\...
1
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1answer
109 views

Torsion coefficient

We went over the Frenet-Serret formulas today in class and the professor wrote $$d\mathbf{\hat{B}}/dt=-\tau{}\mathbf{\hat{N}}.$$ He said that the coefficient is always negative (so that tau is always ...
3
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0answers
40 views

Does this differential geometry problem have a solution? (It's about Frenet formulas)

I have solved a simple exercise, but in solving it I came to a contradiction. I want to know if it's correct. In the exercise I have a curve parameterized by the arc length such that: $\alpha (0)=\...