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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Geometrical meaning scalar triple product $(N,N_1,N_2)$

N is normal vector from Frenet frame. Capital letters are for coefficients of first fundamental form in surface theory, small letters are for the second. Does the isometric invariant $$ \frac{eg-f^2}{\...
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Problems with differential geometry

I have problems with this statement: "Let $\pi: \mathbb{R} \to S^1$ given by $\pi (x) = (\cos(x), \sin(x))$. Let $f\colon[0, l] \to S^1$ a differentiable function where $f(t)=(f_1(t), f_2(t))$ ...
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Curve with a prescribed Frenet frame

Suppose you are given an antisymmetric matrix $X$. Then $A(t)=e^{Xt}$ is a curve of orthonormal matrices, with $A(0)=Id$. Is it possible to construct a curve whose Frenet frame vectors $T,N,B$ are the ...
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Find the tangent, normal and binormal vectors at the point $(1,1,1)

I'm having some trouble with the following question: Let $\alpha:[0,2] \to \mathbb R^3$ with $\alpha(t)=(t,t^2,t)$. Find the tangent, normal and binormal vectors at the point $(1,1,1)$. I first ...
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Arc length parameter relation with fourth derivative

Let non-planar curve $\gamma:I \rightarrow \mathbb{R^{3}}$ with arc length parameter $s$. Find $a,\,b,\,c$ such that $\gamma^{(4)}(s)=a\gamma^\prime(s)+b\gamma^{\prime\prime}(s)+c\gamma^{(3)}(s)$. I ...
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1 vote
1 answer
39 views

Some help with question 8.b, chapter 2.3, O'Neil's Intro to Dif. Geometry

This exercise states the folowing: " For a unit-speed curve $\beta(s) = (x(s),y(s)) \in \mathbb{R}^2$, the unit tangent is T = $\beta' = (x',y')$ as usual, but the unit normal N is defined by a ...
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1 vote
1 answer
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How to derive these general formulas for the Frenet Frame of a curve not parameterized by arc-length?

We may define the Frenet Frame $(T, N, B)$ of a regular curve $\alpha$ as follows: $$T:=\frac{\alpha'}{| \alpha'|}$$ $$N:=\frac{T'}{|T'|}$$ $$B:=T\times N$$ and it can be shown that if $\beta$ is an ...
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Given a curve $\alpha$, how can one derive a formula for the normal vector $N_{\alpha}$?

Let $\alpha:I\rightarrow \mathbb{R}$ be a twice differentiable curve such that $\alpha '(t)$ and $\alpha ''(t)$ are linearly independent for every $t$. Let $$s:=\int_{t_0}^{t}|a'(u)|du$$ and $\beta = \...
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tangent vector and parametrization

Say I use the arc length $s$ for the parametrization of a curve ${\bf r}(s)$. The normalized tangent vector $\hat{\bf t}$ at $s$ is given by $\hat{\bf t}=\frac{d {\bf r}(s)}{ds}$ with $\left|\frac{d {\...
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Which curves have constant torsion?

I know from the Frenet-Serret equations that if a curve $r(s)$ (parametrized by arc-length) has constant torsion $\tau \neq 0$, then $$ b' = \tau n \Rightarrow b \times b' = \tau b \times n = -\tau v \...
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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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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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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 = \...
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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\\...
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1 answer
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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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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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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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2 votes
1 answer
95 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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1 vote
0 answers
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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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$\|\mathbf{r}'\|$ 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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1 answer
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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 ...
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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 ...
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2 votes
1 answer
158 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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0 answers
68 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{\...
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1 answer
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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 ...
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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 ...
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1 vote
1 answer
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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, .....
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1 vote
2 answers
91 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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2 answers
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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] = ...
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1 vote
1 answer
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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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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} = \...
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1 vote
1 answer
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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 ...
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62 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 ...
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3 votes
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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 $(...
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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 $\beta$ ...
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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 ...
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1 answer
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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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1 answer
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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 ...
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1 vote
1 answer
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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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1 vote
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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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0 votes
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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 ...
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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 $(x'' \cdot T)T$ ...
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1 vote
1 answer
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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 ...
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4 votes
2 answers
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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 ...
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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 ...
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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 ...
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2 answers
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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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1 vote
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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}+\...
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2 votes
0 answers
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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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