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).
154
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Parametrizing a self-intersecting tubular surface centred around a space curve to form a Klein bottle?
As part of my math project in school I am trying to derive a parametrization of a klein bottle. I am starting by creating a tubular/canal surface around the space curve p:
$$x(t)=5(1+\sin(t))$$
$$y(t)=...
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Calculating unit vectors B , T , N a (probably ellipse like) curve
$$r(t) = a\cos(t)i+b\sin(t)j+ctk$$
So this is the equation of a curve for which I've had trouble calculating it's unit vectors, especially
$B$ and $N$.
I know that $T = v/|v|$ and that also $B = v × a ...
2
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2
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Curves And Surfaces Exercise (6) of Chapter 1
I am currently reading the book "CURVES AND SURFACES" [ SECOND EDITION 2009 ] by Ros and Montiel.
I am having trouble to solve and understand Exercise (6) of Chapter 1.
It says the following ...
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1
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Properties of Mannheim pairs
Two regular curves $\alpha$ and $\beta$ are considered a Mannheim pair when $N_{\beta} = \pm B_{\alpha}$ and there is a differentiable function $\lambda: I \rightarrow \mathbb{R}$ in order that $$\...
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Riccati equation from strange Frenet–Serret system
If $\kappa(\mathrm{s})$ and $\tau(\mathrm{s})$ are continuous functions, then we can apply to the system of three simultaneous differential equations of first order in $\alpha, \beta, \gamma$.
$$
\...
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Determinant of a Frenet curve
I have this problem:
Let $c$ be a Frenet curve in $\mathbb{R}^n$. Show that
$
\operatorname{Det}\left(c^{\prime}, c^{\prime \prime}, \ldots, c^{(n)}\right)=\prod_{i=1}^{n-1}\left(\kappa_i\right)^{n-i}...
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For a unit-speed curve $\beta$ with unit tangent vector field $T_\beta$, we define the spherical image $\sigma(s)=T_\beta (s)$. Prove that:
For a unit-speed curve $\beta$ with unit tangent vector field $T_\beta$, we define the spherical image $\sigma$ With $\sigma(s)=T_\beta (s)$.
How can I prove that $$\kappa_\sigma=\sqrt{1+(\frac{\tau_\...
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$\beta:I\subset\mathbb{R}\to\mathbb{E}^3$ an arc length parameterized curve, how can I prove following statements:
We know that:
$$\beta:I\subset\mathbb{R}\to\mathbb{E}^3$$ is an arc length parameterized curve
with $$\forall s\in I:\kappa(s)\neq0 \wedge \tau(s)\neq0$$
And knowing that:
$$\Vert\beta(s)\Vert=const.$$...
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3
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182
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The angle between corresponding tangent lines on two Bertrand curves is constant
The angle between corresponding tangent lines on two Bertrand curves is constant and torsions of the two associate Bertrand curves have the same sign and their product is constant
Two distinct ...
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1
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140
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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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1
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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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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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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
$$
b' = \tau n \Rightarrow b \times b' = \tau b \times n = -\tau v \...
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a regular curve with $\kappa(t)>0$ is helix if and only if $\frac{\kappa}{\tau}$ is constant
A curve is said to be helix if its tangent line have a constant angle with a fixed direction. i.e. $\langle T(t),u\rangle$ is constant for some unit vector $u$.
I am trying to prove: a regular curve ...
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91
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Find unit normal vector given speed-, acceleration- and jerk vectors, Calculus III
So we have been given the following:
$$\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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42
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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:
$$\begin{bmatrix}T'\\N'\\ B'\end{bmatrix}=\begin{bmatrix}0&\kappa&0\\-\kappa&0&\tau\\0&-\tau&0\end{bmatrix}\begin{bmatrix}T\\...
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Why is this proof of the Frent-Serret formulae wrong?
I'm aware that similar questions have been asked, but none of them appear relevant to my own question.
By definition and by simple geometric observations, $N,B,T$ are all mutually perpendicular unit ...
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votes
1
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193
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Normal planes and spherical curves
I am interested in the following result:
"If all the normal planes of a curve pass through a particular point, then the curve is contained in a sphere".
My approach:
Let $\alpha: I \to \...
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What is $W$ when $W\times X = X'$ for every $X\in \{T, N, B\}$?
($\times$ denotes the usual cross product)
Let $\alpha: I \to R^3$ be a smooth regular curve with non-zero curvature, parametrized by arc length. Given there exists $W: I \to R^3$ such that $W\times X ...
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Questions about Frenet Frames
A curve $\gamma : I \rightarrow \mathbb{R}^n $ is called a curve of
general type in $\mathbb{R}^n$ if the first $n-1$ derivatives are
linearly independent $\forall t \in I$
A moving (orthonormal) ...
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Projection of moving point onto static curve and respective velocities / Frenet Coordinates
Consider a curve in 2D $\vec{p}(s)$ parameterized by arclength $s$ and the usual local coordinate system on the curve (Frenet Frame with unit vectors $\vec{n}(s)\perp\vec{t}(s)$, no torsion, curvature ...
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$\|\mathbf{r}'\|$ in terms of Frenet-Serret Frame variables
On Wikipedia I find the equation:
$$\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
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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 ...
user242708
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2
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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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1
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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
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166
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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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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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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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2
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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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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
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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} = \...
- 513
1
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1
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113
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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 ...
- 41
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0
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120
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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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0
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363
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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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0
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416
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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 $\beta$ ...
- 6,807
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0
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46
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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
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108
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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
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405
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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
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66
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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:
$$\tau=\frac{...
- 1,829
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0
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103
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Frenet formulas of $\alpha(t)=(t,t^{2},t^{3})$
Given the following curve
$$\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)...
- 1,829
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1
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31
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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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