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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Using Frenet coordinates as a metric

Having two trajectories, one that is my target and one which is the result of a system trying to follow the target trajectory, I get two slightly different trajectories as a result. The first is my ...
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1answer
44 views

in 2D dimensional plane, is it problematic to have Frenet-Serret frame with zero curvature?

I have a Frenet-Serret frame moving on a 2-D plane. As of now, I do not care about the binormal vector. So my equations are given by, \begin{align} \dot{T} = v\kappa N \\ \dot{N} = -v\kappa T \end{...
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1answer
30 views

Frenet- Serret and Darboux frame

I am aware the two frames are different, aside from sharing the same tangent unit vector in their basis. But I was wondering, why or when would one choose to use/work with one other frame and not the ...
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1answer
22 views

Arclength-Parametrized Space Curve Inequality

Give any arclength parametrized space curve $\alpha(s)$ (where space curve just means its codomain is $\mathbb R^3$), I want to show the following inequality: $$\lVert \alpha(s) \rVert ≥ \lvert \...
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39 views

Finding Parametric Equation of Curve with some conditions

Let $S$ be a Sphere (in 3d space ,i.e. $\mathbb{R^3}$) and $\gamma : \mathbb{R} \to S$ be a curve that is parameterized by length. For all $t$ , we have $|\gamma''(t)| = k<1$ and $k$ is a constant. ...
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1answer
42 views

Why this equation is correct? (Frenet frame)

$2(T_a-T_b)•(T'_a-T'_b)=-2T_a•T'_b-2T'_a•T_b$ Is $2T_a•T'_a=0$ ? Please explain me why this equation work.
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1answer
20 views

Frenet Frame along a curve and Riemannian Curvature on $S^2$

I would appreciate some help showing the following statement. Let $\omega: [0,1] \rightarrow S^2$ be a smooth curve with velocity vector $V = \omega'$, speed $v = |V|$ and Frenet frame $\{T,N\}$. ...
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22 views

Toroidal frame system

*Context: I'm following the textbook of Barrett O'Neill about differential geometry. In order to calculate the connection forms of the torous I need the relation between a toroidal frame and the ...
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1answer
52 views

Does the velocity vector have to have unit speed parametrisation in calculations of the frenet frame?

Consider the curvature of a curve $\beta$ at a point s. This is given by $\kappa(s):=|T'(s)|$, where $T(s)=\beta '(s) $. similarly we define the fields in the frenet frame $\{T,N,B\}$ by $$T(s)=\...
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1answer
85 views

Calibrator for unit speed curve

Non-existence of Almost Calibrator of Circle : Assume that $U$ is $\varepsilon$-open ball at origin in $\mathbb{R}^2$. Prove that there does not exist the following function : $f: U\rightarrow \...
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2answers
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How to find the frenet frame of a non-unit speed

I don't understand witch formulas I can use for non-unit speed curves. I know I can use $$\kappa = \frac{||\alpha' \times \alpha''||}{||\alpha'||^3}$$ and $$\tau = \frac{(\alpha' \times \alpha'')\dot ...
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1answer
118 views

Derivative of Frenet-Serret equations

I'm somewhat confused with Frenet-Serret equations and its derivatives. The curve $\gamma(s)$ is parametrized by its arc-length and it's such that the following is true: $$ \frac{1}{k^2} + \left( \...
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1answer
173 views

Frenet–Serret formulas in terms of a curve and cross product

Let $\alpha: I\rightarrow \mathbb R^3$ arc length parametric curve with positive curvature. Show $\exists \ \omega: I\rightarrow \mathbb R^3$ a curve such that $$T'=\omega\times T\quad N'=\omega\times ...
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2answers
53 views

Frenet Formula relating $T$ and $N$.

In the following notes: http://mathematics.stanford.edu/wp-content/uploads/2013/08/Mooney-Honors-Thesis-2011.pdf, the author (on page 5) says that $T$ and $N$ are related by the Frenet formula $\...
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What's the derivate of the Normal vector?

According to frenet formulas the unit tangent vector is given as follows. $ T(t) = \frac{dr/dt}{\lVert r'(t) \rVert}$ And the unit normal vector is given as follows. $ N(t) = \frac{dT/dt}{\lVert T'(...
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How do I extract a curve in xyz from curvature/torsion using Frenet Serret equations?

Somewhat of a continuation of this, opening this scab because I have the same question but the solution was not covered in post and I have been banging my head against the wall for a week trying to ...
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1answer
109 views

T(t)≠0 for all values of t and the tangent line at any given point of the curve always passes through point D. Show that r represents a straight line

Let $r: \mathbb{R}->\mathbb{R}^3$ be a curve in arc-length parametrization such that $T(t)$ does not equal to zero for all values of $t$. Assume that the tangent line at any given point $r(t)$ of ...
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75 views

Condition for a plane curve to intersect its osculating circle

The osculating circle of a curve $\alpha$ at the point $p \in \alpha$ is the circle $\mathbb{S}^1$ which is tangent to $\alpha$ at $p$ and has radius $\frac{1}{k(p)}$. Show that, if $k'(p) \neq 0$, ...
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1answer
685 views

Proving a few properties of Bertrand curves

Here's what I've got so far (and I'm assuming $\alpha$ is a unit speed curve): a) The fact that $\beta(s) = \alpha(s) + r(s)N(s)$ for some scalar function $r$ follows trivially because of the fact ...
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Contact order of a space curve with one of it's tangent lines

Definition $1$: Let $\alpha: I \mapsto \mathbb{R^3}$ and $\beta: \overline{I} \mapsto \mathbb{R^3}$ be two regular curves such that $\alpha(t_0) = \beta(t_0)$, where $t_0 \in I \cap \overline{I}$. $\...
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1answer
31 views

Conditions for a space curve to have constant torsion

Prove that a regular curve $\alpha$, with non vanishing curvature, has constant torsion $\tau = \cfrac{1}{a}, a \neq 0$ if, and only if: $$\alpha(t) = a\left( \int f_1(t) \ dt,\int f_2(t) \ dt, \int ...
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1answer
192 views

Proving that two curves in $\mathbb{R^3}$ with the same binormal vector are congruent

Let $\alpha, \bar{\alpha}: I \mapsto \mathbb{R^3}$ be two regular unit speed curves with non vanishing curvature and torsion. Prove that if the binormal vectors of the curves coincide, i.e $B(s) = \...
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Using local canonical form to approximate a curve (not necessarily unit speed)

Let $\alpha: I \mapsto \mathbb{R^3}$ be a regular curve with non vanishing curvature. Check that if $t_0 \in I$ is fixed, it's possible to choose a cartesian coordinate system and reparameterize the ...
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1answer
58 views

Proving $W(s) = \alpha(s) + pN(s) - p' \sigma B(s)$ is constant, given that $||W(s) - \alpha(s)|| = R^2$

Let $\alpha: I \mapsto \mathbb{R^3}$ be a regular unit speed curve such that $k(s) > 0$ and $\tau(s) \neq 0, \forall s \in I$. Prove that: a) If $\alpha(I)$ is contained in a sphere $S$ ...
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1answer
51 views

Proving a couple things about $\beta(t) = \alpha(t) - s(t)\cos(\theta)u$, where $\alpha$ is a helix. Is this correct?

Let $\alpha: I \mapsto \mathbb{R^3}$ be a helix and $u$ be the constant unit vector whose angle $\theta$ with $\alpha'(t)$ is constant. Let $s(t)$ be the arclength function of $\alpha$ starting at $t =...
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237 views

Whats the relation between the Darboux Frame and the Frenet-Serret on a oriented surface?

i'm studying diferential geomtry and i'm in the part of geodesics, my professor always define a curve that can define the tangent field but for calculating the geodeiscs and normal curvatures at each ...
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1answer
114 views

About the Frenet apparatus: are these expressions wrong?

I'm using Keti Tenenblat's "Introduction to differential geometry" text, and according to the book, we have: $$T'(s) = K(s)N(s)$$ $$N'(s) = -K(s)T(s) - \tau(s)B(s)$$ $$B'(s) = \tau(s)N(s)$$ where $...
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1answer
160 views

Proving binormal vector is the limiting position of perpendicular to tangent lines

"Let $\alpha(s)$ be a regular curve. Verify that the the binormal vector $b(s_0)$ is the limiting position of the perpendicular to the tangent lines to $\alpha$ at $s_0$ and $s_1$ as $s_1$ tends to $...
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1answer
35 views

Proving that two curves that are symmetric about the origin have same curvature and same torsion (up to a sign)

This is an exercise in a differential geometry book I'm studying, and currently I can't fathom why it's there (and it doesn't feel at all intuitive why it would be true). A simple counter example ...
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1answer
46 views

Proving two reparameterizations by arclength differ only by a sign and constant

Let $\alpha(t)$ be a regular curve. Prove that if $\beta(s)$ and $\gamma(\bar{s})$ are two reparameterizations by arclength, then $s = \pm\bar{s} +a$, where $a \in \mathbb{R}$ is a constant. I know ...
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1answer
77 views

Existence of smooth frame along a curve on the manifold

Suppose $M$ is a smooth manifold of dimension $n$. Let $\gamma: [0,1] \mapsto M$ be a smooth curve on it. Assume $\dot{\gamma}(t)$ is never zero. Can we always find smooth vector fields $e_1(t), e_2(t)...
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1answer
32 views

Studying regular space curves when restricted to two differentiable functions

a) Let $\alpha:I \mapsto \mathbb{R^3}$ be a regular curve. Prove that $\forall t_0 \in I$ there is an open interval $J$ that contains $t_0$ on which $\alpha$ is injective. b) Prove that if $\alpha: I ...
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Proving regularity and orthogonality of two curves

Let $\alpha(s) = (x(s)),y(s))$ be a regular plane curve that is parameterized by arclength, and let $n(s)$ be the normal vector and $k(s)$ be the curvature of $\alpha$. Consider the family of curves: ...
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1answer
153 views

Deriving a relationship between curvature, torsion and the curvature of tangent vector

Let $\alpha(s)$ be a regular and biregular curve in $\mathbb{R}^3$ parametrized in arc length with its curvature $k(s)$ and torsion $\tau(s)$. Then we can think of it's tangent vector $T(s)$ as ...
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Apparently disconected concepts of Differential Geometry in basic Mechanics

This questions is already answered in https://physics.stackexchange.com/questions/367797/apparently-disconnected-concepts-of-differential-geometry-on-basic-mechanics/367807#367807[ But I would like ...
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21 views

coordinate frame, computation of coefficients

Let $\mathbf{e_j}$ be a local coordinate frame on the manifold $M$. How do I compute/obtain the coefficients $A_j^k$ so that $$\mathbf{e_j} (v^i) = A_j^k \partial_k (v^i)= A^k_j \frac{\partial v^i}{\...
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1answer
834 views

Curve with constant torsion and curvature is a circular helix.

I am trying to find a proof for the 9th question of section 2.4, from the book Elementary Differential Geometry by Barrett O'Neill. I want to show that a curve $\alpha$ with curvature $\kappa$ and ...
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1answer
68 views

Differentiate curvature, torsion

I want to construct a frenet curve $Y: \Bbb R \to \Bbb R^3$ with constant curvature $K$ and torsion $T$. I figured I'd start with calculating the derivatives of $K$ and $T$ but I don't know how to ...
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Tube surfaces, frenet frame and change of basis

Given a curve $c:[a,b] \to \mathbb{R}^3$ and defining a surface $S:[a,b] \times [0,2 \pi]\to \mathbb{R}^3$ where $S(s,t)=c(s)+R\cdot sin(t)\, N(s)+R\cdot cos(t)\, B(s)$ with being $N(s)$ and $B(s)$ ...
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1answer
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Derivative of normal unit vector in frenet coordinates

I am trying to understand the derivation of some formulas in the paper Motion Control of Wheeled Mobile Robots by C. Samson in Springer Handbook of Robotics (2008). Currently I am stuck at the ...
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60 views

Geometry: Prove the evolute does not contain a line segment

Given a regular curve $\alpha:\mathbb{R}\to\mathbb{E}^2$ with an oriented curvature $\kappa(t) \ne 0$. Prove the evolute does not contain an open subset of a straight line. I'm a bit stumped as to ...
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602 views

Derivative of orthogonal matrix - Generalization of Frenet frame equations

I was studying Differential Geometry and the Frenet Frame equations $\begin{pmatrix} T'\\ N'\\ B' \end{pmatrix} = \begin{pmatrix} 0 & k & 0\\ -k & 0 & \tau\\ 0 & -\tau & ...
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2answers
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Express the velocity of $\gamma$ in terms of the TNB frame of $r$

my question is: Let $r(t)=<cos(t),sin(t),t>$ be a curve and let $N$ be the principal unit normal vector to the curve. Define the curve $\gamma (t)$ by $\gamma (t)=r(t)+N$. Express the velocity ...
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213 views

Torsion of an arc length parametrized curve

Let $c$: $I \to \Bbb{R}^3$ be an arc length parametrized curve with curvature $k(t) \ne$ 0 for all $t \in I$. Show that the torsion of $c$ is given by $\tau(t)$ = $⟨c'(t) \space \times \space c''(t), ...
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1answer
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Let $\alpha(t)$ and $\beta(t)$ be two different curves. $\alpha'(t)=T(t)-k(t)T(t)\lambda(t)$?

I can't understand a step in a solution: Let $\alpha(t)$ and $\beta(t)$ be two different curves. Suppose that $\beta(t)=\alpha(t)+\lambda(t)N(t)$. Then: $$\beta'(t)=\alpha'(t)+\lambda'(t)...
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162 views

Identifying curves using Frenet-Serret apparatus

I have a question about identifying regular curves in $ R^3 $ using Frenet-Serret apparatus. Considering the congruence of curves every unit-speed curves are congruent when they have same curvature ...
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1answer
58 views

Showing another form of a curve $\alpha(s)$ parametrized by arc-length

I have recently come across a question and am looking for some advice as to how to approach it. The question reads: Let $\alpha(s)$, $s=$ arc length, be a curve whose torsion $\tau$ is non-zero ...
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2answers
2k views

What is SIGNED CURVATURE (Simple explanation)

I know what curvature is; positive curvature, negative curvature, zero curvature, i understand. BUT THEN.. Signed curvature? I just can't seem to find something that distinguishes enough difference ...
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161 views

Proving 2 things about Frenet Curves

So here I'm tasked with proving two statements about Frenet Curves. I shall enumerate them now.. I. If γ is an arclength parametrized planar curve, we can regard it as a space curve. Show that this ...
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0answers
109 views

Adapted Frenet Frame of Curve in $\mathbb{R}^3$

Let $\gamma$ be an arclength parameterized Frenet curve. We define the Curvature function to be $$\kappa = \left\lVert\gamma''\right\rVert > 0$$ and the Torsion function to be $$ \tau = \left\...