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).

Filter by
Sorted by
Tagged with
4
votes
1answer
50 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
2answers
33 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 ...
0
votes
0answers
31 views

How to find rotation matrix of a plane curve of varying curvature?

I'm able to find a rotation matrix with respect to a fixed basis for a plane curve of constant curvature (example - circle) or a straight line (zero curvature). But in the case of a sinusoid, ...
0
votes
0answers
11 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
votes
0answers
43 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
votes
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
64 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 ...
0
votes
0answers
26 views

Inequalities in tensor products.

If H and K are hilbert spaces and {xi},{yi} are frames for H and K respectively then tensor product of xi ,yi is a frame for tensor product of H and K.This can be proved for the elementary tensors but ...
4
votes
1answer
328 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
235 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
vote
1answer
55 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
vote
1answer
58 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
votes
0answers
29 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)=\...
1
vote
2answers
82 views

Proving the Frenet-Serret formulae

I have been asked to prove: $$ \frac{d\mathbf{N}}{ds}=-\kappa\mathbf{T}+\tau\mathbf{B} $$ I have been given $\mathbf{N}=\mathbf{B} \times \mathbf{T}$ $$ \frac{d\mathbf{B}}{ds}=-\tau\mathbf{N} $$ ...
5
votes
2answers
227 views

Center and radius of the osculating circle - The limiting position of a circle trough three points

I am stucked on problem 1.7.2.b of Differential Geometry of Curves and Surfaces by Manfredo do Carmo. The problem is similar as this topic, but here the exercise defines the osculator circle, ie, this ...
1
vote
1answer
57 views

A proof that a plane is the osculating plane

I need to prove that, for a curve $\alpha$ at canonical local form, if a plane $P$ is s.t.: 1) $P$ constains the tangent in $0$; 2) For all interval $(-s,s)$ we have points at both sides of $P$. ...
0
votes
0answers
29 views

$\gamma(s)$ is a cylindrical helix if and only if arclength parameterized curve $\beta$ is a cylindrical helix

Consider an arclength parameterized curve $\beta$ in $\mathbb{E}^3$ with $\kappa >0$ and $\tau \neq 0$ and Frenet frame $(T,N,B)$. If we define the curve $\gamma$ as: $$\gamma(s) = \beta(s)+ \frac{...
1
vote
1answer
118 views

Prove that the curve is on a circle

Consider a curve $\alpha: I\rightarrow \mathbb{R}^3$ parameterized by arc length such that all the normals contain a comum point. Prove that the curve is on a circle. Well, we can suppose that ...
0
votes
0answers
232 views

Frenet-Serret Formulas For Arc Length and Regular Parametrization

Frenet-Serret formulas for arc length parametrization are in matrix form where $\widetilde T(s)$, $\widetilde N(s)$, $\widetilde B(s)$ are unit tangent, normal, binormal vectors of a curve $\widetilde ...
2
votes
1answer
196 views

Numerically computing normal, binormal, and tangent directions of non-parametric curve in $\mathbb{R}^{3}$

Let's say that I have numerical data for a curve in $\mathbb{R}^{3}$, but I do not have the parametric equations of the curve; all that I have is a sampling of $N$-many points that lie on this curve, $...
3
votes
1answer
185 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{...
2
votes
1answer
207 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 ...
1
vote
1answer
27 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 \...
1
vote
0answers
40 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. ...
2
votes
1answer
44 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.
0
votes
1answer
61 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 $\{T,N\}$. ...
0
votes
0answers
37 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 ...
1
vote
1answer
65 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)=\...
1
vote
1answer
88 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 \...
1
vote
2answers
420 views

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 ...
1
vote
1answer
275 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( \...
0
votes
1answer
465 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 ...
2
votes
2answers
144 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 $\...
2
votes
0answers
48 views

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'(...
0
votes
0answers
117 views

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 ...
1
vote
1answer
115 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 ...
0
votes
0answers
102 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 $k'(p) \neq 0$, ...
3
votes
1answer
1k 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 ...
3
votes
0answers
58 views

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 $t_0 \in I \cap \overline{I}$. $\...
0
votes
1answer
34 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 ...
1
vote
1answer
378 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) = \...
0
votes
0answers
101 views

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 ...
1
vote
1answer
62 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 $S$ ...
1
vote
1answer
67 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 =...
2
votes
0answers
302 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 ...
1
vote
1answer
186 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 $...
0
votes
1answer
209 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 $...
0
votes
1answer
59 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 ...
2
votes
1answer
47 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 ...
0
votes
1answer
135 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)...