Mathematics

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).

163 questions
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Helices in Lorentz-Minkowski space $\Bbb L^3$.

Context: Consider the Lorentz-Minkowski space $\Bbb L^3$, with the metric: $$ds^2 = dx^2 + dy^2 - dz^2$$ (which I'll denote just by $\langle \cdot, \cdot \rangle$) If $\alpha$ is spacelike and ${\bf N}(s)$ is lightlike for all $s \in I$, the Frenet…
Ivo Terek
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8
votes
2 answers

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 $T = \dot \alpha(s), \tag 4$ $\alpha(s)$ being a unit speed curve in $\Bbb R^3$…
Robert Lewis
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3 answers

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 circle of exercise is whose we call osculator…
Quiet_waters
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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 that the normal lines of $\beta$ and $\alpha$ are…
Matheus Andrade
  • 7,173
5
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1 answer

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 tried to reparametrize this curve by arc length. I…
Eduardo Magalhães
  • 5,765
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The Frenet frame is orthogonal

I have proved $P'=AP$ where $$P= \begin{pmatrix} T \\ N \\B \end{pmatrix}$$ $$A= \begin{pmatrix} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \\ \end{pmatrix} $$ . I am trying to show $P$ is orthogonal by this fact. I try to differentiate…
xyz
  • 1,495
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What's the relation between the Darboux Frame and the Frenet-Serret on a oriented surface?

I'm studying differential geometry and I'm in the part of geodesics, my professor always defines a curve that can define the tangent field but for calculating the geodesics and normal curvatures at each point $\alpha(t)$ of the curve he defines the…
Eduardo Silva
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1 answer

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 another curve which range is contained on the unit…
AlienRem
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4
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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 with $\kappa(t)>0$ is helix if and only if…
xyz
  • 1,495
4
votes
1 answer

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 $\alpha: I \to \mathbb{R}^3$ be the curve (assume it is arc length…
user210089
  • 675
4
votes
2 answers

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 from x and then find an equation that involves…
Nono4271
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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 figure the physics out. $$ \mathbf{a} =…
xue
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2 answers

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 $B(s) = \bar{B}(s)$, they are congruent. I know there is a…
Matheus Andrade
  • 7,173
4
votes
1 answer

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 torsion $\tau$ both constant is a circular helix. My…
user264750
4
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0 answers

Derivative of orthogonal matrix - Generalization of Frenet frame equations

I was studying Differential Geometry and the Frenet Frame equations $\begin{pmatrix} T'\\ N'\\ B' \end{pmatrix} = \begin{pmatrix} 0 & k & 0\\ -k & 0 & \tau\\ 0 & -\tau & 0 \end{pmatrix} \begin{pmatrix} T\\ N\\ B \end{pmatrix}$ made me think about…
Sotiris Simos
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