Use this tag for questions on Frenet frames and the Frenet-Serret formulae. Related tags include (differential-geometry), (multivariable-calculus), and (curves).
Questions tagged [frenet-frame]
164 questions
9
votes
1 answer
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
- 80,301
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
- 72,871
6
votes
1 answer
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
- 6,998
5
votes
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…
656475
- 5,473
5
votes
1 answer
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…
xyz
- 1,473
5
votes
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
- 1,475
5
votes
0 answers
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
- 1,425
5
votes
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
- 4,130
- 1
- 27
- 38
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
- 655
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
- 149
4
votes
1 answer
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
- 185
4
votes
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
- 6,998
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
votes
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…
Sotiris Simos
- 163
4
votes
0 answers
Find the Frenet Frame at each point of the curve
Compute the Frenet Frame at each point of the curve $c(t)=(3t-t^3,3t^2,3t+t^3)$
I first found $c'(t)=(3-3t^2,6t,3+3t^2)$ and checked for arc length parameterization but…
Lauren Bathers
- 395