# 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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### $\beta:I\subset\mathbb{R}\to\mathbb{E}^3$ 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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### 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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### 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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### 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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### 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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### 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 ... 536 views

### 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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### 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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### 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 $\beta$ ...
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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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### 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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### 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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