Prove: The Frenet Formula for Torsion & Curvature

In summary, the conversation discusses a problem involving a regular curve in three-dimensional space and its arc-length parametrization. It is shown that if the torsion is non-zero and there exists a vector Y such that <alpha',Y> is constant, then the ratio of curvature and torsion is also constant. The conversation also explores the converse, where it is proven that if the ratio is constant and torsion is non-zero everywhere, then there exists a vector Y such that <alpha',Y> is constant. The Frenet formula is used in this discussion to prove various relationships between the curve and the vector Y.
  • #1
Dragonfall
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Homework Statement



Suppose [tex]\alpha[/tex] is a regular curve in [tex]\mathbb{R}^3[/tex] with arc-length parametrization such that the torsion [tex]\tau(s)\neq 0[/tex], and suppose that there is a vector [tex]Y\in \mathbb{R}^3[/tex] such that [tex]<\alpha',Y>=A[/tex] for some constant A. Show that [tex]\frac{k(s)}{\tau(s)}=B[/tex] for some constant B, where k(s) is the curvature of alpha.

The Attempt at a Solution



I think the Frenet formula in question that I can use is [tex]n'=-kt-\tau b[/tex], but I can't make it work.
 
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  • #2
What about the converse? Suppose k/tau is constaint and tau is nonzero everywhere, show that there exists a nonzero vector Y such that <a', Y> is constant.
 
  • #3
Ok, I'm pretty rusty at this so bear with me. Take k(s) nonzero (if it is zero, it's not even clear to me how to define the Frenet frame). Since you have arc-length parametrization, alpha'=t (the tangent). So <t,Y> is constant. Now you should be able to prove a bunch of stuff by differentiating <x,Y> where x is various vectors.

i) Show <n,Y>=0.
ii) Show <b,Y> is also constant.
iii) Differentiate <n,Y> and use your favorite Frenet formula.
 

What is the Frenet Formula for Torsion & Curvature?

The Frenet Formula is a mathematical tool used to describe the curvature and torsion of a curve in three-dimensional space. It is commonly used in the field of differential geometry to study the properties of curves in space.

How is the Frenet Formula derived?

The Frenet Formula is derived from the fundamental theorem of the calculus of variations, using the concept of arc length and the curvature of a curve. It was first introduced by Jean Frenet in the 19th century.

What is the significance of the Frenet Formula in mathematics?

The Frenet Formula is important in mathematics as it provides a way to calculate the curvature and torsion of a curve in three-dimensional space. It is also used in various fields such as physics, engineering, and computer graphics to study and analyze curves.

Can the Frenet Formula be extended to higher dimensions?

Yes, the Frenet Formula can be extended to higher dimensions. In fact, there are generalizations of the formula for curves in n-dimensional space, known as the Frenet-Serret formulas.

How is the Frenet Formula applied in real-world applications?

The Frenet Formula has many practical applications, such as in designing roller coaster tracks, analyzing the movement of objects in space, and creating smooth animations in computer graphics. It is also used in medical imaging to study the curvature of blood vessels and in robotics to control the movement of robot arms.

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