Substituting ds by dt in Frenet-Serret Formulas

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In summary, to solve for the curvature (κ) and torsion (τ) in the Frenet-Serret Formulas, we can substitute ds with dt and use the chain rule. We can also use the fact that \dot{s}=\frac{\mathrm{d} s}{\mathrm{d} t}=\left | \frac{\mathrm{d} \vec{r}}{\mathrm{d} t} \right|=\left |\dot{\vec{r}} \right|, where \vec{r}(t) is the parametrization of the curve.
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I want to substitute ds by dt in the Frenet-Serret Formulas where κ is the curvature and is the torsion:
Tangential:[tex]\frac{d\vec{T}}{ds} = κ*\vec{N}[/tex]
Normal:[tex]\frac{d\vec{N}}{ds} = -κ*\vec{T}+τ*\vec{B}[/tex]
Binormal:[tex]\frac{d\vec{B}}{ds} =- τ*\vec{N}[/tex]
I want to substitute [tex]\frac{d\vec{T}}{ds} → \frac{d}{dt} T(t)[/tex] N(t), B(t) and solve for κ and τ.
 
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Just use the chain rule and the fact that
[tex]\dot{s}=\frac{\mathrm{d} s}{\mathrm{d} t}=\left | \frac{\mathrm{d} \vec{r}}{\mathrm{d} t} \right|=\left |\dot{\vec{r}} \right|,[/tex]
where the function [itex]\vec{r}(t)[/itex][/itex] is the parametrization of the curve. Then you have
[tex]\vec{T}=\frac{\dot{\vec{r}}}{\dot{s}}[/tex]
etc.
 

1. What is the Frenet-Serret formula?

The Frenet-Serret formula is a set of three differential equations that describe the motion of a particle in three-dimensional space at a specific point in time. It relates the position, velocity, and acceleration of the particle to the curvature and torsion of the curve that the particle is traveling along.

2. What is the significance of substituting ds by dt in the Frenet-Serret formulas?

Substituting ds by dt in the Frenet-Serret formulas allows us to express the equations in terms of the parameter t, which represents time. This allows us to study the motion of a particle over a specific period of time and understand how its position, velocity, and acceleration change over time.

3. How does the substitution of ds by dt affect the Frenet-Serret formulas?

Substituting ds by dt in the Frenet-Serret formulas changes the equations from being dependent on arc length (ds) to being dependent on time (dt). This means that instead of describing the motion of a particle along a specific curve, the formulas now describe the motion of the particle over a specific period of time.

4. Can the Frenet-Serret formulas be used for any type of curve?

Yes, the Frenet-Serret formulas can be used for any type of smooth curve in three-dimensional space. This includes curves in both two and three dimensions, as well as curves in higher dimensions.

5. How are the Frenet-Serret formulas used in real-world applications?

The Frenet-Serret formulas have a wide range of applications in physics, engineering, and other scientific fields. They are commonly used in the study of motion and dynamics, as well as in the fields of computer graphics, robotics, and biomechanics.

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