# I Frenet Serret Formulas

1. Feb 6, 2017

### Arya Prasetya

Hi, I'm trying to derive the Frenet Serret Formulas, but I am having trouble to understand why, after some checking, that the derivative of binormal vector is:

$\frac{d\hat{b}}{ds}=-\tau\hat{n}$

I understand that, $\hat{t}\wedge\frac{d\hat{n}}{ds}\parallel\hat{n}$ and $|\frac{d\hat{b}}{ds}|=\tau$, but why the negative sign? Isn't it equally possible that it has a positve sign?

2. Feb 6, 2017

### Staff: Mentor

3. Feb 6, 2017

Since $\tau$ is simply a constant of proportionality, I believe the choice is arbitrary, but whatever sign that is used for $\tau$ needs to be consistent with the other Frenet equation involving $\tau$. The equations with their choice of the sign of $\tau$ is apparently somewhat standard. It's a somewhat specialized topic, but I think you might find most of the textbooks use the same sign convention.

4. Feb 7, 2017

### lavinia

$\frac{d\hat{b}}{ds}= \frac{d\hat{t×n}}{ds}= \frac{d\hat{t}}{ds}×\hat{n} + d\hat{t}×\frac{d\hat{n}}{ds} = 0 + \hat{t}×κ\hat{b} = κ\hat{t}×\hat{t}×\hat{n}$

Now use the cross product identity $T×T×N = T(T⋅N) - N(T⋅T)$

Last edited: Feb 7, 2017