Verifying Serret-Frenet equations

Question

I need to verify the Serret-Frenet equations for $ \gamma(t) = (4/5 \cos t, 1-\ \sin t, -3/5 \cos t)$ That is I need to verify $\dot t = \kappa n, \dot n = -\kappa t+ \tau b, \dot b = -\tau n$ Here from the given $\gamma(t)$ I can find $t$ and $\dot t$. But how to find $n$? If I use the relation $n=\dot t/{\kappa}$ won’t it be a circular argument since that is what I need to verify?

Answer 1

For clarity, we need to use different notation for $t$ vector and $t$ as a parameter.

By definition we have that

$$T=\frac{\dot{\gamma(t)}}{\left|\dot{\gamma(t)}\right|}$$

and

$$N=\frac{\dot{T}}{\left|\dot{T}\right|}=\frac{\ddot{\gamma(t)}}{\left|\ddot{\gamma(t)}\right|}$$

with $N\cdot T=0$, indeed

$$\frac d{dt}(T\cdot T)=2T\cdot \dot T=0$$

Refer also to the related

• Tangent, Normal and Binormal vectors
• How to find unit tangent, normal, and binormal vectors?