4

Let's consider the following curve:

$\varphi(t)=\begin{cases} (t,0,e^{-\frac{1}{t^{2}}}) & t>0\\ (0,0,0) & t=0\\ (t,e^{-\frac{1}{t^{2}}},0) & t<0 \end{cases} $

I have to compute the curvature and the torsión of the curve. Well first of all, I computed $\varphi',\varphi'',\varphi'''$ (it was horrible....). This curve is not parametrized by arc length, so I have two options

$i)$ Find that reparametrization

$ii)$ Compute $\varphi',\varphi'',\varphi'''$ and then use the formula for the torsión and curvature. (I computed $\varphi',\varphi'',\varphi'''$ and it was horrible....) But computing a cross product it's something worst than horrible... specially in this problema, I have to compute by three cases , $t<0 , t=0 , t>0$ I think that computing this kind of things it's stupid but it's a homework Dx ... someone knows some tricky way to do this?

Eustass
  • 225

1 Answers1

2

When $t=0$, $\dot \varphi(0)=\ddot \varphi(0)=(1,0,0)$. (You can compute it by direct computation using definition of derivative.)

Thus its curvature is 0 and torsion is not defined.

Furthermore, You can see that $\varphi(t)$ is actually plane curve when $t>0$ or $t<0$. Therefore its torsion is $0$ when $t\neq 0$. Curvature can be computed by direct computation or...

Curvature formular for plane curve $y=f(t)$ is $\frac{|f''(t)|}{(1+f'(t)^2)^{3/2}}$. In this case, $f(t)=e^{-1/t^2}$.

(You can derive this formular from $\gamma(t)=(t,0,f(t))$ and $\kappa=\frac{||\dot\gamma \times \ddot\gamma||}{||\dot\gamma||^3}$ )

quicksilver
  • 642
  • 6
  • 16