Natural parametrization of curves I want to find curvatures and torsions for the following curves but get stuck with their natural parametrizations ($s$ is natural if $|\dot{\gamma}(s)| = 1$). Can anyone help me?
(a) $e^t(\cos t,\sin t,1)$
(b) $(t^3+t,t^3-t,\sqrt{3}t^2)$
(c) $3x^2+15y^2=1, z=xy$
Update:
Here are my attempts on solving (a):
$\dot{\gamma}(t) = (\dot{t}e^t \cos t - \dot{t} e^t \sin t, \dot{t} e^t \sin t + \dot{t}e^t \cos t, \dot{t} e^t)$ which gives $|\dot{\gamma}(t)| = \sqrt{2}\dot{t}e^t=1$ and the solution for this ODE is $t = \ln\frac{\tau}{2}$.
But if I substitute $t$ with $t=\ln\frac{\tau}{2}$ the result will be $\dot{\gamma}(\tau) = (\frac{1}{\sqrt{2}}\cos\ln\frac{\tau}{2} - \frac{1}{\sqrt{2}}\sin\ln\frac{\tau}{2},\frac{1}{\sqrt{2}}\sin\ln\frac{\tau}{2}+\frac{1}{\sqrt{2}}\cos\ln\frac{\tau}{2},\frac{1}{2})$ and $|\dot{\gamma}(\tau)| = \sqrt{\frac{3}{2}}$. So for $|\dot{\gamma}(\tau)| = 1$ we should take $t = \ln\frac{\tau}{3}$. Where is my mistake?
Any help and hints will be very appreciative.

$(a)$
\begin{align*}
  \mathbf{\dot{r}}(t) &= e^{t}(\cos t-\sin t, \sin t+\cos t,1) \\
  |\mathbf{\dot{r}}(t)| &= e^{t}\sqrt{(\cos t-\sin t)^2+(\sin t+\cos t)^2+1} \\
  &= e^{t}\sqrt{3} \\
  s &= \int_{0}^{t} |\mathbf{\dot{r}}(t)| \, dt \\
  &= \sqrt{3}(e^{t}-1) \\
  t &= \ln \left( 1+\frac{s}{\sqrt{3}} \right)
\end{align*}
$(b)$
\begin{align*}
  \mathbf{\dot{r}}(t) &= (3t^2+1, 3t^2-1,2t\sqrt{3}) \\
  |\mathbf{\dot{r}}(t)| &= \sqrt{(3t^2+1)^2+(3t^2-1)^2+12t^2} \\
  &= \sqrt{2(9t^4+6t^2+1)} \\
  &= \sqrt{2}(3t^2+1) \\
  s &= \int_{0}^{t} |\mathbf{\dot{r}}(t)| \, dt \\
  &= \sqrt{2}(t^3+t) \\
  t &= \sqrt[3]{\sqrt{\frac{s^2}{8}+\frac{1}{27}}+\frac{s}{2\sqrt{2}}}-
       \sqrt[3]{\sqrt{\frac{s^2}{8}+\frac{1}{27}}-\frac{s}{2\sqrt{2}}} \\
\end{align*}
$(c)$
\begin{align*}
  \mathbf{r}(t) &=
  \left(
    \frac{\cos t}{\sqrt{3}},
    \frac{\sin t}{\sqrt{15}},
    \frac{\cos t \sin t}{3\sqrt{5}}
  \right) \\
  \mathbf{\dot{r}}(t) &=
  \left(
    -\frac{\sin t}{\sqrt{3}},
    \frac{\cos t}{\sqrt{15}},
    \frac{\cos^2 t-\sin^2  t}{3\sqrt{5}}
  \right) \\
  |\mathbf{\dot{r}}(t)| &=
  \sqrt{\frac{15\sin^2 t+3\cos^2 t+(\cos^2 t-\sin^2 t)^2}{45}} \\
  &= \frac{3-\cos 2t}{3\sqrt{5}} \\
  s &= \int_{0}^{t} |\mathbf{\dot{r}}(t)| \, dt \\
  &= \frac{6t-\sin 2t}{6\sqrt{5}}
\end{align*}
there is no close form for $t$ in terms of $s$.

You still can find $\kappa$ and $\tau$ by keeping  $t$ parametrization
  \begin{align*}
  \kappa &=
  \frac{|\mathbf{\dot{r}} \times \mathbf{\ddot{r}}|}
       {|\mathbf{\dot{r}}|^3} \\
  \tau &=
  \frac{\mathbf{\dot{r}} \cdot \mathbf{\ddot{r}} \times \mathbf{\dddot{r}}}
       {|\mathbf{\dot{r}} \times \mathbf{\ddot{r}}|^2}
\end{align*}

