[Math] Natural parametrization of curves

Question

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.

Answer 1

$(a)$ \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*}