Welcome to mathimatikoi.org forum; Enjoy your visit here.

## Bertrand curves

Differential Geometry
Grigorios Kostakos
Founder
Articles: 0
Posts: 460
Joined: Mon Nov 09, 2015 1:36 am
Location: Ioannina, Greece

### Bertrand curves

We call Bertrand curves, two curves $\overrightarrow{r_1},\overrightarrow{r_2}:I\subset\mathbb{R}\longrightarrow \mathbb{R}^3$ which have the property: For every $t\in I$ the normal lines at $\overrightarrow{r_1}(t),\;\overrightarrow{r_2}(t)$ are identical. Prove that:
(i) the distance between the points $\overrightarrow{r_1}(t)$ and $\overrightarrow{r_2}(t)$ is constant, for every $t\in I$.
(ii) the angle between the tangent lines at $\overrightarrow{r_1}(t),\;\overrightarrow{r_2}(t)$ is constant, for every $t\in I$.
Grigorios Kostakos
Grigorios Kostakos
Founder
Articles: 0
Posts: 460
Joined: Mon Nov 09, 2015 1:36 am
Location: Ioannina, Greece

### Re: Bertrand curves

We give a solution:

WLOG we consider a natural parameter $s$ for the two curves. The normal line of $\overrightarrow{r_1}$ at $\overrightarrow{r_1}(s)$ has equation $\varepsilon_1:\overrightarrow{y}=\overrightarrow{r_1}(s)+\lambda(s)\,\overrightarrow{n_1}(s)\,,\; \lambda\in \mathbb{R}$ and the normal line of $\overrightarrow{r_2}$ at $\overrightarrow{r_2}(s)$ has equation $\varepsilon_2:\overrightarrow{y}=\overrightarrow{r_2}(s)+\mu(s)\,\overrightarrow{n_1}(s)\,,\; \mu\in \mathbb{R}$.
[/centre]
1. Because the two lines are identical, we have that $\overrightarrow{r_1}(s)\in\varepsilon_2\,,\; \overrightarrow{r_2}(s)\in\varepsilon_1$, $\overrightarrow{n_2}(s)=\pm\overrightarrow{n_1}(s)$ and \begin{align*}
\end{align*} So the distance between the points $\overrightarrow{r_1}(s)$ and $\overrightarrow{r_2}(s)$ is constant, for every $s\in I$.
2. For the tangent vectors of $\overrightarrow{r_1}$ at $\overrightarrow{r_1}(s)$ and $\overrightarrow{r_2}$ at $\overrightarrow{r_2}(s)$, we have \begin{align*}\frac{d}{ds}\big(\,\overrightarrow{t_1}\cdot\overrightarrow{t_2}\big)&=\dot{\overrightarrow{t_1}}\cdot\overrightarrow{t_2}+\overrightarrow{t_1}\cdot\dot{\overrightarrow{t_2}}\\
\end{align*} So, the angle between the tangent lines at $\overrightarrow{r_1}(s),\;\overrightarrow{r_2}(s)$ is constant, for every $s\in I$.