Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $||\mathbf{v}(t)||=1,\ \forall t\in (a,b)$. Find all continuous functions $\mathbf{r}:(a,b)\to\mathbb{R}^2$ so that:

$ \mathbf{v}(t_0)=\lim\limits_{t\searrow t_0} \displaystyle \frac{\mathbf{r}(t)-\mathbf{r}(t_0)}{||\mathbf{r}(t)-\mathbf{r}(t_0)||},\forall t_0\in (a,b). $

Is it true that that the only solutions are of the form: $\mathbf{r}(t)=\displaystyle\int f(\tau)\mathbf{v}(\tau)\ \mathrm{d}\tau +\mathbf{c}$, where $f:(a,b)\to (0,\infty)$ is continuous and $\mathbf{c}=(c_1,c_2)$ is a constant?