I have a paper and on that paper I only can read:
Let $f:\mathbb{S^{1}} \to \mathbb{R^2}$ be a function and $f_{\epsilon}=f+\epsilon hn$ where $\mathbb{S^1}$ is the unit circle and $\dot{f}^2=r^2$. Further, we will denote $f_{\epsilon}(x)$ to be $f(\epsilon, x)$. Knowing that $f_{\epsilon}=f+\epsilon hn$ I want to compute some variations. $\delta f, \delta\dot{f}\text{ and }\delta r$.
$\delta f:$ $$\displaystyle f_{\epsilon}=f+\epsilon hn \Rightarrow \frac{f_{\epsilon}-f}{\epsilon}=hn\Rightarrow\delta f=hn$$
$\delta\dot{f}:$ $$\displaystyle \dot{f_{\epsilon}}=\dot{f}+\epsilon\dot{h}n+\epsilon h\dot{n}\Rightarrow \frac{\dot{f}-\dot{f}}{\epsilon}=\dot{h}n-\dot{h}n \Rightarrow \delta\dot{f}=\dot{h}n-h\dot{n}$$
$\delta r:$ $$\displaystyle r_{\epsilon}^{2}=\left(\dot{f}+\epsilon\dot{h}n+\epsilon h\dot{n}\right)^2 \Rightarrow r_{\epsilon}^2=(\dot{f})^2+(\epsilon \dot{h}n)^2+(\epsilon h\dot{n})^2+2(\dot{f}\epsilon\dot{h}n+\dot{f}\epsilon h\dot{n}+\epsilon^2\dot{h}nh\dot{n})\Rightarrow r_{\epsilon}^2=r^2+(\epsilon \dot{h}n)^2+(\epsilon h\dot{n})^2+2(\dot{f}\epsilon\dot{h}n+\dot{f}\epsilon h\dot{n}+\epsilon^2\dot{h}nh\dot{n})\Rightarrow \frac{r_{\epsilon}^{2}-r^2}{\epsilon}=\epsilon((\dot{h}n)^2+(h\dot{n})^2+\dot{h}nh\dot{n})+2(\dot{f}\dot{h}n+\dot{f}h\dot{n})\Rightarrow\frac{r_{\epsilon}-r}{\epsilon}\cdot(r_{\epsilon}+r)=\epsilon((\dot{h}n)^2+(h\dot{n})^2+\dot{h}nh\dot{n})+2(\dot{f}\dot{h}n+\dot{f}h\dot{n})\Rightarrow \delta r(r_{\epsilon}+r)=\epsilon((\dot{h}n)^2+(h\dot{n})^2+\dot{h}nh\dot{n})+2(\dot{f}\dot{h}n+\dot{f}h\dot{n})$$
$$\delta r(r_{\epsilon}+r)=\epsilon((\dot{h}n)^2+(h\dot{n})^2+\dot{h}nh\dot{n})+2(\dot{f}\dot{h}n+\dot{f}h\dot{n})$$
In the last equation I want to make $\epsilon \to 0$ and to say $$2(\delta{r})r=2(\dot{f}\dot{h}n+\dot{f}h\dot{n})\Rightarrow \delta r=\frac{\dot{f}\dot{h}n+\dot{f}h\dot{n}}{r}.$$ This is what I tried. Could you please confirm if my computation are OK and if this is the true meaning of how variations can be computed?