I am wondering if a kind of theorem of tangents from a point to a circumference holds in the sphere.
The situation that I have in mind is the following. Take two geodesics $\gamma_1$ and $\gamma_2$ starting from a point $p\in S^2$ in direction $v_1$ and $v_2$ respectively ($\gamma_i$ parametrised by arc-lenght). follow the two geodesics for $\varepsilon$-time and find two points $x_1$ and $x_2$ whose distance from $p$ is $\varepsilon$. Now in $\gamma_i(\varepsilon)$ we have, considerated as vectors, the frame $\{\gamma_i(\varepsilon),\gamma_i'(\varepsilon), n_i\}$, where $n_i=\gamma_i(\varepsilon)\times\gamma_i'(\varepsilon)$. Now, consider the two geodesics starting respectively from $x_1,x_2$ and directions $n_1,n_2$.
Is it true that the two geodesics meet in a point C?
Is it true that $\mathbf{d_{S^2}(x_1,C)=d_{S^2}(x_2,C)=r}$?
How can we write the parameterization of the piece of circle starting to $\mathbf{x_1}$, ending in $\mathbf{x_2}$, with radius $\mathbf{r}$ and center $\mathbf{C}$?
Bonus: What is the intrinsic curvature of this piece of circle?
Thank you in advance for your comments or responses.