Suppose that I have many points $p_1$,..., $p_n$, all very close to origin 0. For R very, very large, I have circular arcs of radius R, $\alpha_i$, about $p_i$. I do not speak about their parameterization because in my question I will allow the re-parameterization of arcs.
Suppose that:
1) Gauss map is injective on $\cup im(\alpha_i)$ meaning even for different arc, no two normal vector can be the same.
2) Suppose the Gauss map cover most of the circle except for some small intervals of angles $[\theta_i,\theta_{i+1}]$
It can sometimes be obstructed to interpolate these curves together to be a C^2 convex simple, closed, curve. This is because it is easy to construct examples where there are lines which intersect multiple arcs along the same ray.
Question: If I fix a $\theta_i, \theta_{i+1}$ window, for what R can I interpolate these arcs (i.e. introduce smooth small curves connecting them) together to get the $C^2$ smooth convex, simple, closed, curve (allowing re-parameterization of the circular arcs)? Is there an explicit way to perform this interpolation?