I’m interested in getting some estimates for how much area I can see in average if I’m above (let’s say height $h$) of a surface diffeomorphic to a sphere.
Let’s start with a circle of radius $r$. If I’m on a height $h$ (i.e. on a distance of $h+r$ from the center), the horizon is at a distance of $d= \sqrt{2rh + h^2}$ and the length of the perimeter I can see in one direction is $r \arctan (d/r)$ which gives in lowest order of $h$ $x=\sqrt{2rh}$.
If I’m on a sphere, the area I can see is in lowest order of $h$: $A = \pi x^2 = 2 \pi r h$. If I’m on a surface with the two main curvatures $\kappa_{1,2} = 1/r_{1,2}$ the area is in lowest order an ellipse with area $A = 2 \pi \sqrt{r_1 r_2} h$. We know by Gauss-Bonnet that if we integrate the Gauss curvature $\kappa_{1}\kappa_{2}$ over a surface diffeomorphic to a sphere we get $4\pi$.
Putting this together we can see after elementary algebraic manipulation that in lowest order we have $$<A^{-2}>^{-1} = \pi h^2 \cdot \textrm{(Surface area of the deformed sphere),}$$ where by $<x>$ I mean the quantity $x$ averaged over the surface, i.e. take the integral of $x$ over the surface and divide by the area of the surface.
My question is: Can one go beyond the second order in $h$, i.e. $<A^{-2}>^{-1} = \pi \textrm{(Surface area of the deformed sphere)} \cdot h^2 + a h^3 + b h^4 + \ldots$.
Which geometric property is contained in $b, c, \ldots$?
I would also be interested in averages of other powers of the area $A$ one can see from one point hovering at a distance $h$ over the surface as a function of $h$ and geometric properties of the surface considered.