Regions of a general-position great-circle arrangement ($R_N=N(N-1)+2$)

For $N\ge 1$ great circles on a sphere, with no coincident circles or triple intersection, the number of open regions is $R_N=N(N-1)+2$. The first circle gives two regions. Each subsequent circle meets the previous $n-1$ circles at $2(n-1)$ distinct points and is divided into that many arcs, each splitting a previous region. Summing the increments proves the formula. If the circles are $a_i\cdot x=0$, each region has one fixed sign vector $(\mathrm{sgn}(a_i\cdot x){)}_i$. Conversely a realized strict sign vector has a connected region: its cone is an intersection of open half-spaces, and radial projection of line segments connects any two points.