= Regions of a general-position great-circle arrangement
{title2=$R_N=N(N-1)+2$}
For $N\geq1$ <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 $(\operatorname{sgn}(a_i\cdot x))_i$. 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.
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