which is equivalent to defining the top hemisphere using positive kr and the bottom hemisphere using negative kr. In particular, a coordinate (kr, ϕ, θ) in the bottom hemisphere (i.e. where kr > 0 and θ ∈ [π/2, π) if constricted to the conventional intervals) is equivalent to the coordinate (-kr, ϕ + π, π -θ) in the redefined intervals.