Local space begins at one chosen origin rather than surveying every place at once. Each body has an azimuth at that moment, and the geometric construction follows the bearing outward as a great circle.
The azimuth uses atan2 so its quadrant survives: A = π + atan2(sin H, cos H sin φ − tan δ cos φ). A plain arctangent can silently point a line into the opposite quarter of the compass.
These arcs describe initial bearings on a sphere. They are geometric references for observation, not instructions to travel or relocate. This page is an explainer; local-space arcs enter the interactive globe in v1.