Triangulation
moonlight-triangulation
Moonlight's meshes form a lawful monoid — joining two is union, written <> — and carry finite set algebra. Every panel below is emitted by the engine itself, so each operation has a distinct, legible geometric witness.
Union is a normal form.
The operands are meshed alone before their 125 shared sites descend to one 345-site Delaunay result. The gold clasps make the join's local consequences visible instead of burying them in density.
235 sites, meshed alone — the left operand
235 sites, meshed alone — the right operand
The Delaunay normal form of the union — 345 sites. Gold clasps at the crown dips are the only edges introduced by joining the two site sets.
Seventeen stars enter a 282-site sky. Seventy-eight short gold edges bloom around the new sites where interleaved operands already agree locally.
Refinement is a second operation.
The join stays a join. Composed refinement places forty-six Steiner sites exactly where the crown-dip slivers require them, without pretending union itself performs repair.
The same union after refinement. Forty-six gold Steiner sites dissolve the crown-dip slivers; healing remains a second composed operation.
The same operands carry set algebra.
Intersection keeps the lens, difference keeps one crescent, and symmetric difference keeps both crescents while preserving the certified void between them.
125 sites — the shared lens, retriangulated
110 sites — the left crescent, with its boundary carried in as constraints and its interior certified by the engine rather than cropped
220 sites — both crescents without the shared lens; the void between their constraint loops remains genuinely outside the certified mesh
A gap makes the cost visible.
Separated clouds force the join to expose its long Delaunay teeth. The adjacent sweep then shows six actual refinement runs dissolving that zipper as the angle guarantee tightens.
Two clouds that never touch are laced by thirty-four gold cross-edges. The long teeth are the honest Delaunay shape of joining across a gap.
Six complete engine runs, never interpolated. Boundary splits arrive first; as the guarantee tightens, 161 Steiner sites dissolve the zipper by 14°.
