moonlight-triangulation

Part of Moonlight, the sheaf-theoretic computation layer beneath Melusine and Pale Meridian.

moonlight-triangulation carries Delaunay and constrained Delaunay triangulations as a lawful finite-set algebra under canonical observation: a mesh represents its site set, a join returns a valid Delaunay representative, and the result is a triangulation again — so the operations close, compose, and fold. Operations return typed obstructions where the finite arena cannot represent a result.

Delaunay triangulation, constrained Delaunay (CDT), exact rational planar regions and labelled overlay, intrinsic valuations, polygonal Minkowski morphology, exact regular/power geometry, the Voronoi dual, natural-neighbour interpolation, Ruppert refinement, walk point location, convex hull, exact Shewchuk predicates, exact zigzag persistence across non-nested activation depths, and versioned binary serialization.

Persistence rose

Open the animation directly. · Open the static vector poster. · Read the exhibit guide.

Operations

Interpret cell sets as incidence categories. · Compute persistent alpha topology. · Track topology across non-nested activation depths. · Compute weighted alpha persistence.

Exact power cells and affine envelopes

PowerWeight admits a finite signed additive offset through its own PowerWeightError; PowerSite admits its binary64 position once. regularTriangulation constructs the exact lifted upper hull once. It retains every admitted labelled site as its semantic value; visibility, regular faces, and weighted-dual segments, rays, full lines, or collapsed degenerate edges are sealed derived projections. Pure insertion, removal, and batch reweighting return exhaustive typed disposition transitions. Exact conflict-cavity and face-star descent update only affected topology; an internal typed refusal falls back once to the canonical batch constructor. boundedPowerDiagram then clips only regular-neighbour inequalities against the ConvexPolygon; globally hidden sites need no clip, while lower-dimensional sites use the complete HPI oracle. It retains one full-dimensional, lower-dimensional, empty, coincident-equivalent, or coincident-dominated result per label. powerDiagramPlanarLayer is only the derived full-dimensional view. When a regular value is already available, boundedPowerDiagramFromRegular reuses its normalized lifted hull.

upperEnvelope maps exact c0 + cx*x + cy*y forms into the same owner. Its planar result contains only two-dimensional winning regions, with Nothing outside the window; nonconvex windows and holes use the canonical overlay.

Edit and reweight the complete regular-site section. · Lower weighted topology into exact persistence.

Foreign bindings

The C ABI and its Python, TypeScript, and Rust consumers are documented in the foreign-bindings guide. It owns construction, ABI versioning, lifecycle, obstruction semantics, and consumer examples.

Algebraic contract

union is idempotent; commutativity and associativity hold after canonicalize. Structural Eq is resident equality; use siteRelation for support comparison.

Set-operation types

union :: JoinSemilattice annotation
      => Triangulation 'Unconstrained annotation () () ()
      -> Triangulation 'Unconstrained annotation () () ()
      -> Either BuildError (Triangulation 'Unconstrained annotation () () ())

unions :: JoinSemilattice annotation
       => [Triangulation 'Unconstrained annotation () () ()]
       -> Either BuildError (Triangulation 'Unconstrained annotation () () ())

siteRelation
      :: Triangulation leftMode leftAnnotation leftDirected leftUndirected leftFace
      -> Triangulation rightMode rightAnnotation rightDirected rightUndirected rightFace
      -> SiteRelation

intersection
      :: Triangulation 'Unconstrained () () () ()
      -> Triangulation 'Unconstrained () () () ()
      -> Either BuildError (Triangulation 'Unconstrained () () () ())

intersectionWith
      :: (leftAnnotation -> rightAnnotation -> annotation)
      -> Triangulation 'Unconstrained leftAnnotation () () ()
      -> Triangulation 'Unconstrained rightAnnotation () () ()
      -> Either BuildError (Triangulation 'Unconstrained annotation () () ())

difference
      :: Triangulation 'Unconstrained leftAnnotation () () ()
      -> Triangulation 'Unconstrained rightAnnotation () () ()
      -> Either BuildError (Triangulation 'Unconstrained leftAnnotation () () ())

symmetricDifference
      :: Triangulation 'Unconstrained annotation () () ()
      -> Triangulation 'Unconstrained annotation () () ()
      -> Either BuildError (Triangulation 'Unconstrained annotation () () ())

Zigzag composition

The activation surface composes admitted local geometry with exact global reduction through the existing typed obstruction:

activationAlphaPersistence slices =
  activationAlphaZigzag slices >>= activationZigzagIntervals

Each slice yields a canonical labelled subcomplex K_i. Adjacent sections glue through K_i -> K_i ∪ K_{i+1} <- K_{i+1}: the union is the join of labelled subcomplexes, and both legs are checked chain maps satisfying boundary . inclusion = inclusion . boundary. The glued zigzag is therefore a finite path in chain complexes; rational homology and interval decomposition produce its authoritative global barcode. Betti profiles are derived views, while any failed local, overlap, or gluing obligation remains an ActivationZigzagError.

Publication schedules and scale behavior.

Use

Use the broad default library with explicit Vector inputs. It exposes the facade and the safe leaf modules; construction and set algebra retain their typed obstruction in Either:

build-depends:
  base >= 4.19 && < 5,
  moonlight-triangulation >= 1.4 && < 1.5,
  vector >= 0.13 && < 0.14

Use withScopedTriangulation for allocation-free local incidence traversal. Use Handles.Dynamic when a handle must escape. Fixed VertexId/edge/face values are unchecked resident indices and belong only with the mesh that issued them.

SetAlgebra.hs is the smallest compile-checked workflow. The example index adds constrained interiors, alpha-boundary descent, exact overlay and valuations, and polygonal morphology.

For a smaller compile/dependency cone, depend on only the component modules you import:

build-depends:
  moonlight-triangulation:dcel >= 1.4 && < 1.5,
  moonlight-triangulation:build >= 1.4 && < 1.5

Architecture

The resident DCEL is immutable structure-of-arrays over paged copy-on-write storage; local mutation is sealed in ST. Half-edge twins are index complements. Exact predicates use bounded machine-word evaluation and fall through to aligned Integer arithmetic when required.

Consumer libraries

The Cabal manifest owns component visibility and dependencies; this is its consumer-facing projection. Use a public sublibrary directly when its smaller dependency cone matters; trusted implementation units remain private.

Build and validate package components locally.