0072 — Column solids over one planar arrangement

• Status: Accepted
• Date: 2026-09-24
• Deciders: Friedrich Schrödter
• Supersedes: —

Part of axiolid/kernel#120.

Context

Two coaxial booleans and one plane cut were still refused by name after ADR 0071:

• a union of prisms with different spans, and a difference whose tool stops inside the subject (a counterbore, a blind pocket, a slot through the middle heights) — the result is stepped;
• a sloped plane that crosses a prism's top or bottom cap inside the section — part of the old cap survives next to the cut.

None of these is one prism, but all of them are the same shape: the plan is cut into cells, and above each cell the solid occupies a stack of height intervals, each bounded by a plane (flat, or sloped as in ADR 0071). Which intervals a cell carries depends only on which input rings contain it.

boolean_stepped::union_prisms_stepped already returned such a result as bands, but left the ledge faces between bands unbuilt. Building each band with its own overlay and gluing them fails for a structural reason: two overlays round crossing points independently, so the ledge and the band walls would disagree on vertices by an ulp and the shell would not close.

Decision

Build every such result from one exact arrangement of all input rings and one column builder over it.

• axiolid-overlay gains ArcArrangement: every ring's edges split at every crossing, each piece labelled with the rings containing its left and its right side, shared boundary pieces merged and naming every ring that carries them (EdgeSource: ring, edge, direction). regions(pred) links the pieces bounding any membership predicate into outer and hole rings. It reuses the exact predicates and ring assembly of arc_overlay (ADR 0070), which now shares that assembly.
• axiolid-construct::column (crate-private) takes the arrangement, a list of planes and a function from membership mask to height blocks, and emits: a cap per (plane, facing) over the cells where a block ends there; a wall per arrangement piece over each height run where exactly one side is solid, facing the empty side; vertical edges split at every height any face meets at a vertex, so walls meeting there share edges. Faces are grouped into shells through shared edges; each connected shell with positive enclosed volume is one ExactBRep.
• Curved walls stay Cylinder faces; sloped rims keep the ellipse edge and Sinusoid2 pcurve of ADR 0071, built by the same helpers.

The public entry points change behaviour, not signature:

• boolean_prisms_exact, boolean_arc_prisms_exact and their _solids variants build stepped results instead of refusing them. The equal-span cases keep their previous path.
• clip_arc_prism_exact builds a plane crossing a cap.
• Names: a wall is named after the operand edge it lies on (Side(n) counted across that operand's rings, as its own extrusion counts them), a cap after the operand whose own cap lies on that plane (subject first), and the floor or ceiling a difference leaves after the tool cap that made it. Only the tool can leave one: the subject's own opposite cap never bounds a coaxial result.

Still refused, by name:

• A result enclosing a cavity (a tool buried inside the subject). The builder can build the void shell, but tessellate reads only solids()[0].outer and treats voids as boolean intent, so a void would be lost silently downstream. The refusal names the cavity.
• Pieces of solid that touch only along an edge: not a manifold solid.
• Non-coaxial curved booleans and cones, spheres, tori and NURBS. Those need general surface/surface B-rep booleans (OCCT's BOPAlgo scale); the column reduction does not apply and nothing here pretends it does.

Alternatives considered

Option	Why not
One overlay per band, glue the bands	Crossing points round independently per overlay; shared vertices disagree by an ulp and the shell cannot close without a tolerance-based sewing step.
Snap per-band vertices within tolerance	Exactly the tolerance guessing ADR 0070 removed from the planar path.
Keep returning bands and let callers assemble	Every caller would rebuild the ledge logic; the band list is also not a solid, so it cannot enter the exact B-rep pipeline.
General B-rep boolean now	Correct end state for non-coaxial cases, but orders of magnitude more work; the vertical-column family covers the building-model cases #120 names (openings, counterbores, stepped footings, sloped roofs over columns).

Consequences

Positive

• Every vertical-column result shares one vertex table, so faces agree on every vertex by index and the audit checks them against each other.
• One builder replaces three special cases (single prism, sloped cap, bands) for everything it covers; the single-prism paths remain for their existing inputs.
• Found and fixed a measurement bug on the way: exact_properties ignored face orientation, which only showed on solids off z = 0.

Negative / costs

• The builder is crate-private and specialised to vertical columns; a general boolean will need its own topology construction.
• Cavities stay refused until tessellation and measurement handle void shells.

Follow-ups / risks to watch

• Non-coaxial curved booleans and cones/spheres/tori need their own issue.
• boolean_stepped::union_prisms_stepped is now the lighter alternative, kept for callers who want bands; its volumes are checked against the solid's.

Relation to existing code

• crates/algorithms/planar/overlay/src/arrangement.rs, src/exact_arc/arrangement.rs: ArcArrangement.
• crates/algorithms/construction/construct/src/column.rs: the builder.
• crates/algorithms/construction/construct/src/boolean_column.rs: the coaxial and clip adapters.
• crates/algorithms/construction/construct/src/boolean_exact.rs: entry points, now routing stepped and crossing cases to the adapters.
• crates/algorithms/query/measure/src/exact.rs: the orientation fix.

Verification

• 11 column tests (column_booleans.rs), every volume derived by hand from the inputs: a slab-and-tower union (48), a corner notch (114), a blind pocket (104), a round union with the lens subtracted, a counterbore (11 pi), a strip that splits a disc only below z = 1 (one solid, not two), the single- and multi-solid paths agreeing, a hole's wall numbering after the outer ring, and both non-manifold refusals. Every solid passes the geometric audit.
• The clip tests that pinned the crossing-cap refusal now check the built solid against a closed form for the kept volume (disc_excess, the integral of max(0, x - a) over the unit disc).
• boolean_stepped bands and the stepped solid agree on volume.
• 7 arrangement tests: labels against arc_overlay, inclusion-exclusion over three rings, shared pieces carrying both sources, source indices surviving ring reversal, crossings shared exactly.
• A 16-fault mutation probe, scripts/probe_column_mutants.py, catches all 16. Its first run caught 14 of 17: nothing split a wall where it changes sides, and nothing pinned hole-wall numbering, so tests for both were added. The seventeenth fault targeted a naming branch no boolean can reach; the branch was removed. Flipping cap facing by hand fails the column tests, as a check on the probe itself.
• The measurement fix has its own test (a raised unit cube measured 7/3), which fails without the fix.