Regions — from allocation to shape

The region of a space is written as a union of rectangles. The written allocation is not the shape. The shape is a list of convex pieces derived from it.

form.spaces[0].outline
// [ [ {x:0,y:0}, {x:3600,y:0}, {x:3600,y:4500}, {x:0,y:4500} ] ]

Areas, wall positions, roof outlines, where columns stand, projections — every derivation that reads shape goes through this one door.

From allocation to pieces

Each rectangle is first mapped to four counter-clockwise vertices.

{x1,y1,x2,y2}  →  [(x1,y1), (x2,y1), (x2,y2), (x1,y2)]

Then, if a boundary carries a drawn line, the pieces are re-cut by its half-plane.

no line  :  pieces = a copy of the allocation
a line   :  pieces = outside the window + the inside re-cut by the half-plane

Cutting happens exactly once, at the exit of composition. It is not idempotent — a region shrunk by an earlier line shrinks the window of a later one.

The order of cutting is not declaration order

Lines cut in canonical boundary order: the lexicographic order of between, with equal between broken by serialised content. Composition may insert boundaries in the middle; the order does not move.

Cutting in declaration order is forbidden because the canonical form discards declaration order. Cutting by information that is discarded means the same canonical form yields different areas — measured, two crossing lines gave 27.00 m² and 22.50 m² from byte-identical models. Shape must be a function of the canonical form.

The window a line reaches

A line is not an infinite line. Along itself it cuts a limited interval; across itself it cuts a range that contains the other side. The asymmetry is the point — treated as an infinite line it swallows a distant wing and the room vanishes; treated as the bounding box of the segment it degenerates for an axis-parallel line and cuts nothing. Both failures actually happened.

A line is vertical when |Δy| ≥ |Δx| (exactly 45° and a degenerate point count as vertical). For a vertical line the "along" axis is y and the "across" axis is x; for a horizontal line it is the other way round.

Case	Along	Across
Re-dividing two spaces (both sides have a region)	the intersection of the two bounding boxes	the union of the two bounding boxes
Cutting the envelope (one side has no region)	the interval of the segment itself	the whole bounding box of the side that has a region

If either side of the window is not greater than the tolerance EPS, the window has no substance and the line cuts nothing (LIN01).

Which side is kept

Which side of the line survives is not written. It is decided by measuring, whole, the pieces the window touches. Whether a piece is touched is read from bounding-box overlap. Each piece is split by the line, the total area on the left is compared with the total on the right, and a difference below AREA_EPS counts as "no bias".

Cut inside the window only; decide from the whole piece. Measuring every piece lets a wing the line never reaches dominate the sign and the room disappears; measuring only inside the window makes a corner-to-corner line look like a bisection.

• Cutting the envelope — the bias of the side that has a region is the side kept. No bias, no cut
• Two spaces — take the bias of both sides; if exactly one is 0, give it the opposite of the other. If both are 0, or if both point the same way, no cut

Which of a and b was written first has no effect on which side is kept. boundary /L1/room /out and boundary /out /L1/room are two spellings of the same relation, and the side kept is decided by the side that has a region. The direction of a/b matters only for edge and swing — the two things that need "seen from which side". Before this rule existed, reversing the order gave 26 m² and 34 m², and check stayed green.

The operation

The pieces are split into inside and outside the window, and the allocation inside the window is merged before being re-divided across the line.

• Re-dividing two spaces preserves total area — the triangle one loses, the other gains
• Cutting the envelope only shrinks the side with a region — the other side gains nothing

If a half-plane cut leaves fewer than three vertices, or an area at or below AREA_EPS, the piece is treated as never having existed. Without that threshold, hair-thin slivers survive at the end of a chamfer, get read as edges, and grow ghost walls.

A line has no direction

The same two points joined in either order are the same line. Before cutting, the endpoints of every line are ordered by resolved coordinate, ascending in (x, then y) — the same rule canonical JSON uses for the endpoint pair. The spelling (the grid reference) travels with the swap, so a diagnostic quotes exactly what was written, only in the other order.

That canonical start point is the origin for the at: of any opening carried by the line. Without the ordering, canonical JSON stays byte-identical while the door moves — measured, line X1,Y1+2000 X2,Y1+4000 and line X2,Y1+4000 X1,Y1+2000 hashed identically while placing the door at (1500, 2500) and (4500, 3500).

What the cut actually did

What a line did is recorded at the moment it does it. The public type DrawnLine carries it as effect.

Value	Meaning	Diagnostic
"cut"	it really cut the shape	—
"nothing"	it cut nothing — same as the default adjacency line, or the allocation lies outside its reach	LIN03 (warning)
"undetermined"	which side to keep is undecidable — equal bias on both sides, or an exact bisection	LIN01 (error)

It is not recomputed later, because by then the counterpart is the already-cut shape and the population no longer matches. effect is a consequence of derivation, not written composition, so it does not appear in canonical JSON.

Write one

koyu 1.0
unit mm
grid X 0 6000
grid Y 0 6000
level L1 0 h:2700 slab:200

space /L1/a room X1..X2 Y1..Y2 name:隅切りの室
space /out exterior name:外部

boundary /L1/a /out t:150 spec:RC
  line X1,Y2-3000 X1+3000,Y2

The north-west triangle (3000 × 3000 ÷ 2 = 4.5 m²) falls off a 6000 × 6000 allocation.

npx tsx src/cli.ts stats corner.muro

L1
  /L1/a	隅切りの室	room	31.50 m2
  Subtotal 31.50 m2
Total 31.50 m2 (indoor floor area)
  room: 31.50 m2

The allocation is still 36 m². The area comes from the pieces.

Neighbouring pages

• Form — derive(model) and the four promises
• Boundary segments — walls stand on the edges of pieces
• line — how to write a line
• space — how to write an allocation
• LIN diagnostics — what comes out when a cut fails