Displacement

Displacement is how many buildings a road set takes — not parcels — counted by the share of each one it takes. Building i contributes cᵢ = Kᵢ / Aᵢ: the area Kᵢ of its footprint the road corridor covers, over its whole area Aᵢ. Displacement is Σcᵢ; the reported fraction divides by the number of buildings.

Where the real footprint outlines are available, Aᵢ is the footprint. Where only a building's location is known, it is a disk of radius half its nearest-neighbour distance — an estimate of the footprint, measured by the same rule.

This is the cost half of the project's one graded tradeoff — the benefit half is permeability. The figure below prices the cost and nothing else: it says what a road set displaces, never whether that road set is worth building. A road it reports as cheap is not thereby a good road — what the road buys is permeability, and nothing here computes it.

The model drawn literally on block ZAF.9.3.1_1_5810: every one of its 6619 buildings is drawn as its outline, the road corridor runs beneath them, and each building is shaded by the share cᵢ of it the corridor takes. The road drawn here is one road at the 7 m floor, and it costs 91.9 buildings — 1.4% of the block. Drag either end of a road, widen the corridor, or switch the second road on. Switched on where it is baked, the two roads cost 187.3 between them; dragged onto each other, the same two cost 91.9 — exactly what the one road costs, because what is charged is the union of the roads' own buffers, so the overlap is paid for once.

Width is per-road

Each road is buffered by its own width_m / 2 before anything is charged against it — there is no width shared by every road in the set, and no global corridor width anywhere in the model. A narrow lane therefore costs less corridor, and so displaces fewer buildings, than a wide street run along the same line.

Overlap is free, by construction

What a road set is charged against is the union of every individual road's own buffer. Two roads that run side by side — opposing one-way lanes flanking the same median, say — fall inside one shared corridor and are charged for it once; pulling them apart widens the union and costs more. No separate rule against double-charging overlap is needed, and none exists — it falls straight out of buffering each road on its own and only then taking the union, before any distance is measured.

Parcels are not buildings

Block.parcels and Block.building_geometries are two distinct fields on the same block. A parcel is a cell of the tessellated block interior — land, not a structure — and that tessellation is the Voronoi diagram of the building points, so a parcel is normally one building's own share of the block: one cell, one building, as on the block drawn above. Normally, not always — clipping a cell against a ragged block edge can split it into separate lobes, and each lobe becomes a parcel in its own right. A lobe that ends up holding no building is charged nothing, because there is nothing inside it to charge.

What separates the two is therefore not whether a parcel holds a building, but what the charge is measured against. Displacement is charged per building, against that building's own footprint, and never per parcel against parcel area. A road crossing one large parcel out at the sparse edge of a block is charged for the part of the one building standing in it that the road covers, not for how much land it takes; the same road through the packed core is charged, building by building, the share cᵢ of each footprint it covers.

Gap-hugging is free

Because cᵢ is exactly zero for any building the corridor does not cover, a road that hugs the tightest gap it can find between two buildings pays nothing at all for how close it runs. Displacement only ever prices the part of a building a corridor actually covers — there is no charge for merely coming close, and no credit either.

A share, not a verdict

A corridor clipping a building's corner costs that corner, not the whole home. That matters most where methods are compared at equal displacement: a charge for any contact would over-count the methods whose roads graze many buildings, and every comparison would then favour the others by construction.