Space-Time Map Layout Algorithm — Worked Examples

Version: 0.3.0

Companion to: layout-algorithm-specification.md Purpose: Traces two concrete fixtures through every stage of the nine-stage pipeline, producing exact intermediate values at each stage boundary. Intended to be read alongside the specification, one document per monitor.

All values here are confirmed by the project's unit test suite. If any value in this document conflicts with a passing test, the test is authoritative.

Fixtures

Two data files in docs/example-data are used:

meetings.ttl — three Persons, two Activities, states on Alice. Simple fixture: no installations, no mereologically transitions.
aircraft684_diagram.ttl — Systems, component slots, installations, slot co-location. Complex fixture: multiple timebands with Y-position changes.

Fixture 1: meetings.ttl

Input Summary

Entity	Type	Key relationships
`alice`	Person	states: Active (2021-06-01 → 2023-06-01), Retired (2023-06-01 → 2025-02-01)
`bob`	Person	no bounds, no states
`carol`	Person	no bounds, no states
`project-meeting`	Activity	2022-09-01 → 2023-02-01; participations: alice, bob, carol
`workshop`	Activity	2023-04-15 → 2023-11-01; participations: alice, carol

Persons are FusionOfIntegralParts (Person is a subclass), but have no visible internal participants in this diagram → rendered as plain bands. Activities render as fusion outlines spanning their participants.

Stage 1 — Ordering Graph Construction

Sub-step 1.1 — Metric breakpoints:

Seven breakpoints are collected from the ParticularPeriod timestamps:

Index	Timestamp	Starting entities	Ending entities
0	2021-06-01	`alice-state-active`	—
1	2022-09-01	`project-meeting`	—
2	2023-02-01	—	`project-meeting`
3	2023-04-15	`workshop`	—
4	2023-06-01	`alice-state-retired`	`alice-state-active`
5	2023-11-01	—	`workshop`
6	2025-02-01	—	`alice-state-retired`

Note: Activities and states both produce breakpoints. States (like alice-state-active) are temporal parts of individuals and generate breakpoints where they start and end — these affect the internal participation/state track display, not Y positions or transitions.

Sub-steps 1.2–1.3 — Ordering constraints:

No entirelyAfter triples in this fixture. Metric inferred edges are generated from adjacent timestamps (e.g., alice-state-active ends at BP_4, alice-state-retired starts at BP_4 — same timestamp, no inferred edge between them; project-meeting ends at BP_2, workshop starts at BP_3 — adjacent timestamps, inferred edge generated).

Sub-step 1.4 — DAG: No cycles. Pipeline continues.

Sub-step 1.5 — Ordinal breakpoints: All entities have metric bounds. No ordinal breakpoints are injected.

Seven metric breakpoints → eight timebands.

Stage 2 — Timeband Decomposition

includeFirst = true (bob.start === undefined, carol.start === undefined). includeLast = true (bob.end === undefined, carol.end === undefined).

Index	Left BP	Right BP	Notes
0	null	2021-06-01	Left-open; alice not yet active
1	2021-06-01	2022-09-01	alice-state-active begins
2	2022-09-01	2023-02-01	project-meeting active
3	2023-02-01	2023-04-15	gap between activities
4	2023-04-15	2023-06-01	workshop active
5	2023-06-01	2023-11-01	alice-state-retired active; workshop active
6	2023-11-01	2025-02-01	alice-state-retired active; workshop ended
7	2025-02-01	null	Right-open; alice-state-retired ended

Eight timebands. ✓

Stage 3 — Entity State Resolution

Entity	Changes across timebands
`alice`	`participatingInFusions`: adds `project-meeting` in TB_2, removes in TB_3; adds `workshop` in TB_4, removes in TB_6. `activeStatePartIds`: `[alice-state-active]` in TB_1–TB_4; `[alice-state-retired]` in TB_5–TB_6; `[]` in TB_0 and TB_7. No `fusedIntoSlots` ever set.
`bob`	`participatingInFusions`: `project-meeting` in TB_2 only. `activeStatePartIds`: always `[]`.
`carol`	`participatingInFusions`: `project-meeting` in TB_2; `workshop` in TB_4 and TB_5. `activeStatePartIds`: always `[]`.

No installations. All fusedIntoSlots remain [] throughout.

Stage 4 — Co-location Groups

None. No slot-to-slot fusions, no installations. No co-location in any timeband.

Stage 5 — Y-Position Assignment

Settings: minBandHeight=30, paddingV=15, rootGap=5, minInternalRowHeight=20.

Pre-computation — sub-row counts:

alice: max concurrent = 2 (e.g. TB_2: alice-state-active sub-row + project-meeting participation). Height = max(30, 2×20) = 40.
bob: max concurrent = 1 (TB_2: project-meeting participation). Height = max(30, 1×20) = 30.
carol: max concurrent = 1. Height = 30.

Pass 1 — Global tree (all three are roots; alphabetical order):

alice:  height=40, yStart=0                  (cursor starts at 0)
bob:    height=30, yStart=45  (40 + rootGap 5)
carol:  height=30, yStart=80  (45 + 30 + rootGap 5)

No structural parent–child relationships. No internal node expansion needed.

Pass 2 — No co-location groups in any timeband. Assignments are identical to Pass 1 in all timebands.

Final Y layout (identical in all 8 timebands):

Entity	yStart	height
alice	0	40
bob	45	30
carol	80	30

Heights are constant — Stage 6 will detect zero transition groups for this fixture.

Stage 6 — Transition Group Construction

TransitionGroupList = [] (0 groups).

alice, bob, and carol have identical Y positions across all timebands — no yStart changes are detected at any breakpoint. The Activities and States produce breakpoints but do not move any entity's band vertically.

Stage 7 — Horizontal Section Planning

Settings: chevronOffset=20, minPaddingPx=20, minStraightSectionPx=10, paddingBetweenTransitionsPx=20.

alice, bob, and carol are open-started and open-ended (depth 0 via isPartOf). However, their participation entities (e.g. alice's participation-alice-meeting) have isPartOf project-meeting, giving nesting depth 1. Max nesting depth among open-started entities = 1.

paddingLeftPx  = 1 × 20 + 20 = 40
paddingRightPx = 40  (same entities in TB_7)

No TransitionGroup → no transition sections. The 8 straight-A sections flow directly into each other. Activity outline boxes (project-meeting, workshop) span the relevant straight-A sections without needing a horizontal transition zone.

#	Type	timebandIndex	minWidthPx
1	padding-left	null	40
2	straight-A	0	10
3	straight-A	1	10
4	straight-A	2	10
5	straight-A	3	10
6	straight-A	4	10
7	straight-A	5	10
8	straight-A	6	10
9	straight-A	7	10
10	padding-right	null	40

Total totalMinWidthPx = 160 px. Section count = 2 + 8 + 0 + 0 = 10 ✓

Stage 8 — Linear Expansion and Final Emission

This fixture has open-bounded timebands: TB_0 has no left metric boundary (null start), and TB_7 has no right metric boundary (null end). However, the 6 interior timebands (TB_1 through TB_6) are bounded by the 7 metric timestamps spanning from 2021-06-01 to 2025-02-01.

minLinearWidthPx is null (open-ended timebands exist). isNonLinear = true. The interior bounded sections receive width proportional to their duration; the open-ended sections receive minWidthPx.

For a canvas large enough to proportionally scale the interior: fixedWidthPx = 40 + 40 = 80 (paddings only). availableForBounded = (canvas − 80). totalBoundedDurationMs = 2025-02-01 − 2021-06-01 ≈ 113,270,400,000 ms. Each bounded interior section gets width proportional to its duration.

timeRange = { startMs: 2021-06-01, endMs: 2025-02-01 } (from the outermost metric-bounded entities at depth 0).

Stage 9 — Worm Plan Construction

Pass A — Z-order:

No installations ever occur. Structural tree: three independent roots. DFS preorder on alphabetically-sorted roots:

alice  → Z = 0
bob    → Z = 1
carol  → Z = 2

Pass B — Segments (alice, at canvas=160 = totalMinWidthPx):

All sections at minWidthPx. Cumulative left-X:

Section #	Left X
0 (padding-left)	0
1 (SA-TB_0)	40
2 (SA-TB_1)	50
3 (SA-TB_2)	60
4 (SA-TB_3)	70
5 (SA-TB_4)	80
6 (SA-TB_5)	90
7 (SA-TB_6)	100
8 (SA-TB_7)	110
9 (padding-right)	120

alice: yStart=0, height=40 → yPos = 0 + 40/2 = 20. nestingDepthStart = 0, unknownStart = true → no left inset. nestingDepthEnd = 0, unknownEnd = true → xPos = 160 − (0+1) × 10 = 150.

N=10 active sections → 11 segments:

i	xPos	yPos	note
0	0	20	left body face; depth=0, unknownStart
1	40	20	SA-TB_0 left
2	50	20	SA-TB_1 left
3	60	20	SA-TB_2 left
4	70	20	SA-TB_3 left
5	80	20	SA-TB_4 left
6	90	20	SA-TB_5 left
7	100	20	SA-TB_6 left
8	110	20	SA-TB_7 left
9	120	20	padding-right left
10	150	20	right body face; depth=0, unknownEnd

unknownStart = true, unknownEnd = true. Height = 40. Style: defaultBand.

bob: yPos = 45 + 15 = 60. carol: yPos = 80 + 15 = 95. Identical pattern.

Pass C — Style: All three persons → defaultBand.

Fixture 2: aircraft684_diagram.ttl

Input Summary for Fixture 2

Entity	Type(s)	Key Relationships
`aircraft684`	System	no bounds
`port-engine-component`	System, ReplaceableLifespan	`isPartOf aircraft684`
`fuelpump-component`	ReplaceableLifespan	`isPartOf port-engine-component`
`engine329-fuelpump-component`	ReplaceableLifespan	`isPartOf engine329`
`engine329`	Artefact, System	no bounds; installed into `port-engine-component`
`fuelpump739`	Artefact	no bounds; installed into `engine329-fuelpump-component`
`sysCompFusion`	FusionOfIntegralParts	links `fuelpump-component` ↔ `engine329-fuelpump-component`; ends 2020-01-01
`engine329-installed`	Installed	links `engine329` ↔ `port-engine-component`; ends 2020-01-01
`fuelpump739-installed`	Installed	links `fuelpump739` ↔ `engine329-fuelpump-component`; ends 2024-06-01

Stage 1 — Ordering Graph Construction

Sub-step 1.1 — Metric breakpoints:

Index	Timestamp	Ending entities
0	2020-01-01	`sysCompFusion`, `engine329-installed`
1	2024-06-01	`fuelpump739-installed`

Two breakpoints. No explicit entirelyAfter triples. Inferred edge: engine329-installed#end → fuelpump739-installed#end (BP_0 < BP_1). No cycles.

Sub-step 1.5 — No ordinal breakpoints (all entities with bounds have metric breakpoints).

Two metric breakpoints → three timebands.

Stage 2 — Timeband Decomposition

includeFirst = true (engine329.start === undefined, fuelpump739.start === undefined, aircraft684.start === undefined — no metric starts). includeLast = true (same entities have no metric ends).

Index	Left BP	Right BP	Description
0	null	2020-01-01	Both installations and slot fusion active
1	2020-01-01	2024-06-01	engine329 uninstalled; fuelpump739 still installed
2	2024-06-01	null	fuelpump739 uninstalled; all standalone

Three timebands. ✓

Stage 3 — Entity State Resolution

Only the rows that change across timebands are shown in detail:

Entity	TB_0	TB_1	TB_2
`engine329-installed` (isActive)	yes	no	no
`sysCompFusion` (isActive)	yes	no	no
`fuelpump739-installed` (isActive)	yes	yes	no
`engine329` fusedIntoSlots	`[port-engine-component]`	`[]`	`[]`
`fuelpump739` fusedIntoSlots	`[engine329-fuelpump-component]`	`[engine329-fuelpump-component]`	`[]`
`fuelpump-component` activeSlotFusionIds	`[sysCompFusion]`	`[]`	`[]`
`engine329-fuelpump-component` activeSlotFusionIds	`[sysCompFusion]`	`[]`	`[]`

Structural entities (aircraft684, port-engine-component, fuelpump-component, engine329-fuelpump-component, engine329) are always active with fixed mereologicalParentId.

Stage 4 — Co-location Groups

TB_0: Two groups arising from the active installation and slot fusion:

Slot-fusion group (groupId=sysCompFusion):
- Raw members from activeSlotFusionIds: fuelpump-component and engine329-fuelpump-component (both slots in the fusion).
- fuelpump739 is installed into engine329-fuelpump-component (a group member) → Union-Find merges its install pair into this group.
- Final members: { engine329-fuelpump-component, fuelpump-component, fuelpump739 }.
- Both engine329-fuelpump-component and fuelpump-component are ReplaceableLifespan → multiple slots → alphabetically first = engine329-fuelpump-component (canonical).
Install group (groupId=install:port-engine-component:engine329):
- engine329.fusedIntoSlots = [port-engine-component] → pair { port-engine-component, engine329 }.
- Exactly one slot member: port-engine-component (canonical).

TB_1: One group:

fuelpump739.fusedIntoSlots = [engine329-fuelpump-component] → pair { engine329-fuelpump-component, fuelpump739 }.
Canonical: engine329-fuelpump-component (the only slot member).

TB_2: No groups.

Stage 5 — Y-Position Assignment

Settings: minBandHeight=30, paddingV=15, rootGap=5, minInternalRowHeight=20.

Pre-computation: All entities have subRowCount = 0 (no participations or state sub-rows). Height = max(30, 0×20) = 30 for all. Except containers: aircraft684 and port-engine-component expand in Pass 1 to contain their children.

Pass 1 — Global structural tree

The global structural tree (from invariant mereologicalParentId):

Root: aircraft684
  └── port-engine-component  [isPartOf]
        └── fuelpump-component  [isPartOf]

Root: engine329
  └── engine329-fuelpump-component  [isPartOf]

Root: fuelpump739  (leaf)

Installed entities (engine329, fuelpump739) are not tree-children of their host slots. Installations are handled by Pass 2 co-location, not by the structural tree.

Root ordering is alphabetical by IRI: aircraft684, engine329, fuelpump739.

Depth-first placement:

Derivation:

cursor=0
aircraft684 (internal) → cursor += 15 → cursor=15
  port-engine-component (internal) → cursor += 15 → cursor=30
    fuelpump-component (leaf): yStart=30, height=30; cursor += 30+5 → cursor=65
  port-engine-component: recompute(children=[fuelpump-component])
    minChildY=30, maxChildBottom=60
    yStart=30−15=15, height=60+15−15=60
  (port-engine-component is not a root, so cursor is not advanced by port-engine-component itself)
aircraft684: recompute(children=[port-engine-component])
  minChildY=15, maxChildBottom=75
  yStart=15−15=0, height=75+15−0=90
(aircraft684 is root) → cursor = aircraft684.yStart + aircraft684.height + rootGap = 0+90+5=95

cursor=95
engine329 (internal) → cursor += 15 → cursor=110
  engine329-fuelpump-component (leaf): yStart=110, height=30; cursor += 30+5 → cursor=145
engine329: recompute(children=[engine329-fuelpump-component])
  yStart=110−15=95, height=(110+30)+15−95=60
(engine329 is root) → cursor = 95+60+5=160

cursor=160
fuelpump739 (leaf): yStart=160, height=30; cursor=195

Pass 1 global base (before co-location):

aircraft684:                  yStart=0,   height=90
port-engine-component:        yStart=15,  height=60
fuelpump-component:           yStart=30,  height=30
engine329:                    yStart=95,  height=60
engine329-fuelpump-component: yStart=110, height=30
fuelpump739:                  yStart=160, height=30

Pass 2 — TB_0 co-location overrides

The working assignments map starts as a copy of the Pass 1 global base. Groups are processed install groups first, then slot-fusion groups.

Step A — Install group { port-engine-component (canonical, yStart=15, height=60), engine329 (yStart=95, height=60) }:

delta = canonical.yStart − member.yStart = 15 − 95 = −80
shiftSubtree(engine329, −80):
  engine329:                    yStart = 95 + (−80) = 15
  engine329-fuelpump-component: yStart = 110 + (−80) = 30   ← entire subtree shifts
pin(engine329, { yStart: 15, height: 60 })
re-expand ancestors of engine329: none (engine329 is a root)

Assignments after install group:

aircraft684:                  yStart=0,   height=90
port-engine-component:        yStart=15,  height=60
fuelpump-component:           yStart=30,  height=30
engine329:                    yStart=15,  height=60   ← shifted
engine329-fuelpump-component: yStart=30,  height=30   ← shifted with engine329's subtree
fuelpump739:                  yStart=160, height=30

Step B — Slot-fusion group { engine329-fuelpump-component (canonical, now yStart=30), fuelpump-component (yStart=30), fuelpump739 (yStart=160) }:

The canonical (engine329-fuelpump-component) is now at yStart=30 — it was moved by the install group in Step A.

Process fuelpump-component:

delta = 30 − 30 = 0   ← already aligned; no movement

Process fuelpump739:

delta = 30 − 160 = −130
shiftSubtree(fuelpump739, −130): fuelpump739 yStart = 160 + (−130) = 30
pin(fuelpump739, { yStart: 30, height: 30 })
re-expand ancestors of fuelpump739: none (fuelpump739 is a root)

Key insight: The install group runs first and shifts engine329's entire subtree (including engine329-fuelpump-component) from 110→30. By the time the slot-fusion group runs, its canonical member already sits at 30, so fuelpump-component requires no movement (it was always at 30 in the global base). Only fuelpump739 moves in the slot-fusion step.

TB_0 final Y layout (confirmed by tests):

aircraft684:                  yStart=0,   height=90
port-engine-component:        yStart=15,  height=60   ←→ engine329 (co-located; install group)
fuelpump-component:           yStart=30,  height=30   ←→ engine329-fuelpump-component, fuelpump739 (co-located; slot-fusion)
engine329:                    yStart=15,  height=60   [isInstalledSystem=true]
engine329-fuelpump-component: yStart=30,  height=30
fuelpump739:                  yStart=30,  height=30

Containment check: fuelpump-component [30..60] ⊂ port-engine-component [15..75] ⊂ aircraft684 [0..90]. ✓

Pass 2 — TB_1

Same global base. One co-location group: { engine329-fuelpump-component (canonical, yStart=110), fuelpump739 (yStart=160) }.

No install groups. Slot-fusion/install group:

delta = 110 − 160 = −50
shiftSubtree(fuelpump739, −50): fuelpump739 yStart = 160 + (−50) = 110

TB_1 final Y layout (confirmed by tests):

aircraft684:                  yStart=0,   height=90
port-engine-component:        yStart=15,  height=60
fuelpump-component:           yStart=30,  height=30
engine329:                    yStart=95,  height=60   [standalone; no active install]
engine329-fuelpump-component: yStart=110, height=30   ←→ fuelpump739 (co-located)
fuelpump739:                  yStart=110, height=30

Pass 2 — TB_2

No co-location groups. All assignments from global base applied directly.

TB_2 final Y layout (confirmed by tests):

aircraft684:                  yStart=0,   height=90
port-engine-component:        yStart=15,  height=60
fuelpump-component:           yStart=30,  height=30
engine329:                    yStart=95,  height=60
engine329-fuelpump-component: yStart=110, height=30
fuelpump739:                  yStart=160, height=30   [standalone root]

Stage 6 — Transition Group Construction

At breakpoint 0 (2020-01-01)

Y comparison TB_0 → TB_1:

Entity	TB_0 yStart	TB_1 yStart	Changed?
aircraft684	0	0	No
port-engine-component	15	15	No
fuelpump-component	30	30	No
engine329	15	95	Yes
engine329-fuelpump-component	30	110	Yes
fuelpump739	30	110	Yes

Three transitions in one group (breakpointIndex=0):

Entity	fromCenterY	toCenterY	height	direction
engine329	15+30=45	95+30=125	60	exit (fusedIntoSlots shrank)
engine329-fuelpump-component	30+15=45	110+15=125	30	exit (fallback: toCenterY > fromCenterY)
fuelpump739	30+15=45	110+15=125	30	exit (fallback)

verticalTravel values: |125−45|=80, |125−45|=80, |125−45|=80
maxTravel = 80
widthPx = max(50, 80/4) = max(50, 20) = 50
tickEdge = 'left' (all exit)

At breakpoint 1 (2024-06-01)

Y comparison TB_1 → TB_2:

Entity	TB_1 yStart	TB_2 yStart	Changed?
aircraft684	0	0	No
port-engine-component	15	15	No
fuelpump-component	30	30	No
engine329	95	95	No
engine329-fuelpump-component	110	110	No
fuelpump739	110	160	Yes

One transition in one group (breakpointIndex=1):

Entity	fromCenterY	toCenterY	height	direction
fuelpump739	110+15=125	160+15=175	30	exit (fusedIntoSlots shrank)

verticalTravel = |175−125| = 50
widthPx = max(50, 50/4) = 50
tickEdge = 'left'

Stage 7 — Horizontal Section Planning

Settings: chevronOffset=20, minPaddingPx=20, minStraightSectionPx=10, paddingBetweenTransitionsPx=20.

Padding-left depth: In TB_0, fuelpump739 has start === undefined and the deepest chain: installed into engine329-fuelpump-component (+1), which is isPartOf engine329 (+1), which is installed into port-engine-component (+1), which is isPartOf aircraft684 (+1). Depth = 4.

paddingLeftPx = 4 × 20 + 20 = 100

Straight-B after BP_0: fuelpump739 transitions at both BP_0 and BP_1 (consecutive breakpoints) → straight-B is inserted between them.

Padding-right depth: In TB_2, fuelpump739 has no temporal hosting (depth 0, standalone). engine329 has no active installation (depth 0). Deepest chain: fuelpump-component → port-engine-component → aircraft684 = depth 2.

paddingRightPx = 2 × 20 + 20 = 60

#	Type	timebandIndex	breakpointIndex	minWidthPx	Note
1	padding-left	null	null	100	depth=4
2	straight-A	0	null	10
3	transition	null	0	50	BP_0, tickEdge='left'
4	straight-B	null	null	20	fuelpump739 at both BP_0 and BP_1
5	straight-A	1	null	10
6	transition	null	1	50	BP_1, tickEdge='left'
7	straight-A	2	null	10
8	padding-right	null	null	60	depth=2

Total totalMinWidthPx = 310 px. Section count = 2 + 3 + 2 + 1 = 8 ✓

Breakpoint tick placement: BP_0 and BP_1 both have tickEdge='left', so their ticks sit at the left edge of their transition sections:

BP_0 tick at x = 100 + 10 = 110
BP_1 tick at x = 100 + 10 + 50 + 20 + 10 + ... wait, let me compute cumulative lefts:
- Section 1 (padding-left): x=0, width=100 → right at 100
- Section 2 (SA-TB_0): x=100, width=10 → right at 110
- Section 3 (transition BP_0): x=110, tickEdge='left' → tick at x=110
- Section 4 (SB): x=160, width=20 → right at 180
- Section 5 (SA-TB_1): x=180, width=10 → right at 190
- Section 6 (transition BP_1): x=190, tickEdge='left' → tick at x=190
- Section 7 (SA-TB_2): x=240, width=10 → right at 250
- Section 8 (padding-right): x=250, width=60 → total=310

Stage 8 — Linear Expansion and Final Emission

This fixture has open-ended timebands: TB_0 has no left metric boundary (null), and TB_2 has no right metric boundary (null). minLinearWidthPx = null (full linearity impossible; cannot scale proportionally without bounded endpoints). isNonLinear = true.

Only TB_1 is fully metric-bounded (BP_0 = 2020-01-01 to BP_1 = 2024-06-01, duration ≈ 143,280,000,000 ms). The open-ended timebands TB_0 and TB_2 receive actualWidthPx = minWidthPx.

At canvas=310 (totalMinWidthPx): all sections at minimum width. actualWidthPx = minWidthPx throughout.

totalWidthPx = 310. totalHeightPx = max(yStart + height in TB_2) = max of aircraft684 (0+90=90), port-engine-component (15+60=75), fuelpump-component (30+30=60), engine329 (95+60=155), engine329-fuelpump-component (110+30=140), fuelpump739 (160+30=190) = 190 px.

timeRange from depth-0 entities (those with no isPartOf parent): aircraft684, engine329, fuelpump739 — all have undefined metric bounds → timeRange = { startMs: 0, endMs: 0 } (default; no metric timestamps on top-level entities).

Stage 9 — Worm Plan Construction

Pass A — Z-order

Structural tree (from mereologicalParentId):

aircraft684 (root)
  └── port-engine-component
        └── fuelpump-component
engine329 (root)
  └── engine329-fuelpump-component
fuelpump739 (root)

Virtual-installed edges (across all timebands):

engine329 was installed into port-engine-component → add engine329 as virtual-installed child of port-engine-component.
fuelpump739 was installed into engine329-fuelpump-component → add fuelpump739 as virtual-installed child of engine329-fuelpump-component.

Augmented tree (structural children first alphabetically, then virtual-installed alphabetically):

aircraft684 (Z=0)
  └── port-engine-component  [structural] (Z=1)
        ├── fuelpump-component  [structural] (Z=2)
        └── engine329  [virtual-installed] (Z=3)
              └── engine329-fuelpump-component  [structural] (Z=4)
                    └── fuelpump739  [virtual-installed] (Z=5)

DFS preorder: aircraft684=0, port-engine-component=1, fuelpump-component=2, engine329=3, engine329-fuelpump-component=4, fuelpump739=5.

Pass B — Segments (fuelpump739, at canvas=310)

Section boundaries (all at minWidthPx):

#	Section	Width	xLeft
1	padding-left	100	0
2	straight-A (TB_0)	10	100
3	transition (BP_0)	50	110
4	straight-B	20	160
5	straight-A (TB_1)	10	180
6	transition (BP_1)	50	190
7	straight-A (TB_2)	10	240
8	padding-right	60	250

fuelpump739 center Y per timeband: TB_0 30+15=45, TB_1 110+15=125, TB_2 160+15=175.

nestingDepthStart = 4 (fp739 in TB_0: install-hop to engine329-fuelpump-component, then isPartOf to engine329, then install-hop to port-engine-component, then isPartOf to aircraft684). unknownStart = true → segments[0].xPos = 0 + 4 × 10 = 40.

nestingDepthEnd = 0 (fp739 in TB_2: no temporal hosting, standalone root). unknownEnd = true → segments[8].xPos = 310 − (0+1) × 10 = 300.

N=8 active sections → 9 segments:

i	Vertex	xPos	yPos	Note
0	left body face	40	45	pre-baked; depth=4, unknownStart
1	SA-TB_0 left	100	45	T_s = TB_0
2	TR-BP_0 left	110	45	source Y (TB_0 left edge of transition)
3	SB left	160	125	destination Y carried here (TB_1)
4	SA-TB_1 left	180	125	T_s = TB_1
5	TR-BP_1 left	190	125	source Y (TB_1 left edge of transition)
6	SA-TB_2 left	240	175	destination Y carried here (TB_2)
7	PR left	250	175	T_s = TB_2
8	right body face	300	175	pre-baked; depth=0, unknownEnd

The renderer connects consecutive dots: the transition diagonal at BP_0 runs from segment[2] to segment[3] (yPos 45→125). No Y-averaging is needed.

Pass B — Segments (engine329)

engine329 center Y per timeband: TB_0 15+30=45, TB_1 95+30=125, TB_2 95+30=125.

nestingDepthStart = 2 (engine329 in TB_0: install-hop to port-engine-component, then isPartOf to aircraft684). unknownStart = true → segments[0].xPos = 0 + 2 × 10 = 20.

nestingDepthEnd = 0 (engine329 in TB_2: not installed). unknownEnd = true → segments[8].xPos = 310 − 10 = 300.

i	Vertex	xPos	yPos	Note
0	left body face	20	45	depth=2, unknownStart
1	SA-TB_0 left	100	45
2	TR-BP_0 left	110	45	source Y
3	SB left	160	125	destination Y carried here
4	SA-TB_1 left	180	125
5	TR-BP_1 left	190	125	source = destination = 125 (no Y change)
6	SA-TB_2 left	240	125
7	PR left	250	125
8	right body face	300	125	depth=0, unknownEnd

engine329 does not change Y at BP_1 (yStart=95 in both TB_1 and TB_2). Segments 5 and 6 have the same yPos → the BP_1 transition renders as a horizontal rectangle for this entity.

Pass C — Style assignments

Entity	isSystem	isSystemComponent	everHadFusedIntoSlots	Style key
aircraft684	true	false	false	`rootSystem`
port-engine-component	true	true	false	`componentSlot`
fuelpump-component	false	true	false	`componentSlot`
engine329	true	false	true	`installedSystem`
engine329-fuelpump-component	false	true	false	`componentSlot`
fuelpump739	false	false	true	`installedArtefact`

port-engine-component is both isSystem and isSystemComponent; isSystemComponent takes precedence since the componentSlot condition is checked before rootSystem in the classification table.

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Space-Time Map Layout Algorithm — Worked Examples

Fixtures

Fixture 1: meetings.ttl

Input Summary

Stage 1 — Ordering Graph Construction

Stage 2 — Timeband Decomposition

Stage 3 — Entity State Resolution

Stage 4 — Co-location Groups

Stage 5 — Y-Position Assignment

Stage 6 — Transition Group Construction

Stage 7 — Horizontal Section Planning

Stage 8 — Linear Expansion and Final Emission

Stage 9 — Worm Plan Construction

Fixture 2: aircraft684_diagram.ttl

Input Summary for Fixture 2

Stage 1 — Ordering Graph Construction

Stage 2 — Timeband Decomposition

Stage 3 — Entity State Resolution

Stage 4 — Co-location Groups

Stage 5 — Y-Position Assignment

Pass 1 — Global structural tree

Pass 2 — TB_0 co-location overrides

Pass 2 — TB_1

Pass 2 — TB_2

Stage 6 — Transition Group Construction

At breakpoint 0 (2020-01-01)

At breakpoint 1 (2024-06-01)

Stage 7 — Horizontal Section Planning

Stage 8 — Linear Expansion and Final Emission

Stage 9 — Worm Plan Construction

Pass A — Z-order

Pass B — Segments (fuelpump739, at canvas=310)

Pass B — Segments (engine329)

Pass C — Style assignments

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Space-Time Map Layout Algorithm — Worked Examples

Fixtures

Fixture 1: meetings.ttl

Input Summary

Stage 1 — Ordering Graph Construction

Stage 2 — Timeband Decomposition

Stage 3 — Entity State Resolution

Stage 4 — Co-location Groups

Stage 5 — Y-Position Assignment

Stage 6 — Transition Group Construction

Stage 7 — Horizontal Section Planning

Stage 8 — Linear Expansion and Final Emission

Stage 9 — Worm Plan Construction

Fixture 2: aircraft684_diagram.ttl

Input Summary for Fixture 2

Stage 1 — Ordering Graph Construction

Stage 2 — Timeband Decomposition

Stage 3 — Entity State Resolution

Stage 4 — Co-location Groups

Stage 5 — Y-Position Assignment

Pass 1 — Global structural tree

Pass 2 — TB_0 co-location overrides

Pass 2 — TB_1

Pass 2 — TB_2

Stage 6 — Transition Group Construction

At breakpoint 0 (2020-01-01)

At breakpoint 1 (2024-06-01)

Stage 7 — Horizontal Section Planning

Stage 8 — Linear Expansion and Final Emission

Stage 9 — Worm Plan Construction

Pass A — Z-order

Pass B — Segments (fuelpump739, at canvas=310)

Pass B — Segments (engine329)

Pass C — Style assignments