Spatial Expressions
& Geometry
The specification boundary for exact positions, extents, edges, geometric relationships, transformations, and canonical spatial equivalence.
This section translates accepted Foundational Survey Fabric requirements into the specification rules for spatial expressions and geometry. It now incorporates the candidate mathematical design selected through Formal Mathematical Decisions FMD-01 through FMD-09 wherever those decisions directly resolve primitive geometry, exact predicates, boundary inclusion, normalization, and canonical geometric equivalence.
The candidate core now defines exact Point, Segment, and Simple Closed Polygonal Extent (SCPE) geometry over CRPC coordinates, together with deterministic predicates for orientation, incidence, intersection, containment, connectedness, and polygon validity. General Boolean composition, arbitrary exact rotation closure, curve primitives, and richer composite geometry remain outside the current closed candidate core.
Specification Principle One spatial mathematics. Many spatial expressions.
Core Polygonal Geometry Selected
FMD-01 supplies exact CRPC coordinates. FMD-05 selects the minimum primitive geometry set. FMD-06 supplies exact geometry predicates and polygon validation. FMD-07 supplies deterministic canonical normalization. FMD-02 supplies the orientation convention used by polygon winding. FMD-08 supplies the canonical machine representation for the current core.
FMD-03, FMD-04, and FMD-09 provide the origin, Domain envelope, and precision semantics inherited by this geometry. The present integrated core is intentionally polygonal and exact. It does not yet claim full closure for arbitrary Boolean composition, arbitrary exact rotation, curve geometry, or every possible future composite extent.
Exact Spatial Position
A canonical Point is an exact ordered pair (x,y) of valid normalized CRPC coordinates within the Survey Domain.
A Point denotes one exact spatial position. It remains a spatial expression rather than an independently identified foundational entity. Higher layers may attach identity, function, or meaning to a referenced Point without altering the Survey Fabric.
Exact Straight Connection Between Two Points
A canonical Segment is the closed straight spatial set between two distinct valid canonical Points.
Zero-length Segments are invalid. Geometric Segment identity is undirected, although ordered traversal may be used by algorithms such as polygon winding or orientation.
Simple Closed Polygonal Extent
The current canonical extent primitive is the Simple Closed Polygonal Extent (SCPE).
An SCPE is a finite ordered cycle of at least three distinct valid Points whose consecutive Points define valid Segments, whose boundary is simple and non-self-intersecting, whose area is nonzero, and whose denoted spatial set is the polygon boundary together with its interior.
The current SCPE primitive contains one exterior cycle and no interior cycles. Holes are therefore not part of the primitive itself.
Closed Segments · Boundary-Inclusive Extents
Canonical Segments include both endpoints.
Canonical SCPEs include their boundary. A Point may therefore classify exactly as INTERIOR, BOUNDARY, or EXTERIOR relative to a valid SCPE.
Boundary inclusion is geometric only. Higher layers determine what presence on a boundary means for Parcels, Regions, rights, infrastructure, or World membership.
Deterministic Left, Right, or Collinear
For valid Points A, B, and C, orientation is determined exactly by: orient(A,B,C) = (x₂-x₁)(y₃-y₁) - (y₂-y₁)(x₃-x₁).
Positive result means left / counterclockwise, negative means right / clockwise, and zero means collinear under the canonical right-handed Survey frame.
Exact Incidence
A Point lies on a Segment exactly when it is collinear with the Segment endpoints and lies within both exact coordinate bounds of the closed Segment.
No epsilon, tolerance, near-collinearity, or snapping rule is permitted in canonical incidence.
Exact Contact Classification
Canonical Segment intersection shall be evaluated exactly and may classify as: DISJOINT, TOUCH, PROPER_CROSS, COLLINEAR_OVERLAP, or IDENTICAL.
These results are geometric facts only and do not assign higher-layer meaning to the contact.
Exact Interior, Boundary, or Exterior
Point classification against a valid SCPE shall test boundary incidence first and then use an exact parity-based interior test or another Specification-proven equivalent exact algorithm.
Canonical classification shall not depend on floating-point tolerance, rasterization, display resolution, or implementation-specific epsilon.
Simple, Nondegenerate, Hole-Free Cycle
A valid SCPE shall have at least three distinct vertices, no equal consecutive vertices, no zero-length edges, no nonconsecutive duplicate vertices, no forbidden nonadjacent edge intersection, one simple boundary cycle, nonzero exact area, and complete containment within the Survey Domain.
Adjacent edges may meet only at their shared endpoint. Nonadjacent edges may not cross, touch, overlap, or coincide.
Derived Exact Topology
A valid SCPE is connected and path-connected by construction. Its boundary is one connected simple cycle, and its denoted closed polygonal extent is one connected spatial set.
Continuity and connectedness remain derived geometric properties rather than independently stored foundational facts.
Exact Spatial Relationships
The Survey Fabric shall support exact Point containment, SCPE containment, boundary contact, and separation for the current primitive geometry profile.
Such relationships remain mathematical. Semantic containment, ownership, adjacency policy, access, or institutional consequence belongs above the foundation.
Equivalent Geometry, One Spatial Meaning
Two supported geometry expressions are geometrically equivalent when they denote the same exact spatial set.
Different SCPE start vertices, reversed traversal, or exact redundant collinear vertices do not create different geometry when normalization proves the denoted spatial set is unchanged.
One Normal Form for Supported Geometry
Point coordinates shall be CRPC-normalized. Standalone Segment endpoints shall be ordered lexicographically. SCPE normalization shall remove exact redundant collinear middle vertices, enforce counterclockwise traversal, rotate the cycle so the lexicographically least vertex is first, and omit a repeated terminal closure vertex.
Normalization shall be idempotent and geometry-preserving: N(N(G)) = N(G) and geom(N(G)) = geom(G).
Approximate simplification, near-collinear pruning, raster simplification, and tolerance-based shape alteration are prohibited in canonical normalization.
Derived Geometry Must Resolve Back to Canonical Space
Exact translation by CRPC offsets and uniform scaling by positive rational factors preserve CRPC coordinates and are compatible with the current core.
Arbitrary rotation, reflection, mirroring, and more general transformations may be used by derived views, but full canonical closure for arbitrary transformed output is not yet established because such operations may produce coordinates outside the current rational scalar domain.
No transformation may redefine canonical orientation, Pang scale, or authoritative Survey position.
Reference Without Shape Constraint
Higher-layer spatial entities may use supported exact FSF geometry without being forced to align to Survey subdivisions, tiles, cells, or indexing structures.
The Survey Fabric provides exact spatial truth. Higher layers remain responsible for what the geometry represents.
No Foundational Object Identity
Points, Segments, SCPEs, boundaries, and related spatial forms remain exact mathematical expressions rather than independently governed entities with ownership, lifecycle, institutional identity, or semantic role.
Higher layers may use those expressions to define entities appropriate to their own Architecture domains.
Do Not Pretend Full Closure
The current candidate core does not yet define curves, arbitrary interior-ring polygons, unrestricted composite set geometry, or complete Boolean-result closure as primitive canonical forms.
General union, intersection-result geometry, difference-result geometry, arbitrary composition, and richer topology remain Specification 05 and future formal-design concerns. They shall not be invented inside this section merely to broaden the primitive set.
Integrated Candidate Mathematics
CRPC Point; closed Segment; boundary-inclusive hole-free SCPE; exact orientation; exact incidence and Segment intersection; exact Point-in-SCPE; deterministic polygon validity; connectedness; containment; geometric equivalence; deterministic normalization; exact polygonal area compatibility; no epsilon geometry.
Still Open in the Specification
General Boolean/composite geometry closure; unrestricted union/intersection/difference result forms; curve primitives if ever required; richer hole-bearing composite geometry; arbitrary exact transformation closure; and any future geometry proved necessary beyond Point, Segment, and SCPE.
Core Polygonal Geometry Ready for Prototype and Conformance Work
No contradiction was found between Specification 04 and FMD-01 through FMD-09. The current Point / Segment / SCPE core is sufficiently exact to support deterministic implementation, the selected Spatial Ground FEBCM production boundary, and the first FSF Conformance and Reference Vector work.
Full canonical adoption remains contingent on Specification-wide integration, Conformance, Reference Vectors, independent implementation, adversarial testing, traceability, and the applicable adoption act. General composition and transformation closure remain separate open gates.
Primary Derived Findings
This specification section is derived especially from Requirements Findings #24–#33 and #44–#50.
Architectural Boundary The Survey Fabric defines exact geometry and spatial relationship. Higher Architecture domains decide what occupies that geometry and what those relationships mean; this Specification does not prescribe their internal design.