Geometry
Vectors

Point, Segment, and SCPE Are the Foundational Primitives

The candidate geometry model supports:

Point — one exact CRPC coordinate pair.
Segment — the closed straight segment between two valid distinct Points.
SCPE — a finite ordered simple closed polygonal cycle representing its boundary together with its interior.

SCPE is primitive-level hole-free geometry. Curves, raster cells, grids, meshes, and general composite geometry are not part of the foundational primitive set.

Exact Point Equality Requires Exact Coordinate Equality

Reference Vectors shall prove that two Points are equal exactly when their normalized CRPC x-components and y-components are equal.

Visual coincidence, floating proximity, rendering resolution, or epsilon tolerance shall never substitute for canonical Point equality.

Segment Membership Is Exact

The corpus shall include Points lying exactly on a Segment interior, exactly on either endpoint, and strictly off the Segment.

Classification shall use exact orientation and coordinate-range tests without tolerance or snapping.

Intersection Classes Must Be Deterministic

Reference Vectors shall cover the exact Segment intersection classes defined by the Specification, including proper crossing, endpoint contact, collinear overlap where applicable, and disjoint Segments.

Every conforming implementation shall assign the same intersection classification to the same exact Segment pair.

Boundary Contact Must Be Distinguished From Interior Crossing

Where supported geometry touches only through exact boundary contact, Reference Vectors shall distinguish that condition from proper intersection through interiors or from complete separation.

The vector result remains geometric. It does not imply Parcel adjacency, legal contact, ownership, jurisdiction, or any higher-layer consequence.

Interior, Boundary, and Exterior Must Resolve Exactly

The corpus shall include Points that lie strictly inside an SCPE, exactly on its boundary, and strictly outside it.

The candidate Point-in-SCPE rule uses exact horizontal-ray parity with a half-open y convention. No tolerance or rendering heuristic may determine the result.

Polygon Validity Must Follow the Primitive Contract

Reference Vectors shall test valid and invalid SCPE candidates against the governing structural rules: at least three distinct vertices, valid Points, no consecutive duplicate vertices, nonzero area, one closed cycle, no forbidden nonadjacent intersections, and containment within the Survey Domain.

Redundant collinear middle vertices may be valid input where otherwise permitted, but canonical normalization removes them deterministically.

Containment Is an Exact Geometric Predicate

The corpus shall include exact supported cases of Point containment and SCPE containment according to the governing predicate definitions.

Boundary inclusion shall follow the exact closed-set semantics selected by the primitive model rather than implementation preference.

Connected and Disconnected Geometry Must Be Reproducible

Reference Vectors shall include supported cases sufficient to prove deterministic connectedness classification under the candidate geometry model.

Where boundary contact affects connectedness under the defined semantics, exact boundary classification shall control the result.

Equivalent Supported Geometry Must Normalize Identically

Reference Vectors shall demonstrate the canonical normalization rules selected for supported geometry.

For Segments, endpoint order is canonicalized lexicographically. For SCPEs, normalization removes a repeated terminal closure point if present in an accepted input form, removes redundant collinear middle vertices to a fixed point, selects counterclockwise traversal, and chooses the unique lexicographically least start vertex.

The normalization function shall be idempotent: N(N(G)) = N(G).

Same Supported Spatial Set Means the Same Canonical Geometry

Where two valid supported geometry representations denote the same canonical geometric set, their normalized representations shall agree.

For supported canonical geometry, semantic equality is therefore testable through deterministic normalization rather than visual comparison.

Invalid Geometry Must Not Be Repaired Into Canonical Truth

The corpus shall include repeated consecutive vertices, zero-area polygon candidates, self-intersections, invalid Segment definitions, malformed Points, and other cases that violate the primitive contract.

Such inputs shall be rejected or classified according to the Specification. They shall not be silently snapped, simplified, re-ordered into validity, or otherwise repaired unless an exact permitted normalization rule already applies.

Geometry Validity Must Respect the Survey Domain Without Inventing H

Supported canonical geometry must lie within the Survey Domain. The Domain geometry is parameterized as D_H = {(x,y) : -H ≤ x ≤ H and -H ≤ y ≤ H}.

Because the numerical value of H remains open, capacity-independent geometry fixtures and symbolic Domain-relative cases may proceed now, while final numerical edge-dependent geometry fixtures must wait for the Specification to select H.

Predicate Truth Does Not Automatically Define a Canonical Result Geometry

The Specification may determine that two supported geometries intersect, overlap, or stand in another exact relation without yet defining a canonical geometry object representing the full result of an arbitrary Boolean operation.

Reference Vectors shall distinguish predicate conformance from result-geometry closure. They shall not invent union, difference, or arbitrary intersection result forms ahead of the Specification.

Defined Geometry Predicates Must Reproduce Exactly

Independent conforming implementations shall return the same exact validity, equality, orientation, Point-on-Segment, Segment intersection, Point-in-SCPE, containment, connectedness, and normalization results for identical supported inputs.

A mismatch indicates implementation defect, Specification ambiguity, version incompatibility, fixture defect, or a geometry operation tested beyond current mathematical closure.

Geometry Has Moved From Planned Categories to Exact Predicate Proof

The original page correctly deferred executable vectors while the primitive set and predicate mathematics were unresolved. That condition has materially changed.

Point, Segment, SCPE, exact predicates, validity rules, deterministic normalization, and semantic equality now provide a substantial capacity-independent Reference Vector surface. Broader Boolean result geometry remains open and must stay separate from the predicate corpus until the Specification closes that operation layer.

Referenced Requirements

Geometry Vectors will derive especially from Requirements Findings #10, #24–#26, #31–#33, #44–#50, #56, #65–#66, #69, #74–#75, and #78–#83.