Transformation
Vectors

A Restricted Exact Transformation Core Is Defined

The present candidate transformation core supports exact operations that preserve the Foundational Survey Fabric's canonical frame and exact coordinate semantics.

The directly testable core includes translation, positive uniform scaling, identity cases, inverses of those supported transformations, deterministic composition, and orientation preservation.

Arbitrary rotation and reflection are not part of the same closed canonical placement profile.

Exact CRPC Translation Must Reproduce Identically

Given a Point P = (x,y) and an exact translation vector T = (tx,ty), the transformed Point is:

P' = (x + tx, y + ty).

Where all values are valid CRPC and the resulting Point remains within the Survey Domain, independent implementations shall produce the same exact transformed Point.

Supported Geometry Translates Pointwise

Reference Vectors shall include translated Segments and SCPEs produced by applying the same exact translation to every constituent Point.

Translation shall preserve exact relative geometry, orientation, Segment structure, SCPE simplicity where Domain validity is retained, and semantic geometry apart from absolute position.

Uniform Positive Scaling Preserves Orientation

Where supported by the candidate placement model, positive uniform scaling applies one exact positive scale factor to both canonical axes.

Reference Vectors shall prove that the same factor is applied consistently to x and y, preserving handedness and avoiding the distortion introduced by nonuniform scale.

The Pang itself is not redefined. The geometry is transformed within Pang-based canonical space.

Identity Must Return the Original Canonical Geometry

The corpus shall include zero translation and neutral positive uniform scale.

These identity transformations shall return the original canonical Point, Segment, or SCPE exactly after any required canonical normalization.

Translation Must Reverse Exactly

For a valid translation vector T, applying T and then -T shall restore the original canonical geometry exactly.

No rounding or approximate reconstruction is permitted.

Invertible Positive Scale Must Restore Exact Meaning

Where a positive uniform scale factor and its reciprocal are both exactly representable under the governing candidate mathematics, scaling and then applying the exact inverse scale shall restore the original canonical geometry.

Fixtures shall remain within the exact representational and Survey Domain constraints of the Specification.

Supported Transformations Must Compose Deterministically

Reference Vectors shall include valid compositions of exact translation and positive uniform scaling.

Equivalent supported sequences shall produce the same canonical result where the governing mathematics establishes equivalence. Sequence order shall remain significant where the operations do not commute.

Translation and Positive Uniform Scale Preserve Handedness

Reference Vectors shall prove that the exact orientation sign of ordered Point triples is preserved under translation and positive uniform scaling.

A valid orientation-preserving placement shall therefore retain the canonical right-handed frame rather than introduce reflection.

A Valid Transformation Result Must Still Lie Within the Survey Domain

A transformation may be mathematically defined yet produce geometry outside the canonical Survey Domain.

Reference Vectors shall distinguish a valid transformation rule from a valid final Survey geometry result. Resulting Points or extents outside D_H shall fail Survey-domain validity rather than being clipped, wrapped, or repaired.

Exact Supported Transformations Must Remain Exact

Translation and supported positive uniform scaling shall preserve exact CRPC meaning whenever the resulting coordinates remain within the supported exact representation.

Any implementation that substitutes floating approximation for an exact supported result fails the candidate exactness contract.

Reflection Reverses Handedness

Reflection is not an orientation-preserving transformation of the canonical Survey frame.

A mirrored presentation may exist at another layer, but Reference Vectors shall not treat reflection as canonically equivalent to translation plus positive uniform scaling or as a second canonical orientation.

Distorting Affine Transformations Are Outside the Current Placement Profile

Nonuniform scaling and shear are not part of the current orientation-preserving uniform affine placement model.

Reference Vectors shall therefore classify such operations as outside the present canonical placement profile unless a later Specification revision explicitly admits and governs them.

General Exact Rotation Remains Open

Arbitrary rotation can map rational CRPC coordinates to non-rational coordinates. The Specification has not yet selected the universal exact result type, angular unit, and canonical representation needed to close that operation generally.

Reference Vectors shall therefore not publish arbitrary-angle rotated coordinates as final canonical outputs.

Rotation Is Not Assumed Merely Because Transformation Exists

The current production placement rule favors positive uniform scale plus translation and permits rotation only if the final governing frame mathematics compels or explicitly supports it.

Reference Vectors shall preserve that boundary rather than treating rotation as an automatically available canonical placement operation.

Every Defined Transformation Must Reproduce Exactly

Independent conforming implementations shall return the same exact translated, uniformly scaled, composed, inverted, and normalized result for identical supported transformation inputs.

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

Transformation Is Partially Closed, Not Universally Closed

The original page correctly deferred the corpus while transformation notation, scaling rules, and orientation mathematics were unresolved. The current candidate Specification now supports a restricted exact transformation core.

Section 07 can therefore move into executable fixtures for translation, positive uniform scale, identity, inverse, composition, orientation preservation, and Survey-domain validity. Arbitrary rotation, reflection as canonical placement, and distorting affine transformations remain outside the present closed profile.

Referenced Requirements

Transformation Vectors will derive especially from Requirements Findings #10, #15, #38–#43, #48, #50, #53, #57, #67–#68, #74–#75, and #78–#83.