perpendicularity
Enable an AI agent to recognise a perpendicularity relation, evaluate its support and deviation, and determine whether to accept, constrain, correct or further inspect it.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
Researched by: Codex + Grok
Purpose and description
Enable an AI agent to recognise a perpendicularity relation, evaluate its support and deviation, and determine whether to accept, constrain, correct or further inspect it.
Perpendicularity is the geometric orientation of a line, axis or surface at a right angle to a reference line or plane; in product geometry specification it is an orientation tolerance whose zone (two parallel planes or a cylinder) is constrained at 90° to a datum system.
It can be Test perpendicularity using the criterion appropriate to the participating geometry.; Identify missing references, undefined directions or incompatible coordinate representations before evaluating the relation.; Estimate departure from perpendicularity and determine whether uncertainty permits a decision.; Construct or impose a perpendicularity constraint on eligible geometric entities.; Determine an allowed adjustment toward perpendicularity while respecting fixed references and other constraints.; Request additional measurements or proof when the current evidence cannot settle the relation..
Distinguishing features
An exact perpendicularity assertion requires the applicable right-angle or equivalent geometric criterion; merely being nonparallel is insufficient.
A right angle visible in an image does not establish perpendicularity in the represented space unless the projection supports that inference.
For line-to-plane perpendicularity, the line must follow the plane's normal direction; being perpendicular to one selected in-plane line is insufficient.
For plane-to-plane perpendicularity, evaluate the planes' dihedral relation or their normals; perpendicular planes must not be confused with parallel planes whose normals are parallel.
A measured relation satisfying a tolerance is acceptable under that requirement but is not thereby established as mathematically exact.
Scope
+ The participating lines, directions, axes, planes, surfaces or local geometric elements.
+ The geometric context and criterion that give perpendicularity a definite meaning.
+ Exact right-angle relations and evidence supporting or contradicting them.
+ Measured departures from perpendicularity and their uncertainty.
+ Perpendicularity requirements, permissible deviation and resulting decisions.
- Complete shape and dimensional descriptions of the participating objects.
- Parallelism, collinearity and general angular relationships except as comparison cases.
- Abstract orthogonality of functions, signals or other nongeometric entities.
- Whole-system alignment, assembly fit and mechanical performance.
- Instrument calibration and manufacturing processes beyond their effect on this relation.
Characteristics
- Participating geometric entities
- Identified pair of lines, directions, axes, planes, surfaces or local elements Determines what is asserted to be perpendicular and which evaluation criterion applies.
- Geometric context
- Dimension, ambient space, metric or inner product, and coordinate conventions Coordinates alone do not determine whether the relevant directions form a right angle.
- Extent of assertion
- Global; at a specified point; over a specified region; relative to a fitted feature Separates whole-entity perpendicularity from a local or estimated relation.
- Assertion role
- Required; assumed; constructed; derived; observed Separates intended geometry from established or measured geometry.
- Evaluated angle
- Degrees or radians, with the angle convention and geometric elements identified Provides a quantitative basis for testing a right-angle relation without confusing direction, normal and dihedral angles.
- Departure from perpendicularity
- Angular deviation in degrees or radians, or a separately defined spatial deviation in length units Quantifies the departure using the measure actually required by the application.
- Evaluation uncertainty
- Uncertainty interval or bound in the units of the evaluated quantity, with its interpretation Determines whether available evidence can resolve the perpendicularity decision.
- Perpendicularity assessment
- Established exact; meets stated tolerance; fails stated tolerance; indeterminate; undefined Supports an actionable conclusion while distinguishing exactness, acceptance and insufficient evidence.
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 5 bundles · 9 layers · 16 findings · 26 questions.
Geometric participants Identifies precisely what is perpendicular to what and where the assertion applies.
Perpendicularity between directions, lines, planes and local surface elements requires different interpretations.
Participating elements
Resolves the geometric entities underlying the relation.
Entity pair and geometric support
Record the two participating elements and whether each is directly defined or extracted from a physical feature.
- Which two lines, axes, directions, planes or local elements are asserted to be perpendicular? definition
- If an element represents a physical feature, how was its axis, direction or reference plane obtained? provenance
Extent and incidence
Establishes the location, extent and intersection conditions of the assertion.
Locality and intersection
Distinguish a relation at a point or over a region from a global relation, and distinguish perpendicular entities from perpendicular directions.
- Does the assertion apply to entire entities, fitted representatives or tangent elements at specified locations? boundary
- Must the participating entities intersect under the intended definition, or is the assertion only about their directions? definition
Right-angle criterion Defines the geometric meaning and valid test of perpendicularity.
A perpendicularity judgement depends on the geometry and entity pairing, not merely on a coordinate pattern or visual appearance.
Geometry and representation
Establishes the space, metric and representation in which a right angle is evaluated.
Angle-defining context
Record the dimensional and metric assumptions and reconcile the participants' coordinate frames.
- What ambient space and metric or inner product define the angle between the relevant directions? definition
- Are both participants represented in a compatible frame, and does any image projection or transformation preserve the angle being tested? boundary
Pair-specific test
Selects a valid perpendicularity criterion for the identified entity types.
Criterion and degeneracy
Identify the right-angle, direction-normal or normal-normal test and conditions under which it is undefined.
- For this entity pair, should the agent test a right angle between directions, alignment with a plane normal, or a right angle between plane normals? definition
- Does a zero direction, undefined tangent, nonunique normal or inadequately determined plane prevent this test from having a geometric meaning? boundary
Evidence and deviation Connects the assertion to proof, construction or measurement and quantifies any departure.
Intended right angles, proved right angles and measurements near a right angle support different conclusions.
Assertion support
Separates the role of the assertion from the evidence available for it.
Requirement versus established relation
Record whether perpendicularity is requested, assumed, imposed by construction, derived or observed.
- Is perpendicularity a design requirement, a working assumption, a construction constraint, a proved result or a measurement conclusion? definition
- Which derivation, construction record or observations support this assertion, and on what assumptions do they depend? provenance
Departure and resolution
Determines the magnitude of departure and whether available evidence resolves it.
Measured departure
Record the deviation measure, observation extent and uncertainty without treating an estimate as exact geometry.
- What angular or spatial departure was evaluated, in which units, and over what length, area or sample of the feature? measurement
- How do measurement uncertainty, numerical precision and feature fitting affect the reported departure? measurement
- Could unsampled portions or local variation invalidate a conclusion drawn from the fitted direction or plane? boundary
Acceptance and adjustment Relates perpendicularity evidence to an applicable requirement and permissible next actions.
An agent needs a justified decision rule and adjustment constraints before accepting or changing the relation.
Requirement and decision
Makes exactness, tolerance and unresolved assessments explicit.
Applicable acceptance rule
Record the governing perpendicularity requirement, its reference and the decision rule applied to the evidence.
- Does the task require exact perpendicularity, an angular limit or a spatial tolerance relative to a specified reference? definition
- Where is that requirement defined, including any applicable datum, tolerance-zone interpretation or decision convention? provenance
- Given the evidence and uncertainty, is the relation established exact, acceptable, unacceptable, indeterminate or undefined? measurement
Permitted geometric change
Identifies how the relation may be established, corrected or investigated.
Adjustment and recheck
Determine which participant may change, which geometric constraints must remain satisfied and how the result will be verified.
- Which participant is fixed, and what rotations, reorientations or feature changes are permitted while preserving required positions and other geometric relations? action
- What construction, proof or measurement will verify perpendicularity after an adjustment or resolve the current indeterminate assessment? action
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Kinds and varieties
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- Line-to-line perpendicularity in a plane
- Line-to-plane (axis-to-face) perpendicularity
- Plane-to-plane perpendicularity
- Surface perpendicularity as a GPS/GD&T planar tolerance zone
- Axis or median-line perpendicularity as a cylindrical tolerance zone
- Vector orthogonality in an inner-product space
- Construction and layout squareness (wall-to-floor, plumb-to-level)
- Machine-axis squareness in motion geometry
- Which of these kinds and varieties hold for the sense of perpendicularity this model covers, and on what evidence? provenance
Identifiers and schemes
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- ISO 1101 / ASME Y14.5 characteristic symbol - ⊥ - Drawing symbol for the perpendicularity orientation tolerance.
- Unicode - U+27C2 - PERPENDICULAR character ⟂, used for the geometric relation in text.
- Which of these identifiers and schemes hold for the sense of perpendicularity this model covers, and on what evidence? provenance
Standards and regulation
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- ISO 1101 Geometrical product specifications (GPS) - Geometrical tolerancing (International Organization for Standardization)
- ISO 5459 Geometrical product specifications (GPS) - Datums and datum systems (International Organization for Standardization)
- ASME Y14.5 Dimensioning and Tolerancing (American Society of Mechanical Engineers)
- ISO 230-1 Test code for machine tools - geometric accuracy, including squareness of axes (International Organization for Standardization)
- Which of these standards and regulation hold for the sense of perpendicularity this model covers, and on what evidence? provenance
Real-world use
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- A bore axis specified square to a mounting flange so a shaft or fastener seats without edge load.
- CMM, granite-square and autocollimator checks of prismatic parts against a perpendicularity callout.
- Machine-tool acceptance: spindle or linear-axis squareness to a table or to another axis.
- Setting walls, columns and formwork square to a floor, grid or datum line on site.
- Which of these real-world use hold for the sense of perpendicularity this model covers, and on what evidence? provenance
Typical measurements
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- Angular departure from 90° - about 0.5 arcseconds (precision squares and autocollimators) to about 1° (rough construction) - arcsecond or degree
- Perpendicularity tolerance-zone width on a drawing - commonly 0.02-0.5 on precision prismatic parts - mm
- Machine-tool axis squareness - typically 5-30 - µm/m
- Which of these typical measurements hold for the sense of perpendicularity this model covers, and on what evidence? provenance
Failure modes and hazards
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- A pin or hole that is not square to a face binds, galls, or loads on an edge instead of in bearing.
- A flange not perpendicular to a bore leaks at a gasket or misaligns a mating vessel or motor.
- Shaft-to-housing out-of-square edge-loads bearings and shortens life.
- Uncorrected CMM or machine-tool squareness error reports a good part as bad, or the reverse.
- Stacked 'square' plates accumulate angular error until an assembly will not close.
- Which of these failure modes and hazards hold for the sense of perpendicularity this model covers, and on what evidence? provenance
Regional variation
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- US engineering drawings use ASME Y14.5 perpendicularity; ISO GPS (Europe and much of the rest of the world) uses the same symbol with different default datum and independence rules.
- Shop and site language often says 'square' or 'plumb and square' rather than 'perpendicular'; mathematics prefers 'orthogonal' for the inner-product condition.
- Which of these regional variation hold for the sense of perpendicularity this model covers, and on what evidence? provenance
Neighbouring kinds and how to tell them apart
Reported by the breadth pass; each item needs checking against its source before it becomes normative.
- Angularity - Same orientation family, but the specified angle is not 90°; perpendicularity is the right-angle special case.
- Parallelism - Orientation at constant separation (0° between directions) rather than a right angle; a feature can be parallel to one datum and perpendicular to another.
- Position - Locates a feature relative to a datum system and can subsume perpendicularity; a pure perpendicularity control does not locate the feature.
- Flatness - A form control with no datum; a surface can be flat and still lean relative to a reference.
- Orthogonality - The inner-product condition (including function spaces and coordinate axes); perpendicularity in product geometry is the Euclidean 90° case of lines and planes.
- Squareness (shop metrology) - Often the measured deviation of two machine or part axes in µm/m, not the drawing characteristic or its tolerance zone.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of perpendicularity this model covers, and on what evidence? provenance
Sources
- ISO 1101:2017 Geometrical product specifications (GPS) - Geometrical tolerancing - Tolerances of form, orientation, location and run-out - Defines perpendicularity as an orientation tolerance, the associated symbols, and the planar and cylindrical tolerance zones used on technical drawings.
- ASME Y14.5-2018 Dimensioning and Tolerancing - US product-definition rules for the perpendicularity characteristic, datum referencing, and how surface versus axis perpendicularity is applied.
- ISO 230-1 Test code for machine tools - Geometric accuracy of machines operating under no-load or quasi-static conditions - How squareness/perpendicularity of machine-tool axes is measured in micrometres per metre of travel.
- Perpendicular - The Euclidean geometric relation (lines, a line and a plane, two planes) and its equivalence to a vanishing inner product of direction or normal vectors.
What the second pass must settle
- Does the registry intend perpendicularity to include non-Euclidean local geometry, or only Euclidean geometric and physical relations?
- Should nonintersecting lines with perpendicular directions be included as perpendicular lines, or represented only through a relation between their directions?
- Which engineering perpendicularity conventions must this model support, and which datum, tolerance-zone and acceptance details belong in neighbouring models?
- How should perpendicularity between curved surfaces be represented when it holds only at selected points or along an intersection?
- Does an existing Vercy world model already own this concept or its geometric orthogonality meaning, requiring a registry link instead of a separate publication?