line
a length (straight or curved) without breadth or thickness; the trace of a moving point
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 line, assess whether its geometry and continuity support an intended use, and decide which measurements or transformations are valid.
A line is a one-dimensional geometric magnitude - length without breadth or thickness - whose extremities, when it has any, are points, equivalently the locus traced by a point moving in space.
It can be Measure distance along a specified portion and compare it with straight endpoint separation.; Locate points, tangents and local bending where these are defined.; Test closure, continuity, intersections and coincidence under an explicit tolerance.; Trim, split, extend or join portions while recording changes to endpoints and continuity.; Translate, rotate, scale or project the line and assess the resulting geometric changes.; Sample, fit or simplify its representation against an explicit error allowance..
Distinguishing features
A line records extension along its course without assigning intrinsic breadth or thickness; a visible stroke or wire has a cross-section belonging to its carrier.
A line has a course beyond a single location; an isolated point or a collapsed zero-length representation needs explicit treatment as a degenerate case.
A closed line records its boundary course without including the enclosed region.
Curvature does not disqualify a shape under this registry definition; the narrower mathematical meaning of an infinite straight line is only one possible case.
A geometric line does not require a traversal time or speed; those distinguish a motion trajectory from its underlying trace.
Scope
+ Straight and curved lines, with finite, one-sided unbounded or two-sided unbounded extent
+ Endpoints, closure, continuity and self-intersection
+ Length, direction, curvature and changes of direction where defined
+ Placement within a plane, surface or three-dimensional space
+ Geometric representations and their approximation limits
+ Operations that preserve or deliberately change line geometry
- The width, thickness, material and colour of a drawn stroke, wire or other physical carrier
- Surfaces and regions, including the interior a closed line may bound
- Movement speed, elapsed time and the identity of an object following a trajectory
- Network routing, flow and connectivity semantics beyond geometric incidence
- Text lines, queues, product lines and other non-geometric meanings of line
Characteristics
- Geometric form
- straight | curved | piecewise straight | mixed | undetermined Determines which descriptions, measurements and geometric operations apply.
- Extent
- finite | unbounded in one direction | unbounded in both directions | undetermined Prevents a displayed portion from being mistaken for the complete line.
- Arc length
- length units with measured domain; finite value | infinite | undetermined Measures distance along the course and must be distinguished from endpoint separation.
- Endpoint status
- endpoint locations and inclusion status, or no endpoints; distinguish geometric endpoints from display cuts Controls incidence, trimming and whether joining or closure is needed.
- Closure
- open | closed | undetermined, with any closure tolerance Establishes whether the course returns to its start without assuming it encloses a simple region.
- Continuity and regularity
- continuous portions, gaps, corners, cusps and locations where a tangent is undefined Identifies where traversal, smoothing or tangent-dependent operations require special handling.
- Curvature
- inverse length where defined, with location and estimation method Describes local bending and supports judging whether a proposed approximation preserves shape.
- Geometric embedding
- coordinate frame, dimension, units and any supporting plane or surface Makes positions, distances and apparent intersections interpretable.
- Intersection and overlap
- self or other-line intersection locations, shared portions and unresolved near contacts Distinguishes crossing, touching and coincident courses without inventing network junctions.
- Representation fidelity
- positional error bound in length units, or unknown, with reference geometry and method Determines whether sampled or fitted geometry is adequate for a particular decision.
Analytical facets
- substance
- form
- origin
- conceptual
- agency
- inert
- mobility
- not-applicable
- scale
- not-applicable
- affordances
- observable
Also called
+101
Where this came from
oewn:2024 · CC BY 4.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 5 bundles · 9 layers · 16 findings · 24 questions.
Course and extent Establish what geometric course the line denotes and how far that course extends.
An agent must distinguish the complete line from a finite representation and distinguish geometry from traversal history.
Geometric identity
Identify the line's defining course independently of its visible carrier.
Defining course
Record whether the course is straight, curved or composed of portions, and what definition or observation establishes it.
- What rule, construction or observed trace identifies this particular straight or curved course? definition
- Is the record intended to identify the occupied geometric points alone, or does it also distinguish traversal order and repeated passage? boundary
Extent and ends
Separate actual terminations from limits imposed by a drawing, sample or measurement window.
Endpoint and domain status
Record finite or unbounded extent, endpoint inclusion and any restriction to a selected portion.
- Which apparent ends are actual endpoints, and which are clipping or sampling limits? boundary
- What evidence establishes whether the line terminates, continues indefinitely or has an unresolved continuation? provenance
Length and local shape Describe distance along the line and how its direction changes.
Endpoint separation alone cannot describe a curved course, and local shape controls which geometric operations are well-defined.
Along-course distance
Establish the domain and method for measuring length along the line.
Arc length assessment
Record length for a specified portion, including whether the available representation supports a finite value.
- Between which locations or parameter limits is arc length being measured, and in what units? measurement
- Is the reported length exact, estimated from samples, infinite or undetermined, and what supports that assessment? measurement
Direction and bending
Describe tangents, curvature and exceptional locations along the course.
Local shape regularity
Record straight portions, smooth bends, corners and cusps without assigning tangents or curvature where they are undefined.
- Where are tangent direction and curvature defined, and what construction or estimation method supplies them? measurement
- Which changes of direction are intended corners or cusps, and which may be sampling artefacts? provenance
Continuity and incidence Establish how the course connects to itself and meets other geometry.
Gaps, closure, crossings and overlaps change how a line can be followed or partitioned, even when its overall appearance is similar.
Course continuity
Assess uninterrupted connection and return to the starting location.
Gaps and closure
Record known gaps and whether closure is exact, tolerance-based or unresolved.
- Can the intended course be followed continuously, and where do actual or suspected gaps occur? boundary
- Does the course close exactly or only within a stated tolerance, and how was that tested? measurement
Crossings and shared portions
Distinguish geometric intersection, touching, retracing and overlap.
Incidence classification
Record where courses meet or coincide in the actual embedding, separately from apparent contact in a projection.
- Which contacts are intersections in the line's coordinate space, and which arise only in a projected view? measurement
- At each contact, do the courses cross, touch or share a portion, and does any claimed junction meaning belong to another model? boundary
Representation and geometric actions Connect the ideal line to its recorded representation and constrain changes by the intended geometric result.
Agents act on equations, samples or drawings; they must know which features those representations preserve and which proposed edits change the line itself.
Embedding and fidelity
Record the space, representation method and uncertainty needed to interpret the line.
Representation basis
Identify coordinate frame, units, supporting geometry and how an equation, control-point construction or sample represents the intended course.
- In what coordinate frame and units is the line defined, and is it constrained to a plane or surface? definition
- How was the representation obtained, and what positional error or unresolved interpolation exists between recorded points? provenance
Operation conditions
State the conditions and preservation requirements for a proposed geometric operation.
Permitted geometric change
Record whether an operation changes the intended course or only its representation, together with the geometric properties that must remain valid.
- For the proposed trim, join, extension, transformation or simplification, which endpoints, lengths, incidences or closure properties must be preserved? action
- What construction rule and error allowance make the operation determinate, and how will the resulting course be checked? 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.
- straight line
- line segment
- ray
- curve
- closed curve
- geodesic
- polyline
- space curve
- Which of these kinds and varieties hold for the sense of line 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.
- Wikidata - Q37105 - Item labelled line; described as the infinite non-curved geometric object, i.e. the straight line, which is narrower than Euclid's line (straight or curved).
- Euclid Elements - I.Def.2 - Canonical citation for "a line is length without breadth"; I.Def.3 (extremities are points) and I.Def.4 (straight line) sit beside it.
- Which of these identifiers and schemes hold for the sense of line 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 80000-3:2019 Quantities and units - Part 3: Space and time (International Organization for Standardization): defines length, path length/arc length of a rectifiable curve, and distance as shortest path length.
- Euclid, Elements, Book I, Definitions and Postulates (classical geometric canon, not a modern statutory code): defines line and straight line and requires that a straight line can be drawn from any point to any point.
- Which of these standards and regulation hold for the sense of line 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.
- Euclidean construction: drawing the unique straight path between two marked points and extending it.
- CAD, CAM and vector graphics: storing and rendering one-dimensional primitives (segments, polylines, circular arcs, Bézier curves) with no assigned area.
- GIS and mapping: roads, rivers, pipelines and boundaries encoded as linear features rather than polygons.
- Surveying and metrology: measuring length along a laid-out line or along a curve (tape, total station, path length).
- Kinematics and geometric optics: a particle trajectory or a light ray treated as the trace of a moving point.
- Drafting and plotting: ink, graphite or pixel strokes used as physical stand-ins for zero-width lines.
- Which of these real-world use hold for the sense of line 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.
- path length (arc length) of a rectifiable piece - 0 to unbounded; physical drawings typically millimetres to metres, survey lines metres to tens of kilometres - m
- distance between two points of the line (shortest path) - 0 up to the path length; equal to path length if and only if that piece is straight (in Euclidean metric) - m
- curvature - 0 for a straight line; larger for tighter bends - m^-1
- number of extremities - 0 (unbounded line or closed curve), 1 (ray), 2 (segment) - 1
- Which of these typical measurements hold for the sense of line 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.
- Taking a physical stroke (which has breadth) for a mathematical line, then miscomputing incidence, intersection or area.
- Treating a non-rectifiable curve as if it had a finite length (ISO 80000-3 flags fractal examples).
- Collapsing "line" to "straight line" and then drawing or coding a unique join that the geometry does not actually determine (e.g. on a sphere, many great circles).
- Degenerate segment: coincident endpoints, zero length, unstable direction.
- Digital linework errors: undershoots, dangles and false intersections when polylines fail to meet.
- Raster approximation (aliasing) shifting apparent intersections away from the intended locus.
- Which of these failure modes and hazards hold for the sense of line 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.
- In much English school geometry, "line" means the infinite straight line; Euclid's γραμμή (line) includes curves, and "straight line" / "right line" is the species that lies evenly with its points.
- French distinguishes ligne (general line) from droite (infinite straight line); Spanish línea from recta; German Linie from Gerade.
- Wikidata's default English label line for Q37105 matches the school-straight sense, while several other languages label that same item with their word for straight line.
- Which of these regional variation hold for the sense of line 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.
- point - A point has no part and no length (Euclid I.Def.1); a line has one-dimensional extent, and the extremities of a line are points (I.Def.3).
- surface - A surface has length and breadth only (Euclid I.Def.5); a line has length without breadth.
- straight line - A straight (right) line lies evenly with the points on itself (Euclid I.Def.4) and is locally length-minimizing; the researched line includes curves, whose path length between two points exceeds their distance.
- line segment - A segment is bounded by two endpoints and has finite measurable length; an unbounded line has no finite length and no pair of extremities that exhaust it.
- ray - A ray has exactly one extremity and is unbounded the other way; a two-way infinite line has none, a segment has two.
- distance - Distance is the scalar shortest path length between two points (ISO 80000-3 item 3-1.8), not the one-dimensional locus itself.
- physical stroke or mark - A drawn or printed mark has measurable breadth and thickness; a geometric line has none, though the mark is used to stand for it.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of line this model covers, and on what evidence? provenance
Sources
- Euclid, Elements, Book I, Definitions (Dionysius Lardner edition, 1855), Trinity College Dublin - Classical definition of a line as length without breadth; extremities are points; right line as that which lies evenly between its extremities; generation of a mathematical line as the trace of a moving point.
- Proclus and related ancient commentators on Euclid I, cited in Introduction, Springer (ch. citing Friedlein 1873 / Morrow 1992) - Line as length without breadth, as the flowing of a point, and as magnitude extended in one direction; Aristotle on a moving point generating a line; straight line as uniform undeviating flow.
- Geometry, Encyclopædia Britannica, 11th edition (1911), Encyclopædia Britannica / Wikisource - A line as the one-dimensional object with length only; the path of a moving point is a line; boundaries of a line are points.
- line (Q37105), Wikidata - Stable identifier and labels for the infinite non-curved (straight) line; language contrast such as English line / Spanish recta / Chinese 直线.
- ISO 80000-3:2019 Quantities and units - Part 3: Space and time, International Organization for Standardization - Length need not be measured along a straight line; path length / arc length of a rectifiable curve; distance as shortest path length; some curves (e.g. fractals) are not rectifiable.
- Geometric primitive, Wikipedia - Line or curve as the one-dimensional geometric primitive in CAD, GIS and vector graphics, commonly stored as polylines or parametric curves.
- 11 Lines, geometry course notes, Steve J. Trettel - Modern metric characterization of a (straight) line as a curve whose every finite segment is length-minimizing.
What the second pass must settle
- Does this registry entry include rays, disconnected collections of curve portions and branched traces, or should some of these belong to neighbouring models?
- Should two traces occupying the same geometric points count as the same line when their traversal order or repeated coverage differs?
- How should zero-length degeneracies and non-rectifiable curves be represented without treating them as ordinary measurable lines?
- Which application-specific tolerances should govern closure, coincidence and continuity, and how should conflicting tolerances be reconciled?
- When a line is inferred from a physical stroke or wire, what extraction rule establishes its course and uncertainty?