Circle Ontology

FLOW ALGEBRA
ONTOLOGY: Circle Ontology
VERSION: 4.0
DOMAIN: Mathematical Entities / Geometric Entities
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/geometric-entities/circle-ontology-2.html
SOURCE_ALIGNMENT: Independently constructed; page-specific alignment not verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
DATE: 2026-10-13

01. PURPOSE
Represent circles as typed geometric loci, preserving mathematical definitions, dependencies, relationships, conditions and machine-consumable inference rules.

02. CORE ENTITY
ENTITY Circle:
  TYPE: PlaneGeometricLocus
  PARENT: GeometricEntity
  DEFINITION: Set of points in a Euclidean plane at a fixed positive distance r from a centre c.
  PARAMETERS: centre c in R^2; radius r in R with r > 0
  MEMBERSHIP: p in Circle(c,r) iff distance(p,c)=r
  DISTINCTION: Circle denotes boundary; closed disk denotes distance(p,c)<=r.
  DEGENERATE_CASE: r=0 produces a singleton locus, not a nondegenerate circle.

03. TYPED ENTITIES
  Centre: Point2D
  Radius: PositiveReal
  Diameter: LineSegment
  Circumference: NonnegativeRealLength
  Disk: PlanarRegion
  Chord: LineSegment
  Secant: Line
  Tangent: Line
  Arc: ConnectedCircleSubset
  CentralAngle: Angle
  InscribedAngle: Angle
  Sector: DiskSubset
  SegmentRegion: DiskSubset
  ConcentricCircle: Circle
  UnitCircle: Circle
  Circumcircle: Circle
  Incircle: Circle
  CoordinateFrame: EuclideanFrame2D
  Orientation: ClockwiseOrCounterclockwise

04. REPRESENTATIONS
  CARTESIAN: (x-h)^2+(y-k)^2=r^2; centre=(h,k); r>0
  PARAMETRIC: p(t)=(h+r*cos(t),k+r*sin(t)); t in [0,2*pi)
  VECTOR: ||p-c||_2=r
  IMPLICIT: F(x,y)=(x-h)^2+(y-k)^2-r^2=0
  GENERAL_EQUATION: x^2+y^2+Dx+Ey+F=0
  GENERAL_VALIDITY: r^2=(D^2+E^2)/4-F>0; centre=(-D/2,-E/2)
  POLAR_ORIGIN_CENTRED: rho=r
  UNIT_CIRCLE: x^2+y^2=1

05. FUNDAMENTAL INVARIANTS
  DIAMETER_LENGTH: d=2*r
  CIRCUMFERENCE: C=2*pi*r
  DISK_AREA: A=pi*r^2
  CIRCLE_BOUNDARY_AREA: zero in 2D Lebesgue measure
  ECCENTRICITY: e=0 as a conic
  CURVATURE: kappa=1/r
  SYMMETRY: all rotations about centre; all reflections through lines containing centre
  TOPOLOGY: Circle is homeomorphic to S^1

06. CHORDS, ARCS, SECTORS
  CHORD_LENGTH: L=2*r*sin(theta/2), where theta in [0,pi] is the minor central angle
  ARC_LENGTH: s=r*theta, theta in radians and theta>=0
  SECTOR_AREA: A_sector=(r^2*theta)/2, theta in [0,2*pi]
  MINOR_SEGMENT_AREA: A_segment=(r^2/2)*(theta-sin(theta)), theta in [0,pi]
  CHORD_CENTRE_DISTANCE: q=sqrt(r^2-(L/2)^2), 0<=L<=2*r
  INSCRIBED_ANGLE: angle subtending a specified arc is half its central angle, with vertex on the complementary arc
  THALES_THEOREM: angle subtended by a diameter at another point on the circle is 90 degrees

07. INCIDENCE AND INTERSECTION
  POINT_ON: ||p-c||=r
  POINT_INSIDE_DISK: ||p-c||<r
  POINT_OUTSIDE_DISK: ||p-c||>r
  TANGENT_AT_P: (x-p) dot (p-c)=0, given p on circle
  TANGENT_ORTHOGONALITY: radius at tangency perpendicular to tangent
  LINE_CIRCLE_INTERSECTION: 0,1,2 distinct real points depending on distance from centre to line compared with r
  TWO_CIRCLE_INTERSECTION: 0,1,2 distinct points for distinct circles; coincident circles have infinitely many
  TWO_CIRCLE_TWO_POINTS: |r1-r2|<d<r1+r2, d=distance(c1,c2)
  TWO_CIRCLE_EXTERNAL_TANGENCY: d=r1+r2
  TWO_CIRCLE_INTERNAL_TANGENCY: d=|r1-r2| with d>0 and distinct circles
  DISJOINT_EXTERNALLY: d>r1+r2
  STRICT_CONTAINMENT_NO_INTERSECTION: d<|r1-r2|
  CONCENTRIC: c1=c2

08. TRANSFORMATIONS
  TRANSLATION: Circle(c,r) -> Circle(c+v,r)
  ROTATION: Circle(c,r) -> Circle(R*c,r) for origin-centred orthogonal rotation R
  REFLECTION: Circle(c,r) -> Circle(Q*c,r) for reflection Q through origin
  UNIFORM_SCALING: Circle(c,r) -> Circle(s*c,abs(s)*r), s != 0
  ISOMETRY: preserves radius and circumference
  SIMILARITY: preserves circle class and angles, scales lengths uniformly
  GENERAL_AFFINE_TRANSFORMATION: circle generally maps to ellipse; circle only under suitable similarity transformations

09. SEMANTIC RELATIONSHIPS
  Circle IS_A PlaneGeometricLocus
  Circle HAS_CENTRE Point2D
  Circle HAS_RADIUS PositiveReal
  Circle BOUNDS Disk
  Radius CONNECTS Centre TO CirclePoint
  Diameter IS_A Chord
  Diameter PASSES_THROUGH Centre
  Tangent TOUCHES Circle AT TangencyPoint
  Secant INTERSECTS Circle AT TwoDistinctPoints
  Arc IS_PART_OF Circle
  Sector IS_PART_OF Disk
  Circle IS_SPECIAL_CASE_OF ConicSection
  UnitCircle IS_A Circle
  Circumcircle PASSES_THROUGH PolygonVertices
  Incircle TANGENT_TO PolygonSides

10. FORMAL INFERENCE RULES
  R01 IF r>0 AND c in R^2 THEN Circle(c,r) valid.
  R02 IF r<=0 THEN reject nondegenerate Circle declaration.
  R03 IF ||p-c||=r THEN p lies on circle.
  R04 IF ||p-c||<r THEN p is in disk interior.
  R05 IF ||p-c||>r THEN p is outside closed disk.
  R06 IF diameter d known THEN radius=d/2.
  R07 IF radius r known THEN circumference=2*pi*r.
  R08 IF radius r known THEN disk area=pi*r^2.
  R09 IF circumference C known AND C>0 THEN r=C/(2*pi).
  R10 IF disk area A known AND A>0 THEN r=sqrt(A/pi).
  R11 IF p on circle THEN radius vector p-c is normal to tangent at p.
  R12 IF line-centre distance<r THEN secant with two intersection points.
  R13 IF line-centre distance=r THEN tangent with one intersection point.
  R14 IF line-centre distance>r THEN no intersection.
  R15 IF central angle theta in radians THEN arc length=r*theta.
  R16 IF theta in [0,2*pi] THEN sector area=r^2*theta/2.
  R17 IF chord length L in [0,2*r] THEN minor central angle=2*asin(L/(2*r)).
  R18 IF a chord passes through centre THEN chord is diameter.
  R19 IF a chord is diameter THEN length=2*r.
  R20 IF two circles have equal centres THEN concentric.
  R21 IF circles have same centre and radius THEN they are identical loci.
  R22 IF distinct circles have d=r1+r2 THEN externally tangent.
  R23 IF distinct circles have d=abs(r1-r2)>0 THEN internally tangent.
  R24 IF abs(r1-r2)<d<r1+r2 THEN two intersections.
  R25 IF d>r1+r2 THEN circles are externally disjoint.
  R26 IF d<abs(r1-r2) THEN one circle strictly contains the other without boundary intersection.
  R27 IF rigid motion applied THEN image is circle of unchanged radius.
  R28 IF uniform nonzero scaling applied THEN image radius=abs(scale)*r.
  R29 IF r=1 AND c=(0,0) THEN unit circle.
  R30 IF x^2+y^2+Dx+Ey+F=0 AND (D^2+E^2)/4-F>0 THEN real nondegenerate circle.
  R31 IF discriminant-radius-square=0 THEN singleton degenerate locus, not nondegenerate circle.
  R32 IF discriminant-radius-square<0 THEN no real circle locus.
  R33 IF p,q are distinct circle points THEN segment pq is chord.
  R34 IF polygon vertices all on circle THEN circle is a circumcircle of polygon.
  R35 IF circle tangent to all sides of polygon AND contained within polygon THEN incircle.
  R36 IF circle is nondegenerate THEN its topological boundary equals itself.

11. VALIDATION CONSTRAINTS
  V01: radius must be finite and strictly positive.
  V02: centre coordinates must be finite real values.
  V03: coordinate frame and length units must be declared where applicable.
  V04: distinguish circle circumference from disk area.
  V05: distinguish full line, chord segment, radius segment and diameter segment.
  V06: all trigonometric arc/sector formulae use radians.
  V07: arc angles and minor/major arc conventions must be explicit.
  V08: intersection rules must handle coincident circles separately.
  V09: equality checks in floating-point systems require tolerance and error provenance.
  V10: transformation claims must specify mapping class.
  V11: never infer circularity from a generic affine transformation.
  V12: do not conflate a filled disk with its boundary circle.

12. MATHEMATICAL EXAMPLES
  E01: Circle((0,0),1) has equation x^2+y^2=1.
  E02: Circle((2,-3),5) has equation (x-2)^2+(y+3)^2=25.
  E03: Radius 4 -> diameter 8.
  E04: Radius 4 -> circumference 8*pi.
  E05: Radius 4 -> disk area 16*pi.
  E06: Unit circle point (1,0) lies on circle.
  E07: Unit circle point (0,0) lies inside disk, not on circle.
  E08: Unit circle point (2,0) lies outside disk.
  E09: Circle centred at origin radius 3 tangent to x=3.
  E10: Circle centred at origin radius 3 intersected by y=0 at (-3,0),(3,0).
  E11: Radius 2, theta=pi/2 -> arc length pi.
  E12: Radius 2, theta=pi/2 -> sector area pi.
  E13: Radius 5, central angle pi -> diameter chord length 10.
  E14: Two unit circles with centre separation 2 are externally tangent.
  E15: Two unit circles with centre separation 1 intersect at two points.
  E16: Uniform scaling by 3 maps radius 2 to radius 6.

13. SEMANTIC EQUIVALENCE
  CircleBoundary == EuclideanCircleLocus [when ambient plane and metric match]
  Circumference == CirclePerimeter [when denoting boundary length]
  RadiusLength != RadiusSegment
  Circle != ClosedDisk
  Circle != Sphere
  Circle != Ellipse [except ellipse special case with equal semiaxes]
  UnitCircle == Circle((0,0),1) [in standard Euclidean coordinates]

14. KNOWLEDGE GRAPH TRIPLES
  (Circle, rdf:type, PlaneGeometricLocus)
  (Circle, hasCentre, Point2D)
  (Circle, hasRadius, PositiveReal)
  (Circle, bounds, Disk)
  (Diameter, isA, Chord)
  (Diameter, passesThrough, Centre)
  (Tangent, perpendicularTo, RadiusAtTangency)
  (Arc, partOf, Circle)
  (Sector, partOf, Disk)
  (Circle, specialCaseOf, ConicSection)
  (UnitCircle, subclassOf, Circle)

15. CONFIDENCE
  AXIOMATIC_DEFINITIONS: 1.00 (mathematical, given stated assumptions)
  DERIVED_RULES: 1.00 (mathematical, given stated assumptions)
  EMPIRICAL_COORDINATES: attach measurement uncertainty; never assign universal 1.00.
  NUMERICAL_INFERENCE: confidence depends on numerical precision, conditioning and tolerance.
  SOURCE_ALIGNMENT: unverified against linked webpage.

16. FLOW_HASH
  METHOD: SHA-256 over canonical UTF-8 content after normalising line endings to LF.
  STATUS: compute at publication; no unverified hash claimed.
  CHANGE_TRACKING: recompute on each content revision; preserve previous hashes in version history.

17. UPDATE_MODEL
  VERSION: 4.0
  UPDATE_TRIGGER: corrected theorem, source-page revision, taxonomy change, or formal schema revision.
  REVISION_POLICY: preserve stable entity IDs; append change log; validate all rules and examples.
  COMPATIBILITY: reference Point, Line, LineSegment, Angle, Curve, Plane and Polygon ontologies.

18. AI_CONSUMPTION
  INGEST_MODE: typed semantic assertions with provenance.
  ENTITY_ID: geometric-entities/circle-ontology-2
  INFERENCE_POLICY: apply rules only when premises and assumptions are satisfied.
  UNIT_POLICY: enforce compatible units; radians for analytic angle formulae.
  VALIDATION_POLICY: reject invalid radius and ambiguous disk/circle distinctions.
  UNCERTAINTY_POLICY: propagate numerical and measurement uncertainty.
  OUTPUT_POLICY: provide equations, derivations, conditions and counterexamples when relevant.
  PROVENANCE_POLICY: distinguish independently generated formalisation from webpage quotations.

END FLOW ALGEBRA