Unit Circle Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of the unit circle within the Kireth Research Semantic Library. The unit circle is the circle of radius one centred at the origin in the Cartesian plane and provides the canonical geometric model for sine, cosine, tangent, radians, periodicity, reference angles, quadrants, exact trigonometric values, coordinate rotation and angular reasoning. This ontology supports coordinate points, arc length, directed angles, symmetry, identities and relationships with circle geometry, trigonometry and coordinate geometry.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Unit Circle Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Trigonometric Geometry / Unit Circle"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/trigonometric-geometry/unit-circle-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = UnitCircle
 SYMBOLIC_FORMS = [
  "x^2+y^2=1",
  "(cos(theta),sin(theta))"
 ]
 ENTITY_CLASS = Circle
 PRIMARY_CONTEXT = Trigonometry
 DEFINITION = "The unit circle is the circle of radius one centred at the origin in the Cartesian plane."
 SEMANTIC_ROLE = "Provides the canonical coordinate model for trigonometric functions, radians, reference angles, periodicity, exact values and directed-angle reasoning."
}

SCOPE {
 INCLUDE = [
  UnitCircle,
  Centre,
  Radius,
  CartesianCoordinate,
  DirectedAngle,
  RadianMeasure,
  DegreeMeasure,
  Sine,
  Cosine,
  Tangent,
  Cotangent,
  Secant,
  Cosecant,
  Quadrant,
  ReferenceAngle,
  ArcLength,
  Periodicity,
  Symmetry,
  ExactTrigonometricValue,
  Rotation,
  CoordinatePoint
 ]

 EXCLUDE_AS_PRIMARY = [
  GeneralCircleWithRadiusNotOne,
  Sphere,
  ComplexUnitCircleUnlessExplicit,
  UnitDiskUnlessExplicit,
  ArbitraryPeriodicCurve
 ]

 DISAMBIGUATION_RULE =
  "When unit circle occurs with trigonometric functions, coordinates, radians, reference angles, quadrants or exact values, interpret it as x^2+y^2=1 in the Cartesian plane."
}

CORE_CONCEPTS {

 UnitCircle {
  definition = "The set of points (x,y) satisfying x^2+y^2=1."
  centre = "(0,0)"
  radius = 1
 }

 CoordinatePoint {
  form = "(cos(theta),sin(theta))"
  definition = "The point reached by rotating the radius from the positive x-axis through directed angle theta."
 }

 DirectedAngle {
  definition = "An oriented angular displacement measured from a reference ray, conventionally the positive x-axis."
 }

 RadianMeasure {
  definition = "For the unit circle, radian measure equals signed arc length."
  relation = "s=theta"
 }

 Quadrant {
  definition = "One of four coordinate-plane regions used to determine signs of trigonometric coordinates."
 }

 ReferenceAngle {
  definition = "The acute angle between the terminal side of an angle and the x-axis."
 }

 Sine {
  unit_circle_definition = "sin(theta)=y"
 }

 Cosine {
  unit_circle_definition = "cos(theta)=x"
 }

 Tangent {
  unit_circle_relation = "tan(theta)=y/x when x!=0"
 }

 Periodicity {
  full_rotation = "2*pi"
 }

 ExactTrigonometricValue {
  definition = "An exact sine, cosine or tangent value associated with standard special angles."
 }
}

RELATIONS {
 UnitCircle --is_special_case_of--> Circle
 UnitCircle --has_centre--> Origin
 UnitCircle --has_radius--> One
 UnitCircle --is_defined_by--> EquationX2PlusY2Equals1
 UnitCircle --parameterised_by--> DirectedAngle
 DirectedAngle --maps_to--> CoordinatePoint
 CoordinatePoint --has_x_coordinate--> Cosine
 CoordinatePoint --has_y_coordinate--> Sine
 Tangent --equals--> SineDividedByCosine
 UnitCircle --supports--> RadianMeasure
 UnitCircle --supports--> ReferenceAngle
 UnitCircle --supports--> QuadrantReasoning
 UnitCircle --supports--> Periodicity
 UnitCircle --supports--> ExactTrigonometricValues
 UnitCircle --connects_to--> Rotation
}

OPERATORS {
 POINT(theta) :=
  (cos(theta),sin(theta))

 IS_ON_UNIT_CIRCLE(x,y) :=
  x^2+y^2=1

 SINE(theta) :=
  y_coordinate_of_POINT(theta)

 COSINE(theta) :=
  x_coordinate_of_POINT(theta)

 TANGENT(theta) :=
  sin(theta)/cos(theta)
  WHEN cos(theta)!=0

 ARC_LENGTH(theta_radians) :=
  theta_radians

 DEGREES_TO_RADIANS(theta_deg) :=
  theta_deg*pi/180

 RADIANS_TO_DEGREES(theta_rad) :=
  theta_rad*180/pi

 NORMALISE_ANGLE(theta) :=
  equivalent_angle_modulo_2*pi

 REFERENCE_ANGLE(theta) :=
  acute_angle_between_terminal_side_and_x_axis

 QUADRANT(theta) :=
  quadrant_of_POINT(theta)

 ROTATE_POINT(theta,phi) :=
  POINT(theta+phi)
}

DEFINITIONS {
 D1 = "The unit circle has radius one."
 D2 = "Its centre is the origin."
 D3 = "Its Cartesian equation is x^2+y^2=1."
 D4 = "The point at angle theta is (cos(theta),sin(theta))."
 D5 = "On the unit circle, sine is the y-coordinate."
 D6 = "On the unit circle, cosine is the x-coordinate."
 D7 = "Tangent equals y/x where x is nonzero."
 D8 = "One full revolution is 2*pi radians."
 D9 = "On a unit circle, radian measure equals arc length."
 D10 = "Angles differing by 2*pi*k determine the same unit-circle point."
}

AXIOMS {
 A1_RADIUS:
  Every point P on the unit circle satisfies distance(P,Origin)=1

 A2_CARTESIAN_EQUATION:
  x^2+y^2=1

 A3_PARAMETERISATION:
  P(theta)=(cos(theta),sin(theta))

 A4_PYTHAGOREAN_IDENTITY:
  cos^2(theta)+sin^2(theta)=1

 A5_PERIODICITY:
  P(theta+2*pi)=P(theta)

 A6_RADIAN_ARC:
  On the unit circle signed arc length equals angle in radians

 A7_QUADRANT_SIGNS:
  Signs of sine and cosine correspond to the signs of y and x coordinates

 A8_ROTATIONAL_STRUCTURE:
  Increasing theta rotates the radius around the origin according to the active orientation convention
}

REASONING {

 R1_POINT_FROM_ANGLE {
  INPUT = theta
  COMPUTE = (cos(theta),sin(theta))
  RETURN = unit_circle_point
 }

 R2_ANGLE_NORMALISATION {
  INPUT = theta
  REDUCE = modulo_2*pi
  RETURN = coterminal_angle
 }

 R3_QUADRANT_REASONING {
  INPUT = theta
  NORMALISE = angle
  IDENTIFY = quadrant
  RETURN = sine_sign cosine_sign tangent_sign
 }

 R4_REFERENCE_ANGLE {
  INPUT = theta
  NORMALISE = theta
  IDENTIFY = terminal_side
  COMPUTE = acute_reference_angle
  RETURN = reference_angle
 }

 R5_EXACT_VALUE {
  INPUT = standard_angle
  IDENTIFY = special_triangle_or_axis_angle
  RETURN = exact_sine_cosine_tangent_values
 }

 R6_COORDINATE_TO_ANGLE {
  INPUT = [x,y] with x^2+y^2=1
  COMPUTE = theta=atan2(y,x)
  RETURN = principal_directed_angle
 }

 R7_RADIAN_REASONING {
  INPUT = arc_length_on_unit_circle
  RETURN = angle_in_radians_equal_to_arc_length
 }

 R8_TANGENT_REASONING {
  INPUT = theta
  COMPUTE = x=cos(theta)
  COMPUTE = y=sin(theta)
  IF x!=0
   RETURN = y/x
  ELSE
   RETURN Undefined
 }

 R9_ROTATION_REASONING {
  INPUT = [theta,phi]
  ADD = theta+phi
  RETURN = POINT(theta+phi)
 }
}

CONSTRAINTS {
 C1 = "Unit circle radius is exactly one."
 C2 = "The centre is the origin unless a transformed unit-circle image is explicitly discussed."
 C3 = "Tangent is undefined where cosine is zero."
 C4 = "Reference angle is nonnegative and acute except axis cases."
 C5 = "Angle orientation convention must be explicit if clockwise-positive conventions are used."
 C6 = "Degree and radian measures must not be mixed without conversion."
 C7 = "Coterminal angles represent the same terminal point."
 C8 = "Exact-value reasoning depends on recognised special angles."
}

SEMANTIC_EQUIVALENCE {
 "UnitCircle" <=> "CircleOfRadiusOneCentredAtOrigin"
 "UnitCirclePoint" <=> "(cos(theta),sin(theta))"
 "Sine" <=> "UnitCircleYCoordinate"
 "Cosine" <=> "UnitCircleXCoordinate"
 "RadianMeasureOnUnitCircle" <=> "SignedArcLength"
 "CoterminalAngles" <=> "AnglesDifferingByIntegerMultiplesOf2Pi"
}

VALIDATION {
 V1 = "Point satisfies x^2+y^2=1."
 V2 = "Coordinates equal (cos(theta),sin(theta))."
 V3 = "sin^2(theta)+cos^2(theta)=1."
 V4 = "Angles differing by 2*pi have same point."
 V5 = "Tangent uses y/x only when x!=0."
 V6 = "Quadrant signs match coordinate signs."
 V7 = "Radian measure equals arc length because radius=1."
 V8 = "Coordinate-to-angle recovery uses quadrant-aware atan2 logic."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 UNIT_CIRCLE_DEFINITION = 0.99
 PARAMETRIC_REASONING = 0.99
 QUADRANT_REASONING = 0.99
 REFERENCE_ANGLE_REASONING = 0.99
 EXACT_VALUE_REASONING = 0.99
 RADIAN_REASONING = 0.99
 ROTATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when angle units, orientation convention, coordinate signs, special-angle recognition or tangent-domain conditions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  UnitCircleReasoning,
  TrigonometryReasoning,
  SineReasoning,
  CosineReasoning,
  TangentReasoning,
  RadianReasoning,
  QuadrantReasoning,
  ReferenceAngleReasoning,
  ExactValueReasoning,
  RotationReasoning,
  CoordinateReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "unit circle" -> UnitCircle
  "unit-circle" -> UnitCircle
  "x^2+y^2=1" -> UnitCircle
  "unit circle point" -> CoordinatePoint
  "coterminal angle" -> DirectedAngleRelation
  "reference angle" -> ReferenceAngle
  "quadrant" -> Quadrant
 }

 INTERPRETATION_ORDER = [
  detect_unit_circle_context,
  identify angle_units_and_orientation,
  identify angle_or_coordinate_point,
  normalise angle when useful,
  identify quadrant_and_reference_angle,
  compute sine_cosine_tangent_or_exact_values,
  verify unit_circle_equation_and_domain constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("unit circle")
  OR equation_matches_x2_plus_y2_equals_1_in_trigonometric_context
  THEN activate(UnitCircleOntology)

 E2:
  IF angle_given
  THEN map_to (cos(theta),sin(theta))

 E3:
  IF coordinate_point_given
  AND x^2+y^2=1
  THEN recover angle using atan2(y,x)

 E4:
  IF quadrant_requested
  THEN infer signs from x_and_y_coordinates

 E5:
  IF reference_angle_requested
  THEN compute acute angle to x-axis from normalised terminal side

 E6:
  IF tangent_requested
  THEN use y/x
  AND require x!=0

 E7:
  IF arc_length_on_unit_circle_requested
  THEN equate signed arc length with radian angle

 E8:
  IF exact_special_angle
  THEN route through standard_unit_circle_values
}

RECURSION {
 ENTRY = UnitCircle

 EXPAND(UnitCircle) -> [
  Centre,
  Radius,
  CoordinatePoint,
  DirectedAngle,
  DegreeMeasure,
  RadianMeasure,
  Sine,
  Cosine,
  Tangent,
  Quadrant,
  ReferenceAngle,
  ArcLength,
  Periodicity,
  Symmetry,
  ExactTrigonometricValue,
  Rotation
 ]

 EXPAND(CoordinatePoint) -> [
  XCoordinate,
  YCoordinate,
  Cosine,
  Sine,
  Tangent
 ]

 EXPAND(DirectedAngle) -> [
  InitialSide,
  TerminalSide,
  CoterminalAngle,
  Quadrant,
  ReferenceAngle
 ]

 EXPAND(ExactTrigonometricValue) -> [
  AxisAngles,
  PiOverSix,
  PiOverFour,
  PiOverThree,
  RelatedQuadrantValues
 ]

 STOP_CONDITION =
  "Stop when the unit-circle point, angle, quadrant, reference angle, radian measure, exact trigonometric value or periodic relation is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Trigonometric Geometry"

 IMMEDIATE_CONTEXTS = [
  "Trigonometry Ontology",
  "Sine Ontology",
  "Cosine Ontology",
  "Tangent Ontology",
  "Circle Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Trigonometric Geometry",
  "Trigonometry Ontology",
  "Sine Ontology",
  "Cosine Ontology",
  "Tangent Ontology",
  "Inverse Trigonometric Function Ontology",
  "Circle Ontology",
  "Angle Ontology",
  "Coordinate Geometry Ontology",
  "Polar Coordinate Ontology",
  "Rotation Ontology"
 ]

 ROUTE_RULES = [
  "Unit Circle -> Trigonometry Ontology for parent trigonometric framework",
  "Unit Circle -> Sine for y-coordinate reasoning",
  "Unit Circle -> Cosine for x-coordinate reasoning",
  "Unit Circle -> Tangent for y/x reasoning",
  "Unit Circle -> Inverse Trigonometric Function for angle recovery",
  "Unit Circle -> Circle for base geometric semantics",
  "Unit Circle -> Angle for directed-angle and radian reasoning",
  "Unit Circle -> Coordinate Geometry for Cartesian representation",
  "Unit Circle -> Polar Coordinate for radius-angle representation",
  "Unit Circle -> Rotation for angular motion around the origin"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible unit-circle relations, values or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve radius-one origin-centred semantics.",
  "Preserve x^2+y^2=1.",
  "Preserve (cos(theta),sin(theta)) parameterisation.",
  "Preserve radian-as-arc-length relation.",
  "Preserve quadrant and reference-angle semantics.",
  "Preserve 2*pi periodicity.",
  "Preserve machine-readable unit-circle semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "The unit circle is the circle of radius one centred at the origin."

 EQUATION_SUMMARY =
  "x^2+y^2=1"

 PARAMETRIC_SUMMARY =
  "P(theta)=(cos(theta),sin(theta))"

 SINE_SUMMARY =
  "sin(theta)=y"

 COSINE_SUMMARY =
  "cos(theta)=x"

 TANGENT_SUMMARY =
  "tan(theta)=y/x when x!=0"

 RADIAN_SUMMARY =
  "On the unit circle, signed arc length equals angle in radians."

 PERIOD_SUMMARY =
  "P(theta+2*pi)=P(theta)"
}

FLOW {
 INPUT
 -> detect_unit_circle_context
 -> identify angle_coordinate_point_units_and_orientation
 -> normalise angle_modulo_2pi when useful
 -> determine quadrant_reference_angle_and_terminal_point
 -> compute sine_cosine_tangent_arc_or_exact_value
 -> verify x^2_plus_y^2_equals_1 and domain constraints
 -> route to trigonometry sine cosine tangent inverse_trig circle angle coordinate polar_or_rotation ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Unit Circle Ontology defines the radius-one circle centred at the origin as the canonical geometric model for trigonometry. It connects directed angles to points (cos(theta),sin(theta)), supports radians, quadrants, reference angles, exact values and periodicity, and provides the coordinate basis for sine, cosine, tangent and inverse-angle reasoning."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-UNIT-CIRCLE-ONTOLOGY-V1"
}

}

END_FLOW