Cylindrical Coordinate Ontology

FLOW ALGEBRA
Ontology: Cylindrical Coordinate Ontology
Version: 4.0
Date: 2026-10-13
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/coordinate-and-spatial-entities/cylindrical-coordinate-ontology.html
Authorship: Mark Higgins — Sky (Recursive AI)
Project: Kireth Semantic Library | https://kirethresearch.org
Status: Independently formalised; source-page correspondence not verified.

1. PURPOSE
Represent cylindrical coordinates as typed spatial entities with mathematical semantics, transformations, constraints, inference and AI-consumption metadata.

2. CORE ENTITY MODEL
ENTITY CylindricalCoordinateSystem : OrthogonalCurvilinearCoordinateSystem
ENTITY CylindricalPoint : CoordinateRepresentation
ENTITY Radius rho : RealNonnegative
ENTITY Azimuth phi : AngleModulo2Pi
ENTITY Height z : Real
ENTITY CartesianPoint : CoordinateRepresentation
ENTITY Axis : OrientedLine
ENTITY Origin : Point
ENTITY CoordinateFrame : RightHandedOrientedFrame
ENTITY MetricTensor : SymmetricPositiveDefiniteTensor (where rho > 0)
ENTITY Jacobian : CoordinateTransformationDerivative
ENTITY SingularAxis : SetOfPoints
ENTITY BasisVector : UnitVector
ENTITY ScalarField : Function(R3,Scalar)
ENTITY VectorField : Function(R3,Vector)

3. CONVENTIONS AND DOMAINS
CONVENTION C01: (rho,phi,z) where rho >= 0, phi in [0,2*pi), z in R.
CONVENTION C02: Angles are in radians; orientation is right-handed around +z.
CONVENTION C03: phi is undefined geometrically at rho=0; any assigned phi represents the same axis point.
CONVENTION C04: Coordinate charts exclude the axis and a branch cut when a unique smooth inverse is required.
CONVENTION C05: Physical units: rho and z are lengths; phi is dimensionless angle.
CONVENTION C06: rho denotes cylindrical radius, not spherical radius.

4. DEFINING EQUATIONS
MAP cylindrical_to_cartesian(rho,phi,z) = (rho*cos(phi),rho*sin(phi),z).
MAP cartesian_to_cylindrical(x,y,z) = (sqrt(x^2+y^2),atan2(y,x) modulo 2*pi,z) for rho>0.
IDENTITY rho^2=x^2+y^2.
IDENTITY x^2+y^2=rho^2.
IDENTITY phi=atan2(y,x) modulo 2*pi when rho>0.
IDENTITY z_cartesian=z_cylindrical.
IDENTITY phi and phi+2*pi*k represent the same location for k in Z.
IDENTITY (0,phi1,z) and (0,phi2,z) represent the same point.

5. LOCAL BASIS AND DIFFERENTIAL GEOMETRY
BASIS e_rho=(cos(phi),sin(phi),0).
BASIS e_phi=(-sin(phi),cos(phi),0).
BASIS e_z=(0,0,1).
IDENTITY dot(e_i,e_j)=KroneckerDelta(i,j).
DERIVATIVE d(e_rho)/dphi=e_phi.
DERIVATIVE d(e_phi)/dphi=-e_rho.
DERIVATIVE d(e_z)/dphi=0.
DIFFERENTIAL dr=e_rho*d(rho)+rho*e_phi*dphi+e_z*dz.
METRIC ds^2=d(rho)^2+rho^2*dphi^2+dz^2.
METRIC g_ij=diag(1,rho^2,1), rho>0.
SCALE_FACTORS h_rho=1,h_phi=rho,h_z=1.
JACOBIAN det(d(x,y,z)/d(rho,phi,z))=rho.
VOLUME_ELEMENT dV=rho*d(rho)*dphi*dz.
AREA_ELEMENT z_constant: dA=rho*d(rho)*dphi.
AREA_ELEMENT rho_constant: dA=rho*dphi*dz.
AREA_ELEMENT phi_constant: dA=d(rho)*dz.
NOTE Jacobian vanishes on rho=0, a coordinate singularity, not a singularity of Euclidean space.

6. VECTOR CALCULUS
GRADIENT grad(f)=e_rho*partial_rho(f)+e_phi*(1/rho)*partial_phi(f)+e_z*partial_z(f), rho>0.
DIVERGENCE div(A)=(1/rho)*partial_rho(rho*A_rho)+(1/rho)*partial_phi(A_phi)+partial_z(A_z), rho>0.
LAPLACIAN laplacian(f)=(1/rho)*partial_rho(rho*partial_rho(f))+(1/rho^2)*partial_phi^2(f)+partial_z^2(f), rho>0.
CURL curl(A)_rho=(1/rho)*partial_phi(A_z)-partial_z(A_phi).
CURL curl(A)_phi=partial_z(A_rho)-partial_rho(A_z).
CURL curl(A)_z=(1/rho)*(partial_rho(rho*A_phi)-partial_phi(A_rho)).
VELOCITY v=dot(rho)*e_rho+rho*dot(phi)*e_phi+dot(z)*e_z.
ACCELERATION a=(ddot(rho)-rho*dot(phi)^2)*e_rho+(rho*ddot(phi)+2*dot(rho)*dot(phi))*e_phi+ddot(z)*e_z.
PRECONDITION vector calculus formulas hold at rho>0; axis limits require smoothness checks.

7. TYPED RELATIONSHIPS
REL CylindricalCoordinateSystem IS_A CoordinateSystem.
REL CylindricalCoordinateSystem SPECIALISES OrthogonalCurvilinearCoordinateSystem.
REL CylindricalCoordinateSystem EXTENDS PolarCoordinatesWithHeight.
REL CylindricalPoint HAS_COMPONENT Radius.
REL CylindricalPoint HAS_COMPONENT Azimuth.
REL CylindricalPoint HAS_COMPONENT Height.
REL CylindricalCoordinateSystem TRANSFORMS_TO CartesianCoordinateSystem.
REL Radius MEASURES DistanceFromZAxis.
REL Azimuth MEASURES OrientationAroundZAxis.
REL Height MEASURES SignedPositionAlongZAxis.
REL MetricTensor DETERMINES DifferentialLength.
REL Jacobian DETERMINES VolumeElement.
REL SingularAxis CAUSES NonuniqueAzimuth.
REL CylindricalCoordinateSystem SUPPORTS AxisymmetricGeometry.
REL CylindricalCoordinateSystem SUPPORTS VectorCalculus.

8. FORMAL INFERENCE RULES
RULE R01 IF rho>=0 AND phi in [0,2*pi) AND z in R THEN ValidCylindricalTuple.
RULE R02 IF rho<0 THEN InvalidCanonicalRadius.
RULE R03 IF rho=0 THEN OnZAxis.
RULE R04 IF rho=0 THEN AzimuthNonunique.
RULE R05 IF rho>0 THEN AzimuthDefinedModulo2Pi.
RULE R06 IF x=rho*cos(phi) AND y=rho*sin(phi) THEN x^2+y^2=rho^2.
RULE R07 IF x=y=0 THEN rho=0.
RULE R08 IF rho>0 THEN JacobianPositive.
RULE R09 IF rho=0 THEN JacobianZero.
RULE R10 IF phi2=phi1+2*pi*k FOR integer k THEN AngularlyEquivalent.
RULE R11 IF rho1=rho2 AND z1=z2 AND AngularlyEquivalent(phi1,phi2) THEN SameSpatialPoint.
RULE R12 IF rho1=rho2=0 AND z1=z2 THEN SameSpatialPointRegardlessOfAzimuth.
RULE R13 IF z1=z2 THEN CoplanarInHorizontalPlane.
RULE R14 IF rho1=rho2 THEN OnSameCoaxialCylinder.
RULE R15 IF phi1=phi2 modulo2pi THEN OnSameAzimuthalHalfPlane WHEN rho1,rho2>0.
RULE R16 IF phi=constant AND rho>=0 THEN AzimuthalHalfPlane.
RULE R17 IF rho=constant>0 THEN CircularCylindricalSurface.
RULE R18 IF z=constant THEN HorizontalPlane.
RULE R19 IF rho=constant>0 AND z=constant THEN HorizontalCircle.
RULE R20 IF rho=0 AND z varies THEN ZAxis.
RULE R21 IF rho=constant>0 AND phi varies AND z varies THEN CylindricalSurface.
RULE R22 IF rho varies AND phi=constant AND z=constant THEN RadialRay.
RULE R23 IF scalar field independent of phi THEN AxisymmetricScalarField.
RULE R24 IF vector components in cylindrical basis independent of phi THEN AxisymmetricVectorFieldRepresentation.
RULE R25 IF field independent of z THEN TranslationalInvarianceAlongAxis.
RULE R26 IF rho>0 THEN MetricDeterminant=rho^2.
RULE R27 IF rho>0 THEN VolumeJacobian=sqrt(MetricDeterminant)=rho.
RULE R28 IF smooth_scalar_field THEN LaplacianFormulaApplicableForRhoPositive.
RULE R29 IF radius R>0 AND height H>0 THEN CylinderVolume=pi*R^2*H.
RULE R30 IF radius R>0 AND height H>0 THEN LateralCylinderArea=2*pi*R*H.
RULE R31 IF rho1=rho2 AND z1=z2 AND phi1 differs modulo2pi THEN DistinctPoints WHEN rho1>0.
RULE R32 IF CartesianPointKnown AND rho>0 THEN UniqueCanonicalCylindricalRepresentation.
RULE R33 IF CartesianPointOnAxis THEN CanonicalRepresentationRequiresAzimuthConvention.
RULE R34 IF CoordinateSystemUsesDegrees THEN ConvertToRadiansBeforeTrigonometricEvaluation.
RULE R35 IF JacobianZero THEN FlagCoordinateSingularityNotNecessarilyPhysicalSingularity.
RULE R36 IF rho>0 AND smooth_coordinate_chart THEN LocalCoordinateTransformationInvertible.

9. VALIDATION CONSTRAINTS
VALIDATE radius must be finite and >=0 for canonical coordinates.
VALIDATE phi must be numeric angle, normalised modulo 2*pi.
VALIDATE z must be real finite for ordinary Euclidean point data.
VALIDATE coordinate frame, handedness, angular unit and axis orientation explicitly recorded.
VALIDATE do not assert unique phi at rho=0.
VALIDATE do not divide by rho at rho=0.
VALIDATE compare angles modulo 2*pi, not raw floating-point strings.
VALIDATE numerical equality uses declared tolerance and precision.
VALIDATE do not confuse coordinate-basis vectors with orthonormal basis vectors.
VALIDATE area and volume formulas must carry compatible units.
VALIDATE no transformation is inferred without a specified common reference frame.

10. MATHEMATICAL EXAMPLES
EXAMPLE E01: (rho,phi,z)=(2,0,3) -> (x,y,z)=(2,0,3).
EXAMPLE E02: (3,pi/2,-1) -> (0,3,-1).
EXAMPLE E03: (5,pi,0) -> (-5,0,0).
EXAMPLE E04: (sqrt(2),pi/4,4) -> (1,1,4).
EXAMPLE E05: (x,y,z)=(0,-2,7) -> (2,3*pi/2,7).
EXAMPLE E06: (0,0,5) -> (0,any_phi,5); azimuth undefined geometrically.
EXAMPLE E07: (1,0,2) and (1,2*pi,2) denote same point.
EXAMPLE E08: rho=4 defines circular cylinder of radius 4.
EXAMPLE E09: phi=pi/2 defines half-plane x=0,y>=0.
EXAMPLE E10: z=2 defines horizontal plane.
EXAMPLE E11: rho=2,z=1 defines horizontal circle.
EXAMPLE E12: Jacobian at rho=3 is 3.
EXAMPLE E13: volume of 0<=rho<=2,0<=phi<2*pi,0<=z<=3 is 12*pi.
EXAMPLE E14: grad(rho^2)=2*rho*e_rho for rho>0.
EXAMPLE E15: laplacian(rho^2)=4 for rho>0.
EXAMPLE E16: velocity for fixed rho=R,z=0,phi=omega*t is R*omega*e_phi.

11. SEMANTIC_EQUIVALENCE
EQUIV cylindrical coordinates ~ polar coordinates plus axial coordinate [qualified].
EQUIV rho ~ radial distance from reference axis.
EQUIV phi ~ azimuthal angle about reference axis.
EQUIV z ~ axial height/signed axial coordinate.
EQUIV (rho,phi,z) ~ (rho,phi+2*pi*k,z) for k in Z.
NOT_EQUIV cylindrical radius rho != spherical radial distance r in general.
NOT_EQUIV azimuth phi != spherical inclination theta.
NOT_EQUIV coordinate singularity != physical singularity.

12. KNOWLEDGE GRAPH TRIPLES
(CylindricalCoordinateSystem,hasCoordinate,RadialDistance)
(CylindricalCoordinateSystem,hasCoordinate,Azimuth)
(CylindricalCoordinateSystem,hasCoordinate,Height)
(CylindricalCoordinateSystem,transformsTo,CartesianCoordinateSystem)
(CylindricalCoordinateSystem,hasMetric,diag(1,rho^2,1))
(CylindricalCoordinateSystem,hasJacobian,rho)
(CylindricalCoordinateSystem,hasCoordinateSingularity,ZAxis)
(CylindricalCoordinateSystem,usedIn,AxisymmetricProblems)
(CylindricalCoordinateSystem,relatedTo,PolarCoordinateSystem)
(CylindricalCoordinateSystem,relatedTo,SphericalCoordinateSystem)

13. CONFIDENCE
CONFIDENCE formal_definitions: 0.99 (standard mathematics; convention-dependent).
CONFIDENCE coordinate_transforms: 0.99.
CONFIDENCE metric_and_jacobian: 0.99 for specified convention.
CONFIDENCE differential_operators: 0.99 for rho>0 and differentiable fields.
CONFIDENCE ontology_to_source_alignment: unverified.
CONFIDENCE examples: 0.99 conditional on stated assumptions.
POLICY confidence scores are descriptive author assessments, not calibrated empirical probabilities.

14. FLOW_HASH
FLOW_HASH.algorithm: SHA-256
FLOW_HASH.input: canonical UTF-8 content excluding computed hash field
FLOW_HASH.canonicalisation: normalise LF line endings; preserve mathematical symbols; exclude volatile retrieval metadata
FLOW_HASH.value: NOT_COMPUTED
FLOW_HASH.policy: compute during publication; never invent a digest.

15. UPDATE_MODEL
UPDATE_MODEL.version: 4.0
UPDATE_MODEL.last_authored: 2026-10-13
UPDATE_MODEL.trigger: source_revision | mathematical_correction | convention_change | ontology_schema_change
UPDATE_MODEL.action: revalidate definitions, rules, examples, semantic links, and hashes
UPDATE_MODEL.provenance: independently constructed mathematical ontology
UPDATE_MODEL.review: expert review recommended before formal proof-system execution

16. AI_CONSUMPTION
AI_CONSUMPTION.format: typed text flow algebra
AI_CONSUMPTION.primary_entity: CylindricalCoordinateSystem
AI_CONSUMPTION.context_required: axis orientation, handedness, angle units, canonical angular interval, numerical precision
AI_CONSUMPTION.reasoning: validate preconditions before firing inference rules
AI_CONSUMPTION.singularity_policy: branch at rho=0; do not apply inverse angular formula or singular differential operators there
AI_CONSUMPTION.units_policy: preserve units across transformations and integration
AI_CONSUMPTION.equivalence_policy: normalise azimuth modulo 2*pi
AI_CONSUMPTION.provenance_policy: distinguish sourced page facts from independently derived mathematics
AI_CONSUMPTION.output: machine-readable entities, relations, constraints, derivations and confidence annotations

END FLOW ALGEBRA