Cone Ontology

FLOW ALGEBRA
TITLE: Cone Ontology
VERSION: 4.0
DOMAIN: Mathematical Entities / Geometric Entities
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/geometric-entities/cone-ontology-2.html
SOURCE_ALIGNMENT: Independent mathematical formalisation; source page content not verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org

[ENTITY]
ID: GEO.CONE
TYPE: GeometricSolid
LABEL: Cone
DEFINITION: A geometric solid consisting of segments joining a fixed apex to points of a planar base region; in the circular case the base is a disk.
CONTEXT: Euclidean three-dimensional geometry; distinguish solid cone from its lateral surface and infinite conical surface.

[TYPED ENTITIES]
Cone: GeometricSolid
RightCircularCone: Cone
ObliqueCircularCone: Cone
CircularCone: Cone
FiniteCone: Cone
InfiniteConeSurface: Surface
TruncatedCone: Frustum
Apex: Point
BaseRegion: PlanarRegion
BaseCircle: Circle
Axis: Line
Altitude: LineSegment
Height: NonNegativeReal
Radius: NonNegativeReal
SlantHeight: NonNegativeReal
Volume: NonNegativeReal
LateralArea: NonNegativeReal
TotalSurfaceArea: NonNegativeReal
HalfAngle: Angle
Generator: LineSegment

[RELATIONSHIPS]
Cone HAS_APEX Apex
Cone HAS_BASE BaseRegion
Cone HAS_HEIGHT Height
CircularCone HAS_BASE_BOUNDARY BaseCircle
CircularCone HAS_RADIUS Radius
RightCircularCone HAS_AXIS Axis
RightCircularCone HAS_SLANT_HEIGHT SlantHeight
RightCircularCone HAS_LATERAL_AREA LateralArea
RightCircularCone HAS_VOLUME Volume
RightCircularCone IS_A CircularCone
ObliqueCircularCone IS_A CircularCone
TruncatedCone DERIVED_FROM Cone
Cone IS_A GeometricSolid

[FORMAL DEFINITIONS]
F01: For apex A and planar base B, SolidCone(A,B) = {(1-t)A + t b | b in B, 0 <= t <= 1}.
F02: Nondegenerate cone requires A outside the base plane and base region B with positive area.
F03: Height h = perpendicular distance from apex A to base plane.
F04: A right circular cone has its apex on the line through the base-circle centre perpendicular to the base plane.
F05: An oblique circular cone has its apex off that perpendicular line.
F06: Right circular cone: slant height l = sqrt(r^2 + h^2).
F07: Circular cone (right or oblique): V = (1/3) pi r^2 h.
F08: Right circular cone: lateral area L = pi r l.
F09: Right circular cone: total area S = pi r l + pi r^2, including the base.
F10: Right circular cone half-angle theta satisfies tan(theta) = r/h for h > 0.
F11: Right circular cone half-angle theta satisfies sin(theta) = r/l for l > 0.
F12: Right circular cone in coordinates with base at z=0 and apex (0,0,h): x^2+y^2 <= r^2 (1-z/h)^2 for 0 <= z <= h, h>0.
F13: Its lateral surface uses equality in F12.
F14: Infinite right circular double cone with axis z: x^2+y^2 = z^2 tan^2(theta), excluding degenerate theta values.
F15: Similar right circular cones with scale k>0 have lengths multiplied by k, areas by k^2, volumes by k^3.
F16: Frustum of right circular cone: V = (pi h/3)(R^2 + Rr + r^2), with R,r >=0.
F17: Right circular frustum lateral area = pi(R+r)s, s = sqrt(h^2+(R-r)^2).
F18: Cross-section parallel to circular base at distance z from base in right cone has radius r(z)=r(1-z/h), 0<=z<=h.
F19: Corresponding cross-sectional area A(z)=pi r^2(1-z/h)^2.
F20: General cone with measurable planar base area B and apex outside its plane: V=Bh/3.

[VALIDATION CONSTRAINTS]
V01: Geometry context must be specified: solid, lateral surface, or infinite surface.
V02: Lengths r,h,l must be nonnegative; nondegenerate right cone requires r>0 and h>0.
V03: Apply l=sqrt(r^2+h^2) only to a right circular cone.
V04: Apply pi*r*l lateral area only to a right circular cone.
V05: Do not apply right-cone total-area formula to an oblique cone without independent surface calculation.
V06: Area units are squared; volume units are cubed.
V07: h denotes perpendicular altitude, not slant height.
V08: A finite solid cone contains its interior; its lateral surface does not.
V09: A cone and a frustum are not semantically identical.
V10: A circular base region is a disk, whereas its boundary is a circle.

[INFERENCE RULES]
R01: IF shape IS_A RightCircularCone THEN shape IS_A CircularCone.
R02: IF shape IS_A CircularCone THEN shape IS_A Cone.
R03: IF shape IS_A Cone THEN shape IS_A GeometricSolid in the solid-cone context.
R04: IF right_cone(r,h) AND r>0 AND h>0 THEN slant_height=sqrt(r^2+h^2).
R05: IF right_cone(r,h) THEN volume=pi*r^2*h/3.
R06: IF oblique_circular_cone(r,h) THEN volume=pi*r^2*h/3.
R07: IF right_cone(r,h) THEN lateral_area=pi*r*sqrt(r^2+h^2).
R08: IF right_cone(r,h) THEN total_area=pi*r*(r+sqrt(r^2+h^2)).
R09: IF right_cone(r,h) AND h>0 THEN half_angle=atan(r/h).
R10: IF cone(base_area=B,height=h) THEN volume=B*h/3.
R11: IF cone(scale_factor=k) THEN volume_scale=k^3.
R12: IF cone(scale_factor=k) THEN area_scale=k^2.
R13: IF right_cone(r,h) AND r=h THEN half_angle=45 degrees.
R14: IF right_cone(r,h) AND r>0 AND h>0 THEN volume>0.
R15: IF right_cone(r,h) THEN lateral_area<=total_area.
R16: IF right_cone(r,h) THEN total_area-lateral_area=pi*r^2.
R17: IF right_cone(r,h) THEN slant_height>=r.
R18: IF right_cone(r,h) THEN slant_height>=h.
R19: IF parallel_cross_section(z) THEN section_is_disk in solid right-cone context.
R20: IF parallel_cross_section(z) THEN radius=r*(1-z/h) for 0<=z<=h.
R21: IF right_cone(r,h) AND h>0 THEN A(z)=pi*r^2*(1-z/h)^2.
R22: IF frustum(R,r,h) THEN volume=pi*h*(R^2+R*r+r^2)/3.
R23: IF frustum(R,r,h) THEN slant=sqrt(h^2+(R-r)^2).
R24: IF frustum(R,r,h) THEN lateral_area=pi*(R+r)*slant.
R25: IF r=0 OR h=0 THEN flag_degenerate_geometry.
R26: IF user_requests_surface_area THEN clarify inclusion_of_base when unspecified.
R27: IF shape HAS_APEX A AND HAS_BASE B THEN cone_generators_join(A,b) for b on boundary(B).
R28: IF apex lies above base centre along normal THEN circular_cone_is_right.
R29: IF apex does not lie on base-centre normal THEN circular_cone_is_oblique.
R30: IF r,h have compatible length units THEN volume_has_cubic_units.
R31: IF r,h have compatible length units THEN lateral_area_has_square_units.
R32: IF cone volume and base area positive THEN height=3*volume/base_area.
R33: IF right_cone volume and r>0 THEN height=3*volume/(pi*r^2).
R34: IF right_cone volume and h>0 THEN radius=sqrt(3*volume/(pi*h)).
R35: IF lateral_surface_only THEN exclude_base_disk_from_area.
R36: IF surface_equation_is_double_cone THEN do_not_assume_finite_solid_cone.

[MATHEMATICAL EXAMPLES]
E01: r=3,h=4 => l=5, V=12*pi, L=15*pi, S=24*pi.
E02: r=2,h=6 => V=8*pi.
E03: r=5,h=12 => l=13, L=65*pi, S=90*pi.
E04: r=1,h=1 => V=pi/3, theta=45 degrees.
E05: r=3,h=4 => A(z=2)=9*pi/4.
E06: r=3,h=4 => A(z=4)=0.
E07: r=3,h=4 => A(z=0)=9*pi.
E08: r=3,h=4 => cross_section_radius(z=2)=1.5.
E09: r=2,h=3, scale k=2 => new radius=4, height=6, volume factor=8.
E10: frustum R=4,r=2,h=3 => V=28*pi.
E11: frustum R=4,r=2,h=3 => slant=sqrt(13), L=6*pi*sqrt(13).
E12: base_area=15,height=9 => V=45.
E13: right cone r=6,h=8 => l=10, V=96*pi.
E14: right cone r=6,h=8 => L=60*pi, S=96*pi.
E15: r=3,h=4 => tan(theta)=3/4.
E16: r=3,h=4 => sin(theta)=3/5.

[SEMANTIC_EQUIVALENCE]
cone (solid geometry) ~= finite cone solid [when finite and base specified]
right circular cone ~= circular cone with perpendicular axis through base centre
slant height ~= generator length [right circular cone only]
cone surface != cone solid
cone != cylinder
cone != pyramid [unless a broader generalised-cone convention is explicitly adopted]
frustum != full cone
base disk != base circle

[KNOWLEDGE_GRAPH]
(GEO.CONE, rdf:type, GeometricSolid)
(GEO.RIGHT_CIRCULAR_CONE, rdfs:subClassOf, GEO.CONE)
(GEO.OBLIQUE_CIRCULAR_CONE, rdfs:subClassOf, GEO.CONE)
(GEO.CONE, hasPart, GEO.APEX)
(GEO.CONE, hasPart, GEO.BASE_REGION)
(GEO.CONE, hasProperty, GEO.HEIGHT)
(GEO.RIGHT_CIRCULAR_CONE, hasProperty, GEO.SLANT_HEIGHT)
(GEO.CONE, relatedTo, GEO.FRUSTUM)
(GEO.CONE, relatedTo, GEO.CYLINDER)
(GEO.CONE, relatedTo, GEO.CONIC_SECTION)

[CONFIDENCE]
Mathematical definitions and standard formulas: HIGH.
Source-page-specific coverage: UNVERIFIED.
Semantic equivalence: CONTEXT_DEPENDENT.
Inference validation: REQUIRES typed inputs and precondition checks.

[FLOW_HASH]
Algorithm: SHA-256
Canonicalisation: UTF-8; LF line endings; deterministic key ordering for machine records.
Hash value: COMPUTE_ON_DEPLOYMENT; do not treat placeholder as an actual hash.

[UPDATE_MODEL]
Version: 4.0
Review trigger: Source page change, taxonomy revision, mathematical correction, or changed AI-consumption contract.
Procedure: Validate definitions -> check formulas and rule preconditions -> regenerate graph -> compute FLOW_HASH -> publish new version.
Provenance: Mathematical formalisation independently authored; linked webpage not verified.

[AI_CONSUMPTION]
Entity key: GEO.CONE
Preferred label: Cone
Parse formal definitions as typed claims with scope and preconditions.
Never conflate solid cone, lateral surface, infinite double cone, or frustum.
Use h as perpendicular altitude; verify shape is right circular before using slant-area formula.
Return symbolic exact values where possible; attach units to measurements.
Validate all 36 inference rules against explicit assumptions.
Represent unsupported source-page claims as unknown, not verified.
END FLOW ALGEBRA