Finite Planes

• Introduction
  • Planes in Magma
• Construction of a Plane
  • FiniteProjectivePlane< v | X : parameters >: RngIntElt, List → PlaneProj
  • FiniteProjectivePlane< V | X : parameters >: SetIndx, List → PlaneProj
  • FiniteProjectivePlane(W): ModTupFld → PlaneProj
  • FiniteProjectivePlane(F): FldFin → PlaneProj
  • FiniteProjectivePlane(q): RngIntElt → PlaneProj
  • FiniteAffinePlane< v | X : parameters >: RngIntElt, List → PlaneAff
  • FiniteAffinePlane< V | X : parameters >: SetIndx, List → PlaneAff
  • FiniteAffinePlane(W): ModFld → PlaneAff
  • FiniteAffinePlane(F): FldFin → PlaneProj
  • FiniteAffinePlane(q): RngIntElt → PlaneProj
  • Example: Classical Projective Plane of Order 3
• The Point-Set and Line-Set of a Plane
  • Introduction
  • Creating Point-Sets and Line-Sets
    • PointSet(P): Plane → PlanePtSet
    • LineSet(P): Plane → PlaneLnSet
  • Using the Point-Set and Line-Set to Create Points and Lines
    • V . i: PlanePtSet, RngIntElt → PlanePt
    • V ! [a, b, c]: PlanePtSet, SeqEnum → PlanePt
    • V ! [a, b]: PlanePtSet, SeqEnum → PlanePt
    • V ! x: PlanePtSet, Elt → PlanePt
    • Representative(V): PlanePtSet → PlanePt
    • Rep(V): PlanePtSet → PlanePt
    • Random(V): PlanePtSet → PlanePt
    • L . i: PlanePtSet, RngIntElt → PlanePt
    • L ! [a, b, c]: PlaneLnSet, SeqEnum → PlaneLn
    • L ! [m, b]: PlaneLnSet, SeqEnum → PlaneLn
    • L ! S: PlaneLnSet, SetEnum → PlaneLn
    • L ! S: PlaneLnSet, SeqEnum → PlaneLn
    • L ! l: PlaneLnSet, PlaneLn → PlaneLn
    • Representative(L): PlaneLnSet → PlaneLn
    • Rep(L): PlaneLnSet → PlaneLn
    • Random(L): PlaneLnSet → PlaneLn
    • Example: Creating Points and Lines
  • Retrieving the Plane from Points, Lines, Point-Sets, and Line-Sets
    • ParentPlane(V): PlanePtSet → Plane, PlanePtSet, PlaneLnSet
    • ParentPlane(L): PlaneLnSet → Plane, PlanePtSet, PlaneLnSet
    • ParentPlane(p): PlanePt → Plane, PlanePtSet, PlaneLnSet
    • ParentPlane(l): PlaneLn → Plane, PlanePtSet, PlaneLnSet
• The Set of Points and Set of Lines
  • Points(P): Plane → { PlanePt }
  • Lines(P): PlaneLnSet → { PlaneLn}
• The Defining Points of a Plane
  • Support(P): Plane → { Elt }
  • Support(l): PlaneLn → SetEnum
  • Support(P, p): Plane, PlanePt → .
  • Support(p): PlanePt → .
  • Example: Points versus Support of a Plane
• Subplanes
  • sub<P | L>: Plane, List → Plane
  • SubfieldSubplane(P, F): Plane, FldFin → Plane, PlanePtSet, PlaneLnSet
  • Example: Subplane Generated by a Quadrangle
• Structures Associated with a Plane
  • VectorSpace(P): Plane → ModTupFld
  • Field(P): Plane → FldFin
  • IncidenceMatrix(P): Plane → AlgMatElt
  • Dual(P): Plane → Plane, PlanePtSet, PlaneLnSet
  • Example: Vector Space and Field of a Plane
• Numerical Invariants of a Plane
  • Order(P): Plane → RngIntElt
  • NumberOfPoints(P): Plane → RngIntElt
  • # V: PlanePtSet → RngIntElt
  • NumberOfLines(P): Plane → RngIntElt
  • # L: PlaneLnSet → RngIntElt
  • pRank(P): Plane → RngIntElt
  • pRank(P, p): Plane, RngIntElt → RngIntElt
  • Example: Numerical Invariants of PG(2,8)
• Properties of Planes
  • IsDesarguesian(P): Plane → BoolElt
  • IsSelfDual(P): PlaneProj → BoolElt
• Identity and Isomorphism
  • P eq Q: Plane, Plane → BoolElt
  • P ne Q: Plane, Plane → BoolElt
  • IsIsomorphic(P, Q: parameters): Plane, Plane → BoolElt, Map
  • P subset Q: Plane, Plane → BoolElt
• The Connection between Projective and Affine Planes
  • FiniteAffinePlane(P, l): PlaneProj, PlaneLn → PlaneAff, PlanePtSet, PlaneLnSet, Map
  • ProjectiveEmbedding(P): PlaneAff → PlaneProj, PlanePtSet, PlaneLnSet, Map
  • Example: Projective Embedding of an Affine Plane
• Operations on Points and Lines
  • Elementary Operations
    • p eq q: PlanePt, PlanePt → BoolElt
    • p ne q: PlanePt, PlanePt → BoolElt
    • l eq m: PlaneLn, PlaneLn → BoolElt
    • l ne m: PlaneLn, PlaneLn → BoolElt
    • p in l: PlanePt, PlaneLn → BoolElt
    • p notin l: PlanePt, PlaneLn → BoolElt
    • S subset l: { PlanePt}, PlaneLn → BoolElt
    • S notsubset l: { PlanePt}, PlaneLn → BoolElt
    • l meet m: PlaneLn, PlaneLn → PlanePt
    • Representative(l): PlaneLn → PlanePt
    • Rep(l): PlaneLn → PlanePt
    • Random(l): PlaneLn → PlanePt
  • Deconstruction Functions
    • Index(P, p): Plane, PlanePt → RngIntElt
    • Index(P, l): Plane, PlaneLn → RngIntElt
    • p[i]: PlanePt, RngIntElt → FldFinElt
    • l[i]: PlaneLn, RngIntElt → FldFinElt
    • Coordinates(P, p): Plane, PlanePt → [ FldFinElt ]
    • Coordinates(P, l): Plane, PlaneLn → [ FldFinElt ]
    • ElementToSequence(p): PlanePt → [ FldFinElt ]
    • Eltseq(p): PlanePt → [ FldFinElt ]
    • ElementToSequence(l): PlaneLn → [ FldFinElt ]
    • Eltseq(l): PlaneLn → [ FldFinElt ]
    • Set(l): PlaneLn → { PlanePt }
    • Example: Deconstructing Points and Lines
  • Other Point and Line Functions
    • IsCollinear(P, S): Plane, { PlanePt} → BoolElt, PlaneLn
    • IsConcurrent(P, R): Plane, { PlaneLn} → BoolElt, PlanePt
    • ContainsQuadrangle(P, S): Plane, { PlanePt } → BoolElt
    • Pencil(P, p): Plane, PlanePt → { PlaneLn }
    • Slope(l): PlaneLn → FldFinElt
    • IsParallel(P, l, m): Plane, PlaneLn, PlaneLn → BoolElt
    • ParallelClass(P, l): Plane, PlaneLn → { PlaneLn }
    • ParallelClasses(P): PlaneAff → { { PlaneLn } }
    • Example: Point and Line Functions in AG(2,3)
• Arcs
  • kArc(P, k): Plane, RngIntElt → SetEnum
  • CompleteKArc(P, k): Plane, RngIntElt → SetEnum
  • IsArc(P, A): Plane, { PlanePt } → BoolElt
  • IsComplete(P, A): Plane, { PlanePt } → BoolElt
  • Conic(P, S): Plane, { PlanePt } → SetEnum
  • QuadraticForm(S): { PlanePt } → RngMPolElt
  • Tangent(P, A, p): Plane, { PlanePt }, PlanePt → PlaneLn
  • AllTangents(P, A): Plane, { PlanePt} → { PlaneLn}
  • AllSecants(P, A): Plane, { PlanePt} → { PlaneLn}
  • ExternalLines(P, A): Plane, { PlanePt} → { PlaneLn}
  • AllPassants(P, A): Plane, { PlanePt} → { PlaneLn}
  • Knot(P, C): Plane, { PlanePt} → PlanePt
  • Exterior(P, C): Plane, { PlanePt} → { PlanePt}
  • Interior(P, C): Plane, { PlanePt} → { PlanePt}
  • Example: Oval Design from an Arc
• Unitals
  • IsUnital(P, U): Plane, { PlanePt} → BoolElt
  • AllTangents(P, U): Plane, { PlanePt} → { PlaneLn}
  • UnitalFeet(P, U, p): Plane, { PlanePt}, PlanePt → { PlanePt}
  • Example: Hermitian Unital in PG(2,9)
• The Collineation Group of a Plane
  • The Collineation Group Function
    • CollineationGroup(P): Plane → GrpPerm, GSet, GSet, PowMap, Map
    • AutomorphismGroup(P): Plane → GrpPerm, GSet, GSet, PowMap, Map
    • PointGroup(P): Plane → GrpPerm, GSet, GSet, PowMap, Map
    • LineGroup(P): Plane → GrpPerm, PowMap, Map
    • CollineationGroupStabilizer(P, k): Plane, RngIntElt → GrpPerm, GSet, GSet, PowMap, Map
    • CollineationSubgroup(P): Plane → GrpPerm, GSet, GSet, PowMap, Map
  • General Action of Collineations
    • y ^ g: Elt, GrpPermElt → Elt
    • y ^ G: Elt, GrpPerm → GSet
    • Image(g, Y, y): GrpPermElt, GSet, Elt → Elt
    • Orbit(G, Y, y): GrpPerm, GSet, Elt → GSet
    • Orbits(G, Y): GrpPerm, GSet → [ GSet ]
    • Stabilizer(G, Y, y): GrpPerm, GSet, Elt → GrpPerm
    • Action(G, Y): GrpPerm, GSet → Hom(Grp), GrpPerm, GrpPerm
    • ActionImage(G, Y): GrpPerm, GSet → GrpPerm
    • ActionKernel(G, Y): GrpPerm, GSet → GrpPerm
    • Example: G-Sets of the Collineation Group
    • Example: Projective Bundle in PG(2,q)
    • Example: Baer Derivation of an Affine Plane
  • Central Collineations
    • CentralCollineationGroup(P, p, l): Plane, PlanePt, PlaneLn → GrpPerm, PowMap, Map
    • CentralCollineationGroup(P, p): Plane, PlanePt → GrpPerm, PowMap, Map
    • CentralCollineationGroup(P, l): Plane, PlaneLn → GrpPerm, PowMap, Map
    • IsCentralCollineation(P, g): Plane, GrpPermElt → BoolElt, PlanePt, PlaneLn
    • Example: Central Collineations of a Plane
  • Transitivity Properties
    • IsPointTransitive(P): Plane → BoolElt
    • IsTransitive(P): Plane → BoolElt
    • IsLineTransitive(P): Plane → BoolElt
    • Example: Transitivity of a Collineation Group
• Translation Planes
  • BaerDerivation(q2): RngIntElt → PlaneAff, PlanePtSet, PlaneLnSet
  • BaerSubplane(P): PlaneProj → PlaneProj, PlanePtSet, PlaneLnSet
  • OvalDerivation(q: parameters): RngIntElt → PlaneAff, PlanePtSet, PlaneLnSet
• Planes and Designs
  • Design(P): Plane → Dsgn, SetIncPt, SetIncBlk
  • FiniteAffinePlane(D): Inc → Plane, PlanePtSet, PlaneLnSet
  • FiniteProjectivePlane(D): Inc → Plane, PlanePtSet, PlaneLnSet
  • Example: Projective Plane from a Singer Difference Set
• Planes, Graphs, and Codes
  • LineGraph(P): Plane → Grph
  • IncidenceGraph(P): Plane → Grph
  • LinearCode(P, K): Plane, FldFin → Code
  • Example: Incidence Graph and Code of a Plane