Introduction

      Planes in Magma

 
Construction of a Plane
      FiniteProjectivePlane< v | X : parameters > : RngIntElt, List -> PlaneProj
      FiniteProjectivePlane(W) : ModTupFld -> PlaneProj
      FiniteAffinePlane< v | X : parameters > : RngIntElt, List -> PlaneAff
      FiniteAffinePlane(W) : ModFld -> PlaneAff
      Example Plane_Constructors (H154E1)

 
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
            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 ! l : PlaneLnSet, PlaneLn -> PlaneLn
            Representative(L) : PlaneLnSet -> PlaneLn
            Random(L) : PlaneLnSet -> PlaneLn
            Example Plane_points-lines (H154E2)

      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 -> .
      Example Plane_supp (H154E3)

 
Subplanes
      sub<P | L> : Plane, List -> Plane
      SubfieldSubplane(P, F) : Plane, FldFin -> Plane, PlanePtSet, PlaneLnSet
      Example Plane_sub (H154E4)

 
Structures Associated with a Plane
      VectorSpace(P) : Plane -> ModTupFld
      Field(P) : Plane -> FldFin
      IncidenceMatrix(P) : Plane -> AlgMatElt
      Dual(P) : Plane -> Plane, PlanePtSet, PlaneLnSet
      Example Plane_sub (H154E5)

 
Numerical Invariants of a Plane
      Order(P) : Plane -> RngIntElt
      NumberOfPoints(P) : Plane -> RngIntElt
      NumberOfLines(P) : Plane -> RngIntElt
      pRank(P) : Plane -> RngIntElt
      pRank(P, p) : Plane, RngIntElt -> RngIntElt
      Example Plane_invar (H154E6)

 
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 Plane_embedding (H154E7)

 
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
            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 ]
            ElementToSequence(l) : PlaneLn -> [ FldFinElt ]
            Set(l) : PlaneLn -> { PlanePt }
            Example Plane_decon (H154E8)

      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 Plane_elt-other (H154E9)

 
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 }
      Knot(P, C) : Plane, { PlanePt } -> PlanePt
      Exterior(P, C) : Plane, { PlanePt } -> { PlanePt }
      Interior(P, C) : Plane, { PlanePt } -> { PlanePt }
      Example Plane_arcs (H154E10)

 
Unitals
      IsUnital(P, U) : Plane, { PlanePt } -> BoolElt
      AllTangents(P, U) : Plane, { PlanePt } -> { PlaneLn }
      UnitalFeet(P, U, p) : Plane, { PlanePt }, PlanePt -> { PlanePt }
      Example Plane_unital (H154E11)

 
The Collineation Group of a Plane

      The Collineation Group Function
            CollineationGroup(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 Plane_CollineationGSet (H154E12)
            Example Plane_Collineation (H154E13)
            Example Plane_baer (H154E14)

      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 Plane_cent-coll (H154E15)

      Transitivity Properties
            IsPointTransitive(P) : Plane -> BoolElt
            IsLineTransitive(P) : Plane -> BoolElt
            Example Plane_trans (H154E16)

 
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 Plane_designs (H154E17)

 
Planes, Graphs and Codes
      LineGraph(P) : Plane -> Grph
      IncidenceGraph(P) : Plane -> Grph
      LinearCode(P, K) : Plane, FldFin -> Code
      Example Plane_codes (H154E18)

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