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 (H141E1)
The Point-Set and Line-Set of a Plane
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 (H141E2)
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 (H141E3)
Subplanes
sub<P | L> : Plane, List -> Plane
SubfieldSubplane(P, F) : Plane, FldFin -> Plane, PlanePtSet, PlaneLnSet
Example Plane_sub (H141E4)
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 (H141E5)
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
Example Plane_invar (H141E6)
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 (H141E7)
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) : PlanePt -> RngIntElt
Index(P, l) : 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 (H141E8)
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 (H141E9)
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 (H141E10)
Unitals
IsUnital(P, U) : Plane, { PlanePt } -> BoolElt
AllTangents(P, U) : Plane, { PlanePt } -> { PlaneLn }
UnitalFeet(P, U, p) : Plane, { PlanePt }, PlanePt -> { PlanePt }
Example Plane_unital (H141E11)
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, 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 (H141E12)
Example Plane_Collineation (H141E13)
Example Plane_baer (H141E14)
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 (H141E15)
Transitivity Properties
IsPointTransitive(P) : Plane -> BoolElt
IsLineTransitive(P) : Plane -> BoolElt
Example Plane_trans (H141E16)
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 (H141E17)
Planes, Graphs and Codes
LineGraph(P) : Plane -> Grph
IncidenceGraph(P) : Plane -> Grph
LinearCode(P, K) : Plane, FldFin -> Code
Example Plane_codes (H141E18)
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Wed Apr 24 15:09:57 EST 2013