- Introduction
- Construction of a Plane
- The Point-Set and Line-Set of a Plane
- Introduction
- Creating Point-Sets and Line-Sets
- 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 (H150E2)
- Retrieving the Plane from Points, Lines, Point-Sets and Line-Sets
- The Set of Points and Set of Lines
- The Defining Points of a Plane
- Subplanes
- Structures Associated with a Plane
- Numerical Invariants of a Plane
- Properties of Planes
- Identity and Isomorphism
- The Connection between Projective and Affine Planes
- Operations on Points and Lines
- Elementary Operations
- 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 (H150E8)
- 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 (H150E9)
- 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 (H150E10)
- Unitals
- 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 (H150E12)
- Example Plane_Collineation (H150E13)
- Example Plane_baer (H150E14)
- 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 (H150E15)
- Transitivity Properties
- Translation Planes
- Planes and Designs
- Planes, Graphs and Codes
V2.28, 13 July 2023