# The Point-Set and Line-Set of a Plane ## Introduction An affine or projective plane in Magma consists of three objects: the plane $P$ itself, the *point–set* $V$ of $P$, and the *line–set* $L$ of $P$. Although called the point–set and line–set, $V$ and $L$ are not actual Magma sets. They simply act as the parent structures for the points and lines (respectively) of the plane $P$, enabling easy creation of these objects via the `!` and `.` operators. The point–set $V$ belongs to the Magma category `PlanePtSet`, and the line–set $L$ to the category `PlaneLnSet`. In this section, the functions used to create point–sets, line–sets and the points and lines themselves are described. ## Creating Point-Sets and Line-Sets As mentioned above, the point–set and line–set are returned as the second and third arguments of any function which creates a plane. They can also be created via the following two functions. ### `PointSet(P): Plane -> PlanePtSet` Given a plane $P$, return the point–set $V$ of $P$. ### `LineSet(P): Plane -> PlaneLnSet` Given a plane $P$, return the line–set $L$ of $P$. ## Using the Point-Set and Line-Set to Create Points and Lines For efficiency and clarity, the points and lines of a plane are given special types in Magma. The category names for points and lines are `PlanePt` and `PlaneLn` respectively. They can be created in the following ways. ### `V . i: PlanePtSet, RngIntElt -> PlanePt` Given the point–set $V$ of a plane $P$ and an integer $i$, return the $i$-th point of $P$. ### `V ! [a, b, c]: PlanePtSet, SeqEnum -> PlanePt` Given the point–set $V$ of a classical projective plane $P = PG_2(K)$, and elements $a, b, c$ of the finite field $K$, create the projective point $(a : b : c)$ in the plane $P$. ### `V ! [a, b]: PlanePtSet, SeqEnum -> PlanePt` Given the point–set $V$ of a classical affine plane $P = AG_2(K)$, and elements $a, b$ of the finite field $K$, create the point $(a, b)$ in the plane $P$. ### `V ! x: PlanePtSet, Elt -> PlanePt` Given the point–set $V$ of a plane $P$, return the point of $P$ corresponding to the element $x$, which should be coercible into the underlying point set for $P$. (In the case of classical planes, $x$ should be coercible to a vector.) ### `Representative(V): PlanePtSet -> PlanePt` ### `Rep(V): PlanePtSet -> PlanePt` Given the point–set $V$ of a plane $P$, return a representative point of $P$. ### `Random(V): PlanePtSet -> PlanePt` Given the point–set $V$ of a plane $P$, return a random point of $P$. ### `L . i: PlanePtSet, RngIntElt -> PlanePt` Given the line–set $L$ of a plane $P$ and an integer $i$, return the $i$-th line of $P$. ### `L ! [a, b, c]: PlaneLnSet, SeqEnum -> PlaneLn` Given the line set $L$ of a classical plane $P$ defined over a finite field $K$, and elements $a, b, c$ of $K$, create the line $\left$ (i.e. the line given by the equation $ax + by + cz = 0$ if $P$ is projective, or $ax + by + c = 0$ if $P$ is affine). ### `L ! [m, b]: PlaneLnSet, SeqEnum -> PlaneLn` Given the line set $L$ of a classical affine plane $P = AG_2(K)$, and elements $m, b$ of the finite field $K$, create the affine line $y = mx + b$ in $P$. ### `L ! S: PlaneLnSet, SetEnum -> PlaneLn` ### `L ! S: PlaneLnSet, SeqEnum -> PlaneLn` Given the line–set $L$ of a plane $P$ and a set or sequence $S$ of collinear points of $P$, return the line containing the points of $S$. ### `L ! l: PlaneLnSet, PlaneLn -> PlaneLn` Given the line–set $L$ of a plane $P$ and a line $l$ of a (possibly) different plane (generally a subplane of $P$), return the line of $P$ corresponding to $l$. ### `Representative(L): PlaneLnSet -> PlaneLn` ### `Rep(L): PlaneLnSet -> PlaneLn` Given the line–set $L$ of a plane $P$, return a representative line of $P$. ### `Random(L): PlaneLnSet -> PlaneLn` Given the line–set $L$ of a plane $P$, return a random line of $P$. ### `Example: Points Lines (ex-afb4c8)` The following example shows how points and lines of a plane can be created. First we study a classical projective plane. ```magma > P, V, L := FiniteProjectivePlane(5); > V; Point-set of Projective Plane PG(2, 5) > L; Line-set of Projective Plane PG(2, 5) ``` Create the third point of $P$: ```magma > V.3; ( 0 : 0 : 1 ) ``` Create the point $(1:2:3)$ of $P$: ```magma > V![1, 2, 3]; ( 1 : 2 : 3 ) ``` Choose a random point of $P$: ```magma > Random(V); ( 1 : 0 : 0 ) > Random(V); ( 0 : 0 : 1 ) ``` Create the sixth line of $P$: ```magma > L.6; < 1 : 1 : 3 > ``` Create the line of $P$ given by the equation 4x + 3y + 2z = 0: ```magma > L![4, 3, 2]; < 1 : 2 : 3 > ``` Create the line of $P$ containing the points $(0:0:1)$ and $(0:1:0)$: ```magma > L![ V | [0, 0, 1], [0, 1, 0] ]; < 1 : 0 : 0 > ``` Get a representative from the line-set of $P$, and a random line: ```magma > Rep(L); < 1 : 0 : 0 > > Random(L); < 1 : 2 : 4 > ``` Now we look at a non-classical plane. ```magma > V := {2, 4, 6, 8}; > A, P, L := FiniteAffinePlane< SetToIndexedSet(V) | Setseq(Subsets(V, 2)) >; > A: Maximal; Affine Plane of order 2 Points: {@ 2, 4, 6, 8 @} Lines: {6, 8}, {2, 6}, {2, 8}, {2, 4}, {4, 6}, {4, 8} > P; Point-set of Affine Plane of order 2 > L; Line-set of Affine Plane of order 2 ``` Get the third point of $A$: ```magma > P.3; 6 ``` Create the point of $A$ given by the integer 4: ```magma > P!4; 4 ``` Get a representative from the point–set of $A$: ```magma > Rep(P); 2 ``` Get the third line of $A$: ```magma > L.3; {2, 8} ``` Create the line of $A$ containing the integers 2 and 6: ```magma > L![2, 6]; {2, 6} ``` Choose a random line from $A$: ```magma > Random(L); {6, 8} ``` ## Retrieving the Plane from Points, Lines, Point-Sets and Line-Sets The `ParentPlane` function allows you to access the plane to which a point, line, point–set or line–set belongs. ### `ParentPlane(V): PlanePtSet -> Plane, PlanePtSet, PlaneLnSet` The plane $P$ for which $V$ is the point–set. ### `ParentPlane(L): PlaneLnSet -> Plane, PlanePtSet, PlaneLnSet` The plane $P$ for which $L$ is the line–set. ### `ParentPlane(p): PlanePt -> Plane, PlanePtSet, PlaneLnSet` The plane $P$ for which $p$ is a point. ### `ParentPlane(l): PlaneLn -> Plane, PlanePtSet, PlaneLnSet` The plane $P$ for which $l$ is a line.