# Operations on Points and Lines All the usual equality, membership and subset functions are provided along with a collection of deconstruction functions and others. ## Elementary Operations ### `p eq q: PlanePt, PlanePt -> BoolElt` Returns `true` if the points $p$ and $q$ are equal, otherwise `false`. ### `p ne q: PlanePt, PlanePt -> BoolElt` Return `true` if the points $p$ and $q$ are not equal, otherwise `false`. ### `l eq m: PlaneLn, PlaneLn -> BoolElt` Return `true` if the lines $l$ and $m$ are equal, otherwise `false`. ### `l ne m: PlaneLn, PlaneLn -> BoolElt` Return `true` if the lines $l$ and $m$ are not equal, otherwise `false`. ### `p in l: PlanePt, PlaneLn -> BoolElt` Return `true` if point $p$ lies on the line $l$, otherwise `false`. ### `p notin l: PlanePt, PlaneLn -> BoolElt` Return `true` if point $p$ does not lie on the line $l$, otherwise `false`. ### `S subset l: { PlanePt}, PlaneLn -> BoolElt` Given a subset $S$ of the point set of the plane $P$ and a line $l$ of $P$, return `true` if the subset $S$ of points lies on the line $l$, otherwise `false`. ### `S notsubset l: { PlanePt}, PlaneLn -> BoolElt` Given a subset $S$ of the point set of the plane $P$ and a line $l$ of $P$, return `true` if the subset $S$ of points does not lie on the line $l$, otherwise `false`. ### `l meet m: PlaneLn, PlaneLn -> PlanePt` The unique point common to the lines $l$ and $m$. ### `Representative(l): PlaneLn -> PlanePt` ### `Rep(l): PlaneLn -> PlanePt` Given a line $l$ of the plane $P$, return a representative point of $P$ which is incident with $l$. ### `Random(l): PlaneLn -> PlanePt` Given a line $l$ of the plane $P$, return a random point of $P$ which is incident with $l$. ## Deconstruction Functions ### `Index(P, p): Plane, PlanePt -> RngIntElt` Given a point $p$ from the point–set $V$ of a plane $P$, return the index of $p$, i.e. the integer $i$ such that $p$ is `V.i`. ### `Index(P, l): Plane, PlaneLn -> RngIntElt` Given a line $l$, return the index of $l$ in the plane $P$, i.e. the integer $i$ such that $l$ is $L.i$ (where $L$ is the line–set of $P$). ### `p[i]: PlanePt, RngIntElt -> FldFinElt` The $i$-th coordinate of the point $p$, which must be from a classical plane. If $p$ is from a projective plane, then $i$ must satisfy $1 \le i \le 3$; if $p$ is from an affine plane, then $i$ must satisfy $1 \le i \le 2$. ### `l[i]: PlaneLn, RngIntElt -> FldFinElt` The $i$-th coordinate of the line $l$, which must be from a classical plane. The integer $i$ must satisfy $1 \le i \le 3$. Recall that in a classical plane $\left$ (where $a, b, c \in K$) represents the line given by the equation $ax + by + cz = 0$ in a projective plane or $ax + by + c = 0$ in an affine plane. ### `Coordinates(P, p): Plane, PlanePt -> [ FldFinElt ]` Given a point $p = (a:b:c)$ from a classical projective plane $P$ (or $p = (a,b)$ from a classical affine plane $P$), return the sequence $[a, b, c]$ (or $[a, b]$ in the affine case) of coordinates of $p$. ### `Coordinates(P, l): Plane, PlaneLn -> [ FldFinElt ]` Given a line $l = \left$ from a classical plane $P$ (projective or affine), return the sequence $[a, b, c]$ of coordinates of $l$. ### `ElementToSequence(p): PlanePt -> [ FldFinElt ]` ### `Eltseq(p): PlanePt -> [ FldFinElt ]` Given a point $p = (a:b:c)$ from a classical projective plane $P$ (or $p = (a,b)$ from a classical affine plane $P$), return the sequence $[a, b, c]$ (or $[a, b]$ in the affine case) of coordinates of $p$. ### `ElementToSequence(l): PlaneLn -> [ FldFinElt ]` ### `Eltseq(l): PlaneLn -> [ FldFinElt ]` Given a line $l = \left$ from a classical plane $P$ (projective or affine), return the sequence $[a, b, c]$ of coordinates of $l$. ### `Set(l): PlaneLn -> { PlanePt }` The set of points contained in the line $l$. ### `Example: decon (ex-b3b9f2)` The following example illustrates the use of some of the elementary and deconstruction functions on lines and points discussed in the previous two subsections. ```magma > K := GF(4); > P, V, L := FiniteProjectivePlane(K); ``` Create the line $x + z = 0$: ```magma > l := L![1, 0, 1]; > l; < 1 : 0 : 1 > ``` Look at the points on the line $l$: ```magma > Set(l); { ( 0 : 1 : 0 ), ( 1 : w^2 : 1 ), ( 1 : 0 : 1 ), ( 1 : w : 1 ), ( 1 : 1 : 1) } ``` Get the coordinates of the line $l$: ```magma > Coordinates(P, l); [ 1, 0, 1 ] > l[1]; 1 ``` Find the index of the line $l$ in the line–set $L$ of $P$, and check it: ```magma > Index(P, l); 8 > l eq L.8; true ``` Test if a point is on the line $l$: ```magma > V![1, 0, 1] in l; true ``` Test a set of points for containment in $l$: ```magma > S := {V.1, V.2}; > S; { ( 1 : 0 : 0 ), ( 0 : 1 : 0 ) } > S subset l; false ``` Create the line containing the points in $S$: ```magma > l2 := L!S; > l2; < 0 : 0 : 1 > > S subset l2; true ``` And finally, find the point common to the lines $l$ and $l2$: ```magma > p := l meet l2; > p; ( 0 : 1 : 0 ) > p[3]; 0 ``` ## Other Point and Line Functions ### `IsCollinear(P, S): Plane, { PlanePt} -> BoolElt, PlaneLn` Return `true` if the set $S$ of points of the plane $P$ are collinear, otherwise `false`. If the points are collinear, the line which they define is also returned. ### `IsConcurrent(P, R): Plane, { PlaneLn} -> BoolElt, PlanePt` Return `true` if the set $R$ of lines of the plane $P$ are concurrent, otherwise `false`. If the lines are concurrent, their common point is returned as a second value. ### `ContainsQuadrangle(P, S): Plane, { PlanePt } -> BoolElt` Return `true` if the set $S$ of points of a plane $P$ contains a quadrangle. ### `Pencil(P, p): Plane, PlanePt -> { PlaneLn }` The pencil of lines passing through the point $p$ in the plane $P$. ### `Slope(l): PlaneLn -> FldFinElt` The slope of the line $l$ of a classical affine plane $P$. ### `IsParallel(P, l, m): Plane, PlaneLn, PlaneLn -> BoolElt` Return `true` if the line $l$ is parallel to the line $m$ in the affine plane $P$. ### `ParallelClass(P, l): Plane, PlaneLn -> { PlaneLn }` The parallel class containing the line $l$ of an affine plane $P$. ### `ParallelClasses(P): PlaneAff -> { { PlaneLn } }` The partition into parallel classes of the lines of the affine plane $P$. ### `Example: Elt Other (ex-c0212f)` We use the affine plane $AG_2(3)$ to demonstrate some of the above functions. ```magma > A, V, L := FiniteAffinePlane(3); ``` Create the line $y = 2x + 1$ in A, and check its slope: ```magma > l := L![2, 1]; > l; < 1 : 1 : 2 > > Slope(l); 2 ``` Find the lines parallel to $l$: ```magma > ParallelClass(l); { < 1 : 1 : 0 >, < 1 : 1 : 1 >, < 1 : 1 : 2 > } > [Slope(m): m in ParallelClass(l)]; [ 2, 2, 2 ] ``` Get the pencil of lines through a point of $l$: ```magma > p := Rep(l); > p; ( 1, 0 ) > Pencil(A, p); { < 1 : 0 : 2 >, < 1 : 1 : 2 >, < 1 : 2 : 2 >, < 0 : 1 : 0 > } ```