Triangle →
Trilateral
Graph of Function → Mapping
Diagram
A range of
points on a line → A pencil of lines on a
point
Duality Exercise ¤
I say “point”;
you say
“line”.
I say “line”;
you say “point”.
I say “intersection”;
you say “join”.
I say “join”;
you say “intersection”.
Principle of Duality in
Projective Planar Geometry (PPG, also $RP^2$) ¤
Suppose
S is a statement in $RP^2$ and that,
by applying the appropriate dual changes to the words
and concepts of S,
S’ is the corresponding dual statement in $RP^2$.
Principle
of Duality: S is a theorem of
$RP^2$ if and only if S’ is a theorem of $RP^2$.
Application of
Duality in $RP^2$ ¤
S:
Two points, P and Q, determine a unique line, the join
of the points P and Q.
S’: Two lines, p and q, determine a unique point, the
intersection of the lines p and q.
Comments:
• When lines are parallel in Euclidean
Geometry they meet in $RP^2$ at a unique point at
infinity.
• Two distinct points at infinity
determine a unique line, the line at infinity ( the
“horizon” line).
So...¤.
parallel
lines meet on the horizon line.
Duality and
Linear Functions ¤
Suppose
$f: R → R$ is defined by
$f(x) = mx+b$.
The
graph of $f$,
$l_f = \{ (x,y): y =
f(x) , x ∈ R\}$
is a
unique line in the euclidean plane, which we can
consider a line in $RP^2$.
A
point $P$ in Euclidean geometry lies on $l_f$ if
and only if
for some $a ∈ R$, $P$ is the point of
intersection of two lines
$\{(a,y); y ∈ R\}$ and $\{(x,f(a): x ∈ R\}$.
Consider
$l_f$ as a line in $RP^2$.
Then by the
principle of duality, there is a unique point,
$L_f$, in $RP^2$ with dual properties:
$L_f$ has a
(distinguished) pencil of lines passing through it
determined by the function $f$.
A line $p$
passes through $L_f$, if and only if
for some $a ∈
R$, $p$ is the join of two points:
$A$, a point
determined on a line (the $X$ axis) corresponding to the
number $a$, and
$B=f(A)$, a
point determined on a line (the $Y$ axis) corresponding
to the number $f(a)$.
We will call
the point $L_f$ the “focus point” for the linear
function $f$.¤