## Dec. 2, 2013

Martin Flashman
¤
Humboldt State University
Arcata, CA 95521
flashman@humboldt.edu
flashman.neocities.org

#### Example: LF.DA.0  with GeoGebra .

For more on linearity [composition and inverses] go to http://flashman.neocities.org/MD/section-2.1LF.html.

Duality: A pairing of words, concepts and figures.¤
Words:
Point → Line
Line → Point

Intersection → Join
Join → Intersection

Concepts and Figures:
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.

•    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).

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 $f$.¤

• ##### The Focus Point for a Linear Function and Duality ¤

Example: LF.DFP  with GeoGbra

##### Visualizing functions $f: RP^1 → RP^1$ ¤ Many functions $f: R → R$  extend to give functions from $RP^1$ to $RP^1$ . We can visualize these in $RP^2$ by treating the infinite values appropriately. Examples: 1. $f(x) = 4: f(∞) = 4$ (?). 2. $f(x) = x + 4: f(∞) = ∞$. 3. $f(x) = x^2 : f(∞) = ∞$. 4. $f(x) = \frac 1 x : f(0)=∞. f(∞) = 0$.

But in fact $RP^1$ can be visualized as a circle. ¤ This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com 3 December 2013, Created with GeoGebra

Using this we have two more ways to visualize a function $f: RP^1 → RP^1$:
¤

The graph of  $f: RP^1 → RP^1$ is a curve on  $RP^1 × RP^1$- the torus!
Example: Graph of $f$ on the Torus.

The mapping diagram of $f$, a “surface” with boundaries a circle and a subset of a circle.¤
Examples: The Mapping Diagram of $f: RP^1 → RP^1$.

Bonus: A 3D Mapping Diagram for a Linear Function of 2 Variable ¤