Duality in Projective Geometry



	Duality in Projective Geometry In Projective Geometry, there is a dual notion between pointsand lines in the plane. That is, the space of all lines in the projective plane, ${\bf P^{\ast}}$ ,is a space satisfying all the axioms required of a projective plane. Moreover the two spaces are exactly equivalent, in that if the term ``line'' isexchanged with ``point'' and ``collinear'' with ``coincident'' and ``intersection'' with ``join'' ect. then there is no way to tell a difference between thegeometry we started with and the geometry afterwards. Duality is usually denoted with an $\ast$ used as a superscript. For example, we defined $\rho(x,y)$ to be an operation on pairs of collinear triples of points in the plane,that maps each of these pairs to a line according to Pappus' Theorem. Thuswe can define $\rho^{\ast}$ to be its dual. Then $\rho^{\ast}(x,y)$ is an operation on pairs of coincident triples of lines in the plane, that map each of these pairs to a point according to the dual to Pappus' theorem. Let l₁, l₂, and l₃be three distinct coincident lines and m₁, m₂,and m₃ be three more distinct coincident lines incidentat a different point. Then we define n₁ to be the uniqueline containing the two points $l_{2} \cap n_{3}$ and $l_{3} \cap n_{2}$ .Similarly we can define $n_{2}=\overline{(l_{1} \cap n_{3}) (l_{3} \cap n_{1})}$ and $n_{3}=\overline{(l_{1} \cap n_{2}) (l_{2} \cap n_{1})}$ .The dual to Pappus' Theorem states that these three lines are coincident. The applet below demonstrates the dual to Pappus' theorem. alt="Yourbrowser understands the <APPLET> tag but isn't running the applet, forsome reason." Your browser is completely ignoring the <APPLET> tag!Get Netscape. You can move the blue and the green points which in turnmove the blue or the green lines that these points are incident on. Thered lines indicate the lines constructed by the dual to Pappus' theorem and the point represents $\rho^{\ast}(x,y)$ . Next Step: Observation Two

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