The Real Projective Plane $RP^2$


  • Consider $R^3 = \{ (x,y,z) : x,y,z \in R\} $ as a vector space over $R$.
  • Definition:  $ v, w \in R^3 - {(0,0,0)}$ , $v$ ~ $w$ iff there is $c \ne 0$ where $w = cv$.
  • FACT:  ~ is an equivalence relation.
  • Notation:  $[v]$ ~ $=\{ w: w$ ~ $v\} =$ < $ x,y,z$ >  for $v =(x,y,z) \in R^3$.
  • Definition: The real projective plane: $RP^2=\{[v]$~ $ :v \in R^3\ -\{(0,0,0)\} $}
  • N.B.  The Euclidean plane in $RP^2$: $R^2$ "=" {< $ x,y,1$ >  $: x,y \in R\}$ 
  • An ideal point in $RP^2$: < $x,y,0$ > $\in RP^2$
  • The ideal points in the real projective plane: $RP^2 - R^2 = $ {< $x,y,0$ >  $: x,y \in R\} $

    • The Euclidean Plane $R^2$ of $RP^2$ in $R^3$
    Ideal Points of $RP^2$  in $R^3$