Projective geometry

• The projective plane ${\cal P}^2$ is the set of all pairs $\{{\bf x},{\bf -x}\}$ of antipodal points in ${\cal S}^2$ .
• Two alternative definitions of ${\cal P}^2$ , equivalent to the preceding one are
  1. The set of all lines through the origin in $\mathbb{R}^3$ .
  2. The set of all equivalence classes of ordered triples $(x_1,x_2,x_3)$ of numbers (i.e., vectors in $\mathbb{R}^3$ ) not all zero, where two vectors are equivalent if they are proportional.
• The space ${\cal P}^2$ can be thought of as the infinite plane tangent to the space ${\cal S}^2$ and passing through the point $(0,0,1)^t$ .
  

• Let $\pi : {\cal S}^2 \rightarrow {\cal P}^2$ be the mapping that sends ${\bf x}$ to $\{{\bf x},{\bf -x}\}$ . The $\pi$ is a two-to-one map of ${\cal S}^2$ onto ${\cal P}^2$ .
• A line of ${\cal P}^2$ is a set of the form $\pi {\bf l}$ , where ${\bf l}$ is a line of ${\cal S}^2$ . Clearly, $\pi {\bf x}$ lies on $\pi {\bf l}$ if and only if $\xi^t {\bf x} = 0$ .
• Homogeneous coordinates: In general, points of real $n$ -dimensional projective space, ${\cal P}^n$ ,

  are represented by $n+1$ component column vectors $(x_1,\ldots,x_n,x_{n+1}) \in {\mathbb{R}}^{n+1}$ such that at least one $x_i$ is non-zero and $(x_1,\ldots,x_n,x_{n+1})$ and $(\lambda x_1,\ldots,\lambda x_n,\lambda x_{n+1})$ represent the same point of ${\cal P}^n$ for all $\lambda \neq 0$ .

• $(x_1,\ldots,x_n,x_{n+1})$ is the homogeneous representation of a projective point.