title: Oriented projective geometry
text: # Oriented projective geometry
  
  Oriented projective geometry is an oriented version of real projective geometry.
  
  Whereas the real projective plane describes the set of all unoriented lines through the origin in $R^3,$ the oriented projective plane describes lines with a given orientation. There are applications in computer graphics and computer vision where it is necessary to distinguish between rays light being emitted or absorbed by a point.
  
  Elements in an oriented projective space are defined using signed homogeneous coordinates. Let $\mathbb {R} _{*}^{n}$ be the set of elements of $\mathbb {R} ^{n}$ excluding the origin.
  
  * Oriented projective line, $\mathbb {T} ^{1}$: $(x,w)\in \mathbb {R} _{*}^{2}$, with the equivalence relation $(x,w)\sim (ax,aw)\,$ for all $a>0$.
  * Oriented projective plane, $\mathbb {T} ^{2}$: $(x,y,w)\in \mathbb {R} _{*}^{3}$, with $(x,y,w)\sim (ax,ay,aw)\,$ for all $a>0$.
  
  These spaces can be viewed as extensions of euclidean space. $\mathbb {T} ^{1}$ can be viewed as the union of two copies of $\mathbb {R}$, the sets (x,1) and (x,-1), plus two additional points at infinity, (1,0) and (-1,0). Likewise $\mathbb {T} ^{2}$ can be viewed as two copies of $\mathbb {R} ^{2}$, (x,y,1) and (x,y,-1), plus one copy of $\mathbb {T}$ (x,y,0).
  
  An alternative way to view the spaces is as points on the circle or sphere, given by the points (x,y,w) with
  
      $x^2+y^2+w^2=1.$
  
  ## Oriented real projective space
  
  Let n be a nonnegative integer. The (analytical model of, or canonical) oriented (real) projective space or (canonical) two-sided projective space $\mathbb {T} ^{n}$ is defined as
  
      $\mathbb {T} ^{n}=\{\{\lambda Z:\lambda \in \mathbb {R} _{>0}\}:Z\in \mathbb {R} ^{n+1}\setminus \{0\}\}=\{\mathbb {R} _{>0}Z:Z\in \mathbb {R} ^{n+1}\setminus \{0\}\}.$
  
  Here, we use $\mathbb {T}$ to stand for two-sided.
  
  ### Distance in oriented real projective space
  
  Distances between two points $p=(p_{x},p_{y},p_{w})$ and $q=(q_{x},q_{y},q_{w})$ in $\mathbb {T} ^{2}$ can be defined as elements
  
      $((p_{x}q_{w}-q_{x}p_{w})^{2}+(p_{y}q_{w}-q_{y}p_{w})^{2},\mathrm {sign} (p_{w}q_{w})(p_{w}q_{w})^{2})$
  
  in $\mathbb {T} ^{1}$.
  
  ## Oriented complex projective geometry
  
  Let n be a nonnegative integer. The oriented complex projective space ${\mathbb {CP} }_{S^{1}}^{n}$ is defined as
  
      ${\mathbb {CP} }_{S^{1}}^{n}=\{\{\lambda Z:\lambda \in \mathbb {R} _{>0}\}:Z\in \mathbb {C} ^{n+1}\setminus \{0\}\}=\{\mathbb {R} _{>0}Z:Z\in \mathbb {C} ^{n+1}\setminus \{0\}\}$. Here, we write $S^{1}$ to stand for the 1-sphere.
  