Spherical geometry

Before we introduce projective geometry let us briefly consider spherical geometry ( ${\cal S}^2$ ), which is the geometry on the surface of a sphere.

• The space ${\cal S}^2$ :
  $${\cal S}^2 = \left\{ {\bf x} \in \mathbb{R}^3 : \vert\vert x \vert\vert = 1 \right\}$$

• lines in ${\cal S}^2$ : If one begins at a point in ${\cal S}^2$ and travels straight ahead on the surface, one will trace out a great circle. Viewed as a set in $\mathbb{R}^3$ this is the intersection of ${\cal S}^2$ with a plane through the origin. We will call this great circle a line in ${\cal S}^2$ :
  Let $\xi$ be a unit vector. Then,
  $${\bf l} = \left\{ {\bf x} \in {\cal S}^2 : \xi^t {\bf x} = 0\right\}$$
  is the line with pole $\xi$ .
  

• Two points ${\bf p}$ and ${\bf q}$ are antipodal if ${\bf p} = -{\bf q}$ .
• Lines in ${\cal S}^2$ cannot be parallel. Any two lines intersect at a pair of antipodal points.
• A point on a line:
  $${\bf l} \cdotp {\bf x} = 0$$    or $${\bf l}^T {\bf x} = 0$$    or $${\bf x}^T {\bf l} = 0$$

• Two points define a line:
  $${\bf l} = {\bf p} \times {\bf q}$$

• Two lines define a point:
  $${\bf x} = {\bf l} \times {\bf m}$$