8. Three-dimensional projective space

All the ideas discussed above can be adapted to three-dimensional space. An overview:

(1)

Homogeneous coordinates are $(x,y,z,s)_h$. The names of each point differ by nonzero scalar factors. $pt (x,y,z,s)$ is an ordinary point if $s \neq 0$ and is a point at infinity if $s = 0$.

(2)

Each point at infinity corresponds to a family of parallel lines in R$^3$. These lines meet in P$_3$ at their point at infinity.

(In R$^3$, two lines are parallel if they lie in the same plane and do not intersect. Two lines that do not intersect and do not lie in the same plane are said to be skew.)

(3)

Real projective 3-space P$_3$ consists of R$^3$ together with points at infinity.

(4)

A projective transformation $T:$   P$_3 \rightarrow$   P$_3$ is a transformation obtained by multiplying the homogeneous coordinates of each point by a nonsingular $4 \times 4$ matrix $A$.

(5)

Five points in P$_3$ are said to be in general position if no four are coplanar. If $P_1,\dots P_5$ are in general position and $Q_1,\dots Q_5$ are in general position, then there exists a projective transformation $T:$   P$_3 \rightarrow$   P$_3$ such that $T(P_{1}) = Q_1,\dots, T(P_{5}) = Q_5$.