Projective geometry

A point in projective $n$-space, ${\cal P}^n$, is given by a $(n+1)$-vector of coordinates ${\tt x}=[x_1 \ldots x_{n+1}]^\top$. At least one of these coordinates should differ from zero. These coordinates are called homogeneous coordinates. In the text the coordinate vector and the point itself will be indicated with the same symbol. Two points represented by $(n+1)$-vectors ${\tt x}$ and ${\tt y}$ are equal if and only if there exists a nonzero scalar $\lambda$ such that $x_i = \lambda y_i$, for every $i$ $(1 \leq i \leq n+1)$. This will be indicated by ${\tt x} \sim {\tt y}$.

Often the points with coordinate $x_{n+1}=0$ are said to be at infinity. This is related to the affine space ${\cal A}$. This concept is explained more in detail in section 2.3.

A collineation is a mapping between projective spaces, which preserves collinearity (i.e. collinear points are mapped to collinear points). A collineation from ${\cal P}^m$ to ${\cal P}^n$ is mathematically represented by a $(m+1)\times(n+1)$-matrix ${\bf H}$. Points are transformed linearly: ${\tt x} \mapsto {\tt x}' \sim {\bf H}{\tt x}$. Observe that matrices ${\bf H}$ and $\lambda {\bf H}$ with $\lambda$ a nonzero scalar represent the same collineation.

A projective basis is the extension of a coordinate system to projective geometry. A projective basis is a set of $n+2$ points such that no $n+1$ of them are linearly dependent. The set ${\tt e}_l = [0 \ldots 1 \ldots 0]^\top$ for every $l$ $(1 \leq l \leq n+1)$, where 1 is in the $l$th position and ${\tt e}_{n+2}=[1 1 \ldots 1]^\top$ is the standard projective basis. A projective point of ${\cal P}^n$ can be described as a linear combination of any $n+1$ points of the standard basis. For example:

$${\tt m}=\sum^{n+1}_{l=1} \lambda_l {\tt e}_l$$

It can be shown [44] that any projective basis can be transformed via a uniquely determined collineation into the standard projective basis. Similarly, if two set of points ${\tt m}_1, \ldots , {\tt m}_{n+2}$ and ${\tt m}'_1, \ldots , {\tt m}'_{n+2}$ both form a projective basis, then there exists a uniquely determined collineation ${\bf T}$ such that ${\tt m}'_l \sim {\bf T} {\tt m}_l$ for every $l$ $(1 \leq l \leq n+2)$. This collineation ${\bf T}$ describes the change of projective basis. In particular, ${\bf T}$ is invertible.