Affine stratum

The next stratum is the affine one. In the hierarchy of groups it is located in between the projective and the metric group. This stratum contains more structure than the projective one, but less than the metric or the Euclidean strata. Affine geometry differs from projective geometry by identifying a special plane, called the plane at infinity.

This plane is usually defined by $W=0$ and thus ${\tt\Pi}_\infty = [0 \,0 \,0 \,1]^\top$. The projective space can be seen as containing the affine space under the mapping ${\cal A}^3 \rightarrow {\cal P}^3: [X \, Y \, Z]^\top \mapsto [X \, Y\, Z\, 1]^\top$. This is a one-to-one mapping. The plane $W=0$ in ${\cal P}^3$ can be seen as containing the limit points for $\Vert {\tt M}\Vert \rightarrow \infty$, since these points are $[ \frac{X}{\Vert{\tt M}\Vert} \, \frac{Y}{\Vert{\tt M}\Vert} \,\frac{Z}{\Vert{\... ...\frac{1}{\Vert{\tt M}\Vert}]^\top \sim [X_\infty \, Y_\infty \, Z_\infty \, 0 ]$. This plane is therefore called the plane at infinity ${\tt\Pi }_\infty$. Strictly speaking, this plane is not part of the affine space, the points contained in it can't be expressed through the usual non-homogeneous 3-vector coordinate notation used for affine, metric and Euclidean 3D space.

An affine transformation is usually presented as follows:

$$\left[\begin{array}{c} X' \\ Y' \\ Z' \end{array}\right] = \... ...a_{34} \end{array} \right] \mbox{ with } \det(a_{ij}) \neq 0$$

Using homogeneous coordinates, this can be rewritten as follows ${\tt M}' \sim {\bf T}_{A} {\tt M}$ with

$${\bf T}_{A} \sim \left[\begin{array}{cccc} a_{11} & a_{12} ... ..._{33} & a_{34} \\ 0 & 0 & 0 & 1\end{array} \right] \enspace .$$ (B23)

An affine transformation counts 12 independent degrees of freedom. It can easily be verified that this transformation leaves the plane at infinity ${\tt\Pi }_\infty$ unchanged (i.e. ${\tt\Pi}_\infty \sim {\bf T}_A^{-\top}{\tt\Pi}_\infty$ or ${\bf T}_A^\top {\tt\Pi}_\infty \sim {\tt\Pi}_\infty$). Note, however, that the position of points in the plane at infinity can change under an affine transformation, but that all these points stay within the plane ${\tt\Pi }_\infty$.

All projective properties are a fortiori affine properties. For the (more restrictive) affine group parallelism is added as a new invariant property. Lines or planes having their intersection in the plane at infinity are called parallel. A new invariant property for this group is the ratio of lengths along a certain direction. Note that this is equivalent to a cross-ratio with one of the points at infinity.