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Consider non-zero elements of a real vector space \(E\simeq\RR^n\), equipped with the colinearity equivalence relation:
\[x \sim y \iff \exists \lambda \in \RR^\star,\quad x = \lambda y\]The projective space \(\PS(E)\) is the quotient space for \(\sim\). In other words, \(\PS(E)\) is the space of vector lines of \(E\). The dimension of the projective space \(\PS(E)\) is \(n-1\). In particular:
Generally, we will denote \(\PS^n=\PS\block{\RR^{n+1}}\).
Consider an affine hyperplane \(H\) of \(E\) that does not contain the origin. Then, any vector line of \(E\) that is not parallel to \(H\) (i.e. vector lines of \(\vec{H}\)) can be uniquely associated with a point of \(H\): the point where the line intersects \(H\). The remaining vector lines are parallel to \(H\) and are elements of \(\PS\block{\vec{H}}\), by definition. We obtain the following decomposition of projective spaces:
\[\PS\block{E} = H \cup \PS\block{\vec{H}}\]Equivalently, up to isomorphism:
\[\PS^n \simeq \RR^n \cup \PS^{n-1}\]For \(E\simeq\RR^n\), consider the affine hyperplane where the last coordinate is \(1\):
\[H=\{x \in E, \ e_n^Tx = 1\}\]where \(e_n = (0, \dots, 1)^T\) is the \(n\)-th canonical basis vector. In particular, the direction space \(\vec{H} = \{x\in E,\ x_n = 0\}\) is called the points at infinity, and any vector line not in \(\vec{H}\) corresponds to a unique point of \(H\), obtained by dividing by the last coordinate.
Given \(n\) linearly independent lines in \(E=\RR^n\), one can extract a basis \(\block{e_i}_i\) of \(E\) by picking an arbitrary non-zero vector on each line. Given this basis, one can reconstruct any vector from its coordinates, and therefore any vector line. However, it is important that the reconstructed line does not depend on which basis vector was chosen to represent each line. For instance, given a line \(\lambda v\), one may choose \(v\) or \(2v\) as a representant, which would change the reconstructed vector (assuming fixed coordinates), thus affecting the reconstructed line as well. In other words, a system of \(n\) lines is insufficient to reconstruct any other given line: we need more information.
Formally, given coordinates \(x_i\) and a basis \(\block{e_i}_i\), a reconstructed vector is \(x = \sum_i x_i \lambda_i e_i\) and we would like to make sure that changing any \(\lambda_i\) produces a result colinear with \(x\):
\[\sum_i x_i \lambda_i e_i \sim \sum_i x_i e_i\]In block notation, if we assemble the \(\block{\lambda_i}_i\) into a diagonal matrix \(L=\diag\block{\lambda_i}\) and the basis vectors into a matrix \(B\), we require that there exists some non-zero \(\lambda\) such that, for all non-zero \(x\):
\[BLx = \lambda Bx\]Since the above must hold for all non-zero \(x\) and \(B\) is invertible, the only solution is to have all \(\lambda_i\) equal to \(\lambda\). It can be checked that the condition:
\[\forall x \neq 0 \quad Lx = \lambda x\]is equivalent to:
\[L\ones = \lambda \ones\]and it suffices to enforce our colinearity condition on \(B\ones\). Therefore, we need to somehow store the direction of \(B\ones\) alongside \(BL\) in order to “calibrate” \(L\). Conversely, given any set of \(n\) lines \(BL\) and a direction vector \(d\) for \(B\ones\), one can recalibrate matrix \(L\) as above to enforce the colinearity condition. It is then possible to recover any vector line from its \(\block{x_i}_i\), called the projective coordinates.
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