Projective Bases

A projective basis for ${I\!\!P}^{n}$ is any set of n+2 points of ${I\!\!P}^{n}$, no n+1 of which lie in a hyperplane. Equivalently, the $(n+1)\times(n+1)$ matrix formed by the column vectors of any n+1 of the points must have full rank.

It is easily checked that $\{(1, 0,\ldots, 0)^{\top}, (0, 1, 0,\ldots, 0)^{\top}, \ldots,(0,\ldots, 0, 1)^{\top},(1,\ldots, 1)^{\top}\}$forms a basis, called the canonical basis. It contains the points at infinity along each of the n coordinate axes, the origin, and the unit point $(1,\ldots, 1)^{\top}$. Any basis can be mapped into this standard form by a suitable collineation.

Property: A collineation on ${I\!\!P}^{n}$ is defined entirely by its action on the points of a basis.

A full proof can be found in [23]. We will just check that there are the right number of constraints to uniquely characterize the collineation. This is described by an $(n+1)\times(n+1)$ matrix A, defined up to an overall scale factor, so it has (n + 1)²-1= n(n+2) degrees of freedom. Each of the n+2 basis point images $A b_{i}\sim b'_{i}$ provides n constraints (n+1 linear equations defined up a common scale factor), so the required total of n(n+2)constraints is met.

Exercise 2.3   : Consider three non-aligned points a_i in the plane, and their barycenter g. Check that in homogeneous coordinates (x,y,1), we have

$$g\sim\sum_{i=1}^{3} a_{i}$$

An analogous relation holds for the unit point in the canonical basis.