A projective basis for
is any set of n+2 points of
,
no n+1 of which lie in a hyperplane. Equivalently, the
matrix formed by the column vectors of any n+1 of
the points must have full rank.
It is easily checked that
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
.
Any basis can be
mapped into this standard form by a suitable collineation.
Property: A collineation on
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
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
provides n constraints (n+1 linear equations
defined up a common scale factor), so the required total of n(n+2)constraints is met.