In the mathematical
field of projective geometry
, a projective frame
is an ordered collection of points in projective space
which can be used as reference points to describe any other point in that space. For example:
- Given three distinct points on a projective line, any other point can be described by its cross-ratio with these three points.
- In a projective plane, a projective frame consists of four points, no three of which lie on a projective line.
In general, let KPn denote n-dimensional projective space over an arbitrary field K. This is the projectivization of the vector space Kn+1. Then a projective frame is an (n+2)-tuple of points in general position in
KPn. Here general position means that no subset of n+1 of these points lies in a hyperplane (a projective subspace of dimension n−1).
Sometimes it is convenient to describe a projective frame by n+2 representative vectors v0, v1, ..., vn+1 in Kn+1. Such a tuple of vectors defines a projective frame if any subset of n+1 of these vectors is a basis for Kn+1. The full set of n+2 vectors must satisfy linear dependence relation
However, because the subsets of n
+1 vectors are linearly independent, the scalars λj
must all be nonzero. It follows that the representative vectors can be rescaled so that λj
=1 for all j
+1. This fixes the representative vectors up to an overall scalar multiple. Hence a projective frame is sometimes defined to be a (n
+ 2)-tuple of vectors which span Kn+1
and sum to zero. Using such a frame, any point p
may be described by a projective version of barycentric coordinates
: a collection of n
+2 scalars μj
which sum to zero, such that p
is represented by the vector