# Projective frame

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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 **K***P*^{n} denote *n*-dimensional projective space over an arbitrary field **K**. This is the projectivization of the vector space **K**^{n+1}. Then a projective frame is an (*n*+2)-tuple of points in general position in
**K***P*^{n}. 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 *v*_{0}, *v*_{1}, ..., *v*_{n+1} in **K**^{n+1}. Such a tuple of vectors defines a projective frame if any subset of *n*+1 of these vectors is a basis for **K**^{n+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 *λ*_{j} must all be nonzero. It follows that the representative vectors can be rescaled so that *λ*_{j}=1 for all *j*=0,1,...,*n*+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 **K**^{n+1} and sum to zero. Using such a frame, any point *p* in **K***P*^{n} may be described by a projective version of *barycentric coordinates*: a collection of *n*+2 scalars *μ*_{j} which sum to zero, such that *p* is represented by the vector