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 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

\lambda_0 v_0 + \lambda_1 v_1 + \cdots +\lambda_n v_n + \lambda_{n+1} v_{n+1} = 0.

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=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 Kn+1 and sum to zero. Using such a frame, any point p in KPn 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

\mu_0 v_0 + \mu_1 v_1 + \cdots +\mu_n v_n + \mu_{n+1} v_{n+1}.
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