Projective frame

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 inK"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 Kⁿ⁺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:$lambda\_0\; v\_0\; +\; lambda\_1\; v\_1\; +\; cdots\; +lambda\_n\; v\_n\; +\; lambda\_\{n+1\}\; v\_\{n+1\}\; =\; 0.$However, because the subsets of "n"+1 vectors are linearly independent, the scalars "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:$mu\_0\; v\_0\; +\; mu\_1\; v\_1\; +\; cdots\; +mu\_n\; v\_n\; +\; mu\_\{n+1\}\; v\_\{n+1\}.$