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 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=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

Famous quotes containing the word frame:

    A set of ideas, a point of view, a frame of reference is in space only an intersection, the state of affairs at some given moment in the consciousness of one man or many men, but in time it has evolving form, virtually organic extension. In time ideas can be thought of as sprouting, growing, maturing, bringing forth seed and dying like plants.
    John Dos Passos (1896–1970)