|
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 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 : 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 : ==References== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Projective frame」の詳細全文を読む スポンサード リンク
|