Projective frame
Encyclopedia
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:
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
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, a projective frame is an ordered collection of points in projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
which can be used as reference points to describe any other point in that space. For example:
- Given three distinct points on a projective lineProjective lineIn mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
, any other point can be described by its cross-ratioCross-ratioIn geometry, the cross-ratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line...
with these three points. - In a projective planeProjective planeIn mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
, 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
Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...
of points in general position
General position
In algebraic geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible...
in
KPn. Here general position means that no subset of n+1 of these points lies in a hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
(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
Barycentric coordinates (mathematics)
In geometry, the barycentric coordinate system is a coordinate system in which the location of a point is specified as the center of mass, or barycenter, of masses placed at the vertices of a simplex . Barycentric coordinates are a form of homogeneous coordinates...
: a collection of n+2 scalars μj which sum to zero, such that p is represented by the vector