Reciprocity (projective geometry)
Encyclopedia
A reciprocity is a collineation
from a projective space
onto its dual space
, taking points to hyperplane
s (and vice versa) and preserving incidence
.
If it can be represented as a homography
, it is called a correlation
.
Collineation
In projective geometry, a collineation is a one-to-one and onto map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. All projective linear transformations induce a collineation...
from a 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....
onto its dual space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
, taking points to 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...
s (and vice versa) and preserving incidence
Incidence (geometry)
In geometry, the relations of incidence are those such as 'lies on' between points and lines , and 'intersects' . That is, they are the binary relations describing how subsets meet...
.
If it can be represented as a homography
Homography
Homography is a concept in the mathematical science of geometry.A homography is an invertible transformation from a projective space to itself that maps straight lines to straight lines...
, it is called a correlation
Correlation (projective geometry)
A correlation is a duality from a projective space to itself...
.