In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a duality. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry.
Principle of duality A projective plane may be defined axiomatically as an incidence structure, in terms of a set of points, a set of lines, and an incidence relation that determines which points lie on which lines. These sets can be used to define a plane dual structure.
Interchange the role of "points" and "lines" in
to obtain the dual structure , where is the converse relation of . is also a projective plane, called the dual plane of .
If and are isomorphic, then is called self-dual. The projective planes for any field (or, more generally, for every division ring (skewfield) isomorphic to its dual) are self-dual. In particular, Desarguesian planes of finite order are always self-dual. However, there are non-Desarguesian planes which are not self-dual, such as the Hall planes and some that are, such as the Hughes planes.
In a projective plane a statement involving points, lines and incidence between them that is obtained from another such statement by interchanging the words "point" and "line" and making whatever grammatical adjustments that are necessary, is called the plane dual statement of the first. The plane dual statement of "Two points are on a unique line" is "Two lines meet at a unique point". Forming the plane dual of a statement is known as dualizing the statement.
If a statement is true in a projective plane , then the plane dual of that statement must be true in the dual plane . This follows since dualizing each statement in the proof "in " gives a corresponding statement of the proof "in ".
The principle of plane duality says that dualizing any theorem in a self-dual projective plane produces another theorem valid in .
The above concepts can be generalized to talk about space duality, where the terms "points" and "planes" are interchanged (and lines remain lines). This leads to the principle of space duality.
These principles provide a good reason for preferring to use a "symmetric" term for the incidence relation. Thus instead of saying "a point lies on a line" one should say "a point is incident with a line" since dualizing the latter only involves interchanging point and line ("a line is incident with a point").
The validity of the principle of plane duality follows from the axiomatic definition of a projective plane. The three axioms of this definition can be written so that they are self-dual statements implying that the dual of a projective plane is also a projective plane. The dual of a true statement in a projective plane is therefore a true statement in the dual projective plane and the implication is that for self-dual planes, the dual of a true statement in that plane is also a true statement in that plane.
Dual theorems As the real projective plane, , is self-dual there are a number of pairs of well known results that are duals of each other. Some of these are:
Desargues' theorem ⇔ Converse of Desargues' theorem Pascal's theorem ⇔ Brianchon's theorem Menelaus' theorem ⇔ Ceva's theorem
Dual configurations
Not only statements, but also systems of points and lines can be dualized.
A set of points and lines is called an configuration if of the lines pass through each point and of the points lie on each line. The dual of an configuration, is an configuration. Thus, the dual of a quadrangle, a (43, 62) configuration of four points and six lines, is a quadrilateral, a (62, 43) configuration of six points and four lines.
The set of all points on a line, called a projective range, has as its dual a pencil of lines, the set of all lines on a point, in two dimensions, or a pencil of hyperplanes in higher dimensions. A line segment on a projective line has as its dual the shape swept out by these lines or hyperplanes, a double wedge..
Duality as a mapping
Plane dualities
A plane duality is a map from a projective plane to its dual plane (see above) which preserves incidence. That is, a plane duality will map points to lines and lines to points ( and ) in such a way that if a point is on a line (denoted by ) then . A plane duality which is an isomorphism is called a correlation. The existence of a correlation means that the projective plane is self-dual.
The projective plane in this definition need not be a Desarguesian plane. However, if it is, that is, with a division ring (skewfield), then a duality, as defined below for general projective spaces, gives a plane duality on that satisfies the above definition.
In general projective spaces
A duality of a projective space is a permutation of the subspaces of (also denoted by with a field (or more generally a skewfield (division ring)) that reverses inclusion,Some authors use the term "correlation" for duality, while others, as shall we, use correlation for a certain type of duality. that is: implies for all subspaces of . Dembowski uses the term "correlation" for duality. Consequently, a duality interchanges objects of dimension with objects of dimension ( = codimension ). That is, in a projective space of dimension , the points (dimension 0) correspond to hyperplanes (codimension 1), the lines joining two points (dimension 1) correspond to the intersection of two hyperplanes (codimension 2), and so on.
Classification of dualities The dual of a finite-dimensional (right) vector space over a skewfield can be regarded as a (right) vector space of the same dimension over the opposite skewfield . There is thus an inclusion-reversing bijection between the projective spaces and . If and are isomorphic then there exists a duality on . Conversely, if admits a duality for , then and are isomorphic.
Let be a duality of for . If is composed with the natural isomorphism between and , the composition is an incidence preserving bijection between and . By the Fundamental theorem of projective geometry is induced by a semilinear map with associated isomorphism , which can be viewed as an antiautomorphism of . In the classical literature, would be called a reciprocity in general, and if it would be called a correlation (and would necessarily be a field). Some authors suppress the role of the natural isomorphism and call a duality.for instance When this is done, a duality may be thought of as a collineation between a pair of specially related projective spaces and called a reciprocity. If this collineation is a projectivity then it is called a correlation.
Let denote the linear functional of associated with the vector in . Define the form by:
is a nondegenerate sesquilinear form with companion antiautomorphism .
Any duality of for is induced by a nondegenerate sesquilinear form on the underlying vector space (with a companion antiautomorphism) and conversely.
Homogeneous coordinate formulation
Homogeneous coordinates may be used to give an algebraic description of dualities. To simplify this discussion we shall assume that is a field, but everything can be done in the same way when is a skewfield as long as attention is paid to the fact that multiplication need not be a commutative operation.
The points of can be taken to be the nonzero vectors in the ()-dimensional vector space over , where we identify two vectors which differ by a scalar factor. Another way to put it is that the points of -dimensional projective space are the 1-dimensional vector subspaces, which may be visualized as the lines through the origin in .Dimension is being used here in two different senses. When referring to a projective space, the term is used in the common geometric way where lines are 1-dimensional and planes are 2-dimensional objects. However, when applied to a vector space, dimension means the number of vectors in a basis, and a basis for a vector subspace, thought of as a line, has two vectors in it, while a basis for a vector space, thought of as a plane, has three vectors in it. If the meaning is not clear from the context, the terms projective or geometric are applied to the projective space concept while algebraic or vector are applied to the vector space one. The relation between the two is simply: algebraic dimension = geometric dimension + 1. Also the - (vector) dimensional subspaces of represent the ()- (geometric) dimensional hyperplanes of projective -space over , i.e., .
A nonzero vector in also determines an - geometric dimensional subspace (hyperplane) , by . When a vector is used to define a hyperplane in this way it shall be denoted by , while if it is designating a point we will use . They are referred to as point coordinates or hyperplane coordinates respectively (in the important two-dimensional case, hyperplane coordinates are called line coordinates). Some authors distinguish how a vector is to be interpreted by writing hyperplane coordinates as horizontal (row) vectors while point coordinates are written as vertical (column) vectors. Thus, if is a column vector we would have while . In terms of the usual dot product, . Since is a field, the dot product is symmetrical, meaning .
A fundamental example A simple reciprocity (actually a correlation) can be given by between points and hyperplanes. This extends to a reciprocity between the line generated by two points and the intersection of two such hyperplanes, and so forth.
Specifically, in the projective plane, , with a field, we have the correlation given by: points in homogeneous coordinates lines with equations . In a projective space, , a correlation is given by: points in homogeneous coordinates planes with equations . This correlation would also map a line determined by two points and to the line which is the intersection of the two planes with equations and .
The associated sesquilinear form for this correlation is: , where the companion antiautomorphism . This is therefore a bilinear form (note that must be a field). This can be written in matrix form (with respect to the standard basis) as: , where is the identity matrix, using the convention that is a row vector and is a column vector.
The correlation is given by:
Geometric interpretation in the real projective plane
This correlation in the case of can be described geometrically using the model of the real projective plane which is a "unit sphere with antipodesthe points of a sphere at opposite ends of a diameter are called antipodal points. identified", or equivalently, the model of lines and planes through the origin of the vector space . Associate to any line through the origin the unique plane through the origin which is perpendicular (orthogonal) to the line. When, in the model, these lines are considered to be the points and the planes the lines of the projective plane , this association becomes a correlation (actually a polarity) of the projective plane. The sphere model is obtained by intersecting the lines and planes through the origin with a unit sphere centered at the origin. The lines meet the sphere in antipodal points which must then be identified to obtain a point of the projective plane, and the planes meet the sphere in great circles which are thus the lines of the projective plane.