In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines.
This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, point and line are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the following definition, which is more often encountered in modern textbooks.
Using linear algebra, a projective space of dimension is defined as the set of the vector lines (that is, vector subspaces of dimension one) in a vector space of dimension . Equivalently, it is the quotient set of by the equivalence relation "being on the same vector line". As a vector line intersects the unit sphere of in two antipodal points, projective spaces can be equivalently defined as spheres in which antipodal points are identified. A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane.
Projective spaces are widely used in geometry, allowing for simpler statements and simpler proofs. For example, in affine geometry, two distinct lines in a plane intersect in at most one point, while, in projective geometry, they intersect in exactly one point. Also, there is only one class of conic sections, which can be distinguished only by their intersections with the line at infinity: two intersection points for hyperbolas; one for the parabola, which is tangent to the line at infinity; and no real intersection point of ellipses.
In topology, and more specifically in manifold theory, projective spaces play a fundamental role, being typical examples of non-orientable manifolds.
Motivation
As outlined above, projective spaces were introduced for formalizing statements like "two coplanar lines intersect in exactly one point, and this point is at infinity if the lines are parallel". Such statements are suggested by the study of perspective, which may be considered as a central projection of the three dimensional space onto a plane (see Pinhole camera model). More precisely, the entrance pupil of a camera or of the eye of an observer is the center of projection, and the image is formed on the projection plane.
Mathematically, the center of projection is a point of the space (the intersection of the axes in the figure); the projection plane (, in blue on the figure) is a plane not passing through , which is often chosen to be the plane of equation , when Cartesian coordinates are considered. Then, the central projection maps a point to the intersection of the line with the projection plane. Such an intersection exists if and only if the point does not belong to the plane (, in green on the figure) that passes through and is parallel to .
It follows that the lines passing through split in two disjoint subsets: the lines that are not contained in , which are in one to one correspondence with the points of , and those contained in , which are in one to one correspondence with the directions of parallel lines in . This suggests to define the points (called here projective points for clarity) of the projective plane as the lines passing through . A projective line in this plane consists of all projective points (which are lines) contained in a plane passing through . As the intersection of two planes passing through is a line passing through , the intersection of two distinct projective lines consists of a single projective point. The plane defines a projective line which is called the line at infinity of . By identifying each point of with the corresponding projective point, one can thus say that the projective plane is the disjoint union of and the (projective) line at infinity.
As an affine space with a distinguished point may be identified with its associated vector space (see ), the preceding construction is generally done by starting from a vector space and is called projectivization. Also, the construction can be done by starting with a vector space of any positive dimension.
So, a projective space of dimension can be defined as the set of vector lines (vector subspaces of dimension one) in a vector space of dimension . A projective space can also be defined as the elements of any set that is in natural correspondence with this set of vector lines.
This set can be the set of equivalence classes under the equivalence relation between vectors defined by "one vector is the product of the other by a nonzero scalar". In other words, this amounts to defining a projective space as the set of vector lines in which the zero vector has been removed.
A third equivalent definition is to define a projective space of dimension as the set of pairs of antipodal points in a sphere of dimension (in a space of dimension ).
Definition Given a vector space over a field , the projective space is the set of equivalence classes of under the equivalence relation defined by if there is a nonzero element of such that . If is a topological vector space, the quotient space is a topological space, endowed with the quotient topology of the subspace topology of . This is the case when is the field of the real numbers or the field of the complex numbers. If is finite dimensional, the dimension of is the dimension of minus one.
In the common case where , the projective space is denoted (as well as or , although this notation may be confused with exponentiation). The space is often called the projective space of dimension over , or the projective -space, since all projective spaces of dimension are isomorphic to it (because every vector space of dimension is isomorphic to ).
The elements of a projective space are commonly called points. If a basis of has been chosen, and, in particular if , the projective coordinates of a point P are the coordinates on the basis of any element of the corresponding equivalence class. These coordinates are commonly denoted , the colons and the brackets being used for distinguishing from usual coordinates, and emphasizing that this is an equivalence class, which is defined up to the multiplication by a non zero constant. That is, if are projective coordinates of a point, then are also projective coordinates of the same point, for any nonzero in . Also, the above definition implies that are projective coordinates of a point if and only if at least one of the coordinates is nonzero.
If is the field of real or complex numbers, a projective space is called a real projective space or a complex projective space, respectively. If is one or two, a projective space of dimension is called a projective line or a projective plane, respectively. The complex projective line is also called the Riemann sphere.
All these definitions extend naturally to the case where is a division ring; see, for example, Quaternionic projective space. The notation is sometimes used for .Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) Foundations of Translation Planes, p. 506, Marcel Dekker If is a finite field with elements, is often denoted (see PG(3,2)).
Related concepts
Subspace Let be a projective space, where is a vector space over a field , and
be the canonical map that maps a nonzero vector to its equivalence class, which is the vector line containing .
Every linear subspace of is a union of lines. It follows that is a projective space, which can be identified with .
A projective subspace is thus a projective space that is obtained by restricting to a linear subspace the equivalence relation that defines .
If and are two different points of , the vectors and are linearly independent. It follows that: There is exactly one projective line that passes through two different points of , and A subset of is a projective subspace if and only if, given any two different points, it contains the whole projective line passing through these points.
In synthetic geometry, where projective lines are primitive objects, the first property is an axiom, and the second one is the definition of a projective subspace.
Span Every intersection of projective subspaces is a projective subspace. It follows that for every subset of a projective space, there is a smallest projective subspace containing , the intersection of all projective subspaces containing . This projective subspace is called the projective span of , and is a spanning set for it.
A set of points is projectively independent if its span is not the span of any proper subset of . If is a spanning set of a projective space , then there is a subset of that spans and is projectively independent (this results from the similar theorem for vector spaces). If the dimension of is , such an independent spanning set has elements.
Contrarily to the cases of vector spaces and affine spaces, an independent spanning set does not suffice for defining coordinates. One needs one more point, see next section.
Frame
A projective frame or projective basis is an ordered set of points in a projective space that allows defining coordinates. More precisely, in an -dimensional projective space, a projective frame is a tuple of points such that any of them are independent; that is, they are not contained in a hyperplane.
If is an -dimensional vector space, and is the canonical projection from to , then is a projective frame if and only if is a basis of and the coefficients of on this basis are all nonzero. By rescaling the first vectors, any frame can be rewritten as such that ; this representation is unique up to the multiplication of all with a common nonzero factor.
The projective coordinates or homogeneous coordinates of a point on a frame with are the coordinates of on the basis . They are only defined up to scaling with a common nonzero factor.
The canonical frame of the projective space consists of images by of the elements of the canonical basis of (that is, the tuples with only one nonzero entry, equal to 1), and the image by of their sum.
Projective geometry
Projective transformation
Topology
A projective space is a topological space, as endowed with the quotient topology of the topology of a finite dimensional real vector space.
Let be the unit sphere in a normed vector space , and consider the function
that maps a point of to the vector line passing through it. This function is continuous and surjective. The inverse image of every point of consist of two antipodal points. As spheres are compact spaces, it follows that:
For every point of , the restriction of to a neighborhood of is a homeomorphism onto its image, provided that the neighborhood is small enough for not containing any pair of antipodal points. This shows that a projective space is a manifold. A simple atlas can be provided, as follows.
As soon as a basis has been chosen for , any vector can be identified with its coordinates on the basis, and any point of may be identified with its homogeneous coordinates. For , the set
is an open subset of , and
since every point of has at least one nonzero coordinate.
To each is associated a chart, which is the homeomorphisms
such that
where hats means that the corresponding term is missing.
These charts form an atlas, and, as the transition maps are analytic functions, it results that projective spaces are analytic manifolds.