Projective geometry

The projective geometry is a branch of geometry. It arose from the perspective view of three-dimensional objects in two-dimensional plane. In contrast to the " ordinary " Euclidean geometry, there are no parallels in projective geometry.

The mathematical structures that are investigated in projective geometry, called projective geometries, see the section axiomatic access below.

A projective geometry of a finite number of points of incidence is a finite structure, and is analyzed as such in the finite geometry. Here then are those finite projective geometries with the geometric points as points and one of the types of line, plane ... as blocks even block plans.

Projective geometry as an extension of the affine geometry

The oldest access to projective geometry goes from the space of our experience, that of Euclidean geometry in three dimensions - or because lengths and angles for the following considerations do not play a role, more generally, an affine geometry. There are parallels, ie straight lines, which lie in a plane but do not intersect, and parallel planes. The existence of such parallels is the intellectual capture of Euclidean space required - already Euclid they therefore demanded especially in a parallel postulate, but it does not meet the immediate perception: If you follow parallel lines with the look, they seem to converge and infinite in "Remote " to hit.

The projective geometry takes this into account by supplementing the objects of affine geometry by remote elements: For each class of parallel lines known as a remote point (also infinitely distant point or improper point) is defined, which indicates the direction of this line. All remote points of a plane form a line at infinity ( at infinity or improper line). ( In the three-dimensional line at all is such a line at infinity for a class of parallel planes. Together form the remote level). The common way of speaking, that parallel lines meet at infinity, thus receives - in addition to their practical declaration to the perspective drawing - even a precise mathematical sense.

Due to this expansion, the parallel postulate no longer always applies, but there are two straight lines which lie in the same plane intersect, then at one point. For two non-parallel straight lines this is their cut from the Euclidean geometry known intersection of two parallel lines her far point and an ordinary straight line in a plane and the line at this level in the far point of the line.

For a strictly mathematical structure of the distant points with the classes of parallel lines can be identified. Something vivid you can also say: A remote point "is" the direction of a class of parallel lines. According to " is " in three-dimensional space a line at the direction of a plane and parallel to it in the affine space levels.

The term " projective " geometry derives from the fact that central projections are an important tool and object of study of projective geometry. A central projection of a point (the center ) of a plane ( which does not contain ) can now be applied to each point ( except for). In Euclidean geometry there is no picture for points that lie on the plane parallel to through.

An important concept in this context, the cross-ratio of four points, which lie on a straight line. It will not change as projective mappings.

Axiomatic access

As the geometry in a strictly axiomatic form is summed up in the second half of the 19th century and then set out to vary the axioms systematically, it made ​​sense to replace the parallel axiom by defining that two straight lines lying in a plane always intersect need. However, this is incompatible with the arrangement Axiom II.3.

But one is limited to the incidence axioms, the result is very simple and highly symmetric systems of axioms, which also include the laws of the known projective space.

Such an axiom system that only with the basic concepts " point ", " line" and " incidence " makes do is:

An incidence structure satisfying these axioms then is, a projective geometry.

The first axiom is a summary of the incidence axioms I.1 and I.2.

The second axiom replaces the axiom of parallels. If suitable defines the term " level " in the context of other axioms, it just says that two straight lines of a plane always intersect. If we replace it by the simpler ( and stronger ) axiom

Is the name of the corresponding structure is a projective plane.

The Reichhaltigkeitsaxiome 3 and 4 replace the Hilbert axiom I.8. Structures that satisfy only the axioms 1 to 3, but not 4, are called degenerate projective geometries. ( There are all projective planes. )

Since both the arrangement Axiom III.4 and the completeness axiom V.2 missing, finite models for projective geometries are possible.

The simplest example is the non- degenerate Fano plane, which consists of seven points, and seven lines; in the picture are the "points" the points marked thick, the " straight lines" are the lines and the circle.

A set of points of a projective space, which always contains all the points on which ( according to Axiom 1 unique) straight line connecting two different points, ie linear quantity. Linear quantities play the role of projective subspaces in projective geometry, you also writes, therefore, if a linear amount.

• The simplest (though not the smallest ) type of a linear set is a point set, ie the set of points on a line.
• An arbitrary point set of the space generates a well-defined minimum linear amount

• , And for each point, i.e., a minimal generating system or a point -based. The number of elements of such a point of the base is independent of the choice of the base point. The number is, the projective dimension, it may be a natural number, or more generally an infinite cardinal number, in the latter case is called the linear quantity often infinite dimensional.

These three types of linear quantities satisfy (together with the most a straight line passing through two different points of the linear quantities and limited to this subtree incidence), the first three incidence axioms (more or less trivial) but not the 4 are thus degenerate projective spaces. A linear set that contains three points that do not lie on a common straight line, also satisfies the fourth incidence axiom and is therefore itself a projective space. The dimension of this linear amount is then at least 2 Note to the fact that the term level is to understand axiomatic in the above description and not directly related to the concept of dimension for linear quantities. Eye degenerated projective planes, the linear quantities are in a projective space always belong to one of the three types mentioned above and therefore have a linear quantities projective dimension. The total area is of course also a linear quantity and has a well-defined dimension accordingly.

Additional axioms

Closure properties

As an additional axioms two classic lock sets, the set of Desargues and the set of Pappus are particularly important: These axioms are respectively equivalent to the fact that the geometry can koordinatisieren over a frequency determined by the axioms of class Ternärkörpern:

• Just when the theorem of Desargues in each two-dimensional linear amount of space, the space is coordinatizable by a skew field. These conditions are always satisfied for at least three-dimensional spaces. This last statement is a set of David Hilbert.
• Just when the theorem of Pappus in each two-dimensional linear amount of space, the space is coordinatizable by a commutative field. These conditions are not always fulfilled for three - and höherdimendimensionale.

The closure rates were (implicitly ) as the rates that apply in the real two - or three-dimensional geometry, proved by mathematicians, after which they are named. Implicitly, therefore, because it was in their time neither an axiomatic description of the modern algebraic concept of body not even the body of the real numbers. A modern "non- closure axiom " is the Fano axiom. It is in the study of quadrics of great importance. For these studies, one usually also the axiom of Pappus demand. Applies also the Fano axiom, then the coordinates of the body of the room does not have the characteristic 2, that is, a quadratic equation has "mostly" different sense no or two solutions and you can, for example, in a conic section between tangents and non tangents.

Order properties and topological properties

A projective space is located when a separation relationship is defined on each line, that this relation is preserved at any projectivities. The separation relationship sets the above Hilbert projective affine arrangement continued: Is there a point affine between the points, then the pair of points separates the point from the farthest point of the ( projectively closed ) line. The inter-relationship on the affine line satisfies the axiom of Pasch. Making out the order topology on an arbitrary straight line, the product topology ( the dimension of the affine space ), then this is for the room due to the axiom of Pasch an " acceptable " Topology: The affinities of the space with respect to this topology steadily.

This topology can now ( first on each line) continue by making the affine quantities of intermediate points ( "open intervals" ) for an arbitrary choice of the remote point on the base of a topology on the projective line, and provides the space with the corresponding product topology. This is a projective plane to a topological projective plane and a higher-dimensional space is (more precisely: the set of its points) to a topological space in which the projectivities are homeomorphisms.

Such an arrangement of affine and projective spaces is only possible ( necessary condition ), if applicable in a Koordinatenternärkörper: If at any Beklammerung this " sum " with more than one summand ( in Ternärkörper the associative law for addition does not apply, is a loop) is then. It follows for every space arranged: He and his coordinate space is infinite. If the room is additionally desarguessch, so satisfies the Desargues closure axiom, then its coordinate Chief body has the characteristic 0

Generally one can define a topology on a topological space also axiomatic illustrated for the two dimensional case in the article Topological projective plane. Every projective space can be in terms of the receivables shown there, at least to a topology, namely the discrete topology. This is usually not a " interesting " topology.

On projective spaces over skew fields or bodies such as the field of complex numbers and the real Quaternionenschiefkörper, the finite-dimensional vector spaces over a arranged lower body ( in the examples), one can in affine neck (more precisely actually: in the group of projective Perspektivitäten with a fixed introduce fixed-point hyperplane and any centers on this hyperplane ) a topology: This group, the affine group of translations is a ( left ) vector space over and thus over, thus can the order topology, derived from the arrangement of the straight, on the affine and projective space over transferred.

Homogeneous coordinates

The distance elements of the projective plane may be represented in a coordinate system: Each point in the Euclidean plane is represented by a pair of coordinates (x, y) in the Cartesian coordinate system. If a point along a line through the origin is always further away from the origin, then the ratio is x: y constant. In order to describe this shift to formally infinite, an additional coordinate z is introduced, and thus transferred to the 3- dimensional space for this ratio. The original, two-dimensional coordinates to win back through the picture.

The smaller | z | is at a fixed X and Y, the further the point described above is removed from the origin. For z = 0 the mapping is not defined, in this case there is no lying in the finite pixel in the Euclidean plane. The point (x, y, 0) represents in our representation corresponds exactly to that remote point which lies in the direction defined by ( x, y) origin line at infinity. It should be noted that (x, y, z) and (tx, ty, tz ) ( T is a real number other than zero ) are represented by the same point on the Euclidean plane. The number triples (x, y, z ) is called homogeneous coordinates of a point of the projective plane.

The introduction of homogeneous coordinates can also imagine vividly:

If you cut a line bundle with a projective plane that does not pass through the support point of the bundle, so you can assign to each point of the plane passing through the straight line of the bundle him and vice versa.

Choose the origin of a three dimensional Cartesian coordinate system as a carrier of the bundle and on the plane E with the equation parallel to the x- axis and y- axis coordinate system (x '/ y'), it can be the points in the plane E by the direction vectors the line through the origin represent. Since all vectors represent with the same straight line direction, they also define the same point in the plane E. All bundles straight run parallel to the plane E, ​​direction vectors have to, define the remote points of the plane E. For all other points, they lie in the finite, one obtains their usual coordinates in ( x ' / y') by the system. For the normalization of the direction vectors results in exactly the position vector of the corresponding point in the plane E.

Linear equation in homogeneous coordinates

If the normal vector of a plane passing through the origin plane and the position vector of a point X lying in this plane, that is the plane equation:

Such a plane intersects the plane E in a straight line. Therefore, this is also a linear equation in homogeneous coordinates. Each of different number triple so represents both a point and a straight line of the plane

Is, the two planes are parallel, and the equation is the equation of the line at the level of E

If you hold in the equation the normal vector of fixed and varied, we obtain the equation of a point set. If you hold firmly and varies to the normal vector, we obtain the equations of all lines that pass through the fixed point X, ie a pencil of lines.

Projective spaces of higher ( even infinite ) dimensions can be constructed analogously. You meet all the above axioms 1 to 4

Properties

In the following, under a projective space we understand a structure of points and lines with an incidence relation satisfying the above axioms of Veblen -Young and in which there are two disjoint lines; the projective planes are therefore excluded. Then, the following rates shall apply:

In each of projective space of dimension, the set of Desargues applies: If O, A, B, C, A ', B ', C ' different points, so that O, A, A', O, B, B 'and O, C, C, BC ' determine three different lines, the three points of intersection of AB with A'B ' lie with B'C ' and CA with C'A ' on a straight line. With the help of this proposition can be shown: Every projective space can be described by homogeneous coordinates in a left vector space V over a division ring K. The left vector space V is at least four-dimensional, its dimension can also be an arbitrary infinite cardinal number. The skew field K is commutative, ie a body precisely when in the geometry of this space shall be the set of Pappus ( Pascal ). This is always the case (because finite skew field by the theorem of Wedderburn are necessary commutative ) in finite Desargues planes.

Are of interest in synthetic geometry especially the " nichtdesarguesschen " planes in which the set of Desargues does not apply, in particular the finite among them. The order of a finite projective plane is decreased by 1 the number of points on a, so everyone straight. It is an unproven conjecture that every finite projective plane of prime power order ( such as the Desargues planes). A set of Bruck and Ryser excludes many orders. He says: If n = 4k 1 or 4k 2 order of a projective plane, then n is the sum of two square numbers. The following figures are not orders projective planes 6, 14, 21, 22, 30, 33, 38, 42, 46, ...

With great use of computers has been shown that no projective plane of order 10 exists. The smallest orders for which the question of the existence or non-existence is unresolved, 12, 15, 18, 20 The smallest order of a projective plane is nichtdesarguesschen 9, compare the section Examples of order 9 in the article Ternärkörper.