# Why is the incidence geometry important

## 8 On the concept of projective space

§8 On the concept of projective space In this paragraph we want to summarize the different points of view at the start of projective geometry. First we give an overview of the association-theoretical characterization of projective spaces (8.1). Finally, with the incidence geometric approach, we suggest what is probably the purest geometric path (8.2). The analytical models that are important for practical computing are summarized in 8.3. Finally, the connections between these different approaches should at least be traced. For this purpose, some results are presented in Section 8.4, which are examined in more detail in the lecture “Fundamentals of Geometry”. What all approaches have in common is that we limit ourselves to finite dimensional spaces. 8.1 An overview of the lattice-theoretical definition In 7.4, projective spaces were introduced as irreducible complementary modular and elongated lattices. The following diagram summarizes the individual stations for this definition. Joins (2.1) Joins with N and E (2.3) Irreducible joins (7.1) Longitudinal joins (3.2) (L) Complementary joins (5.1) (K) Modular joins nde (4.2) (M) dimensional lattice (3.4) projective dimensional axion (3.5) (P) distributive lattices (4.1) (DT) Bollean algebras (4.1) KML lattice (5.4) Boolean algebras (4.1) projective spaces ( 7.4) In connections we find two connections that are fully symmetrical in their properties, which we interpret geometrically as “connect” and “cut”. Due to the length (L), the limitation is to finite dimensions, but additional conditions are required for a geometrically meaningful concept of dimensions: First of all, it must be ensured that all rows between two elements have the same length. length (dimensional association), furthermore objects of different dimensions should incise with a sufficient number of points (projective dimensional axiom (P)). The weakened distributivity in the form of modularity (M), together with (L), provides a formulaic version for associations with a geometrically satisfactory dimension term. By adding the complementarity (K) it is finally ensured that the dressing elements are clearly defined by the points incised with them. Finally, irreducibility ensures that straight lines intersect with at least three points. This is important for projective spaces because every “actual” straight line has a distant point, after which at least two are left to determine it. Subsets of the set of points IP (L) which determine structural elements are those which, with two different points P and Q each, also contain all points of the connecting straight line P⊔Q. The following appear as projective spaces of small dimensions: • The empty space (contains only one element, the empty set). • Zero-dimensional spaces contain two elements as a lattice, the one element is then the only point of this space. • Projective straight lines contain a zero element, at least three points and, as a single element, the straight line which incurs with all points. • The smallest projective plane contains 7 points. For the construction one starts from a projective straight line with three points (P1, P2, P3), then there must be a 4th point P4 outside this straight line. The straight lines connecting this point with the three of the first straight lines each need a new point, then you have to combine these points into a straight line. The incidence diagram (cf. § 1) is much clearer for construction than the Hate diagram given in Figure 8.1a). E G123 G247 P1 P2 G346 P3 G256 G357 G145 P4 P5 P6 G167 P7 N Hasse diagram of the smallest projective plane It is clear that the previously indicated combinatorial strategy for constructing projective planes for larger numbers of points (the number of points on a straight line decreases by one is also referred to as the order of the level) becomes unclear. This explains why, even for the relatively small order 12, it is not yet known whether there is a projective plane of this order. For the 10th order one leads a non-existence proof with massive computer use. 84 § 8 on the concept of projective space 8.2 85 Incidence geometric definition Geometries are characterized in that incidences exist or do not exist between their objects. For a geometry we therefore need at least one set of objects and the description of the incidence or non-incidence between them. Within our approach based on association theory, the incidence was described using the abstinence relation “⊑”. Incidences are characterized by pairs of objects. There is the possibility of already observing the hierarchy, that is, to translate the speaking style "lies on" into mathematics. Another approach assumes that z. B. a point not only lies on a straight line (incised with it), but that this straight line also contains the point (i.e. also incised with it), the incidence in this case is seen symmetrically. This procedure (see BEUTELSPACHER / ROSENBAUM [BR] p. 5) leads to the following very rudimentary general 8.2.1 definition (geometry): A geometry is a pair Γ = (Ω, I), where Ω is a Set (geometric objects) and I⊆ΩxΩ is a two-digit relation (called incidence relation) on Ω, which is reflexive and symmetric, so we have for I (ReI) (x, x) ∈ I for all x∈Ω and (SyI) (x, y) ∈ I ⇒ (y, x) ∈ I for all x, y ∈ Ω. Instead of (x, y) ∈ I we also write xIy (“x incised with y”). You can see immediately that the geometries introduced in section 1 meet this definition. But also many more general examples belong to the geometries, the most economical one results if one chooses the empty set for Ω and thus also for I. As soon as Ω is not empty, because of the reflexivity this also applies to I. In particular, the incidence relation I of geometries always includes the equality relation. One can be satisfied with the equality relation for I as the greatest incidence relation, or one can refine it further - this results in the geometries that are actually of interest. If (Ω, I) is a geometry and Ω 'is a subset of Ω, then Γ' = (Ω ', I') with the incidence relation induced by I on Ω '(intersection of I with Ω'xΩ') is also a geometry . In this way, a straight line with its points becomes a “partial geometry” of a plane. The diversity of the objects of a geometry (cf. points, straight lines, planes, circles ...) can be grasped with the following term. 8.2.2 Definition (flag): Let Γ = (Ω, I) be a geometry. Then a subset of Ω whose elements are pairwise incident is also called the flag of this geometry. A flag F is called maximal if there is no x∈Ω \ F, so that F ∪ {x} is also a flag. 8.2.3 a) Examples: Let Γ be the affine plane over the real numbers (AG (2, R)). If P is a point and g is a straight line through this point, then {P}, {g}, {P, g} are flags, only the last one is maximal. We have not included the plane E itself as an element in the geometry85 N. Christmann: Projektive Geometrie WS 2007/2008 b) c) 86, it is given by the set Ω. In incidence geometry, Ω is taken as a proxy for the entire space. In Hilbert's structure of geometry, points, straight lines and planes appear as object types of spatial geometry. Maximum flags in this case are sets of the type {P, g, α}, where P is a point, g is a straight line through P and α is a plane which includes g. In Figure 8.2a a “geometry of descent” is indicated: The dots stand for people, a direct connection is entered for “is a child of” from top to bottom. Two of the people are related when there is an up or down path from one to the other. In this geometry there are maximum flags of lengths 2, 3 and 4, i.e. in particular those of different lengths. Fig. 8.2a A “geometry of descent” In the last example, there are problems in assigning the role of points, straight lines, etc. to objects. This is not least due to the fact that we find maximum flags of different lengths. We exclude such geometries with the following 8.2.4 definition (rank of a geometry): A geometry Γ = (Ω, I) is called of rank r if Ω is divided into a partition Ω1, ...., Ωd (i.e. into a disjoint decomposition of non-empty subsets) in such a way that every maximal flag of Γ from every class of the partition contains exactly one element. The elements of class i (i = 1,…, d) are also referred to as elements of type i. In a geometry of rank r, every maximal flag has exactly r elements. In the example of a projective incidence plane, the set of points and the set of straight lines form a corresponding decomposition (see note in 8.2.3a), which means that the geometry is rank 2. In example 8.2.3b) there is a geometry of rank 3. We cannot assign any rank in the sense of 8.2.4 to the geometry of descent 8.2.3b). For this, the relationships would have to be suitably supplemented by other people. 86 § 8 on the concept of projective space 87 If Γ = (Ω, I) is a geometry of rank r, then Γi '= (Ωi', Ii ') with Ωi': = Ω \ Ωi and Ii as induced incidence relation is one Geometry of rank r-1. So one can construct a geometry of rank 2 from any geometry of rank r. This corresponds to the fact that you can already describe all elements in the visual geometry by specifying the points and straight lines or hyperplanes (cf. corresponding representation set for irreducible KML connections). In the visual space (cf. 8.2.3b), planes can be identified by the fact that they contain at least three points that are not on a straight line and that with two different points P and Q, all points of the associated straight connecting line P⊔Q belong to the plane ; ren. So you can do without levels as basic objects in the construction. This is particularly useful if you want to examine geometries of different dimensions together. The geometries of rank 2 are therefore of particular importance. 8.2.5 Lemma: If Γ = (Ω, I) is a geometry of rank r, then no flag contains two elements of the same type. Proof: Assume that two elements x and x 'of the same type would intersect with each other. Then F = {x, x ’} would be a flag. This can be supplemented to a maximum flag, but this then contained two elements of the same type in contradiction to the requirement that Γ = (Ω, I) is a geometry of rank r. In particular, this lemma implies that two different straight lines can have a point in common, but cannot intersect with each other. We introduce a separate name for geometries of rank 2. 8.2.6 Definition (incidence structure): If Γ = (Ω, I) is a geometry of rank 2, this is also called the incidence structure. The elements of type 1 are mostly referred to as points, those of type 2 as blocks, under additional conditions also as straight lines. If IP is the set of points and IB the set of blocks of Γ = (Ω, I), then one also writes Γ = (IP, IB, I). With an incidence structure (Ω, I), we have Ω = IP∪IB with IP∩IB = {}. Sometimes the definition of Ω is not used and the set triple (IP, IB, I) is used directly. Then one usually drops the reflexivity and symmetry for the incidence and explains them simply as the relation between IP and IB (subset of IPxIB), see also the note at the beginning of this section. We now want to state the axioms that we require of an incidence structure when blocks are denoted as straight lines. The term incidence structure also includes graphs (sometimes also called multigraphs), in which the corners and edges form the object types. Multiple edges (two corners connected to more than one edge) and loops (an edge begins and ends in the same corner) can occur. We want to exclude these cases in the geometry in the narrower sense, here there should be exactly one straight line through two different points and every straight line at least two (in projective spaces even contain three) points. 8.2.7 Definition (incidence geometry): An incidence structure (IP, IB, I) is called incidence geometry or incidence space if the following incidence axioms apply: (I1) For two different points P and Q there is exactly one (connecting line P⊔Q of P and Q) block g with PIg and QIg. (I1 ') Each block g is incised with at least two different points, so for each g∈B there are P, Q∈IP with P ≠ Q and PIg and QIg. 8.2.8 a) Examples: For the incidence geometries (IP, IB, I) according to & rsquo; 8.2.7, neither IP nor IB may be empty, because maximum flags according to 8.2.6 must contain two elements. This excludes the following cases of degeneration, which are permitted with procedures other than incidence geometries: With empty space, all three sets IP, IB I are empty, with one-point space IP consists of exactly one point P, IB is empty and I = { (P, P)}. b) The smallest structure that we allow as incidence geometry (IP, IB, I) has two different points and exactly one straight line that connects these points. c) The following Figure 8.2b shows that geometries with undesirable properties are still permitted (straight lines with different numbers of points) P P1 P2 P3 P4 g P5 P6 P7 P8 P9 Figure 8.2b: An incidence geometry with straight lines with different numbers of points Out (I1) it follows in particular that two different straight lines have at most one point in common. The projective planes already introduced in § 1 can now also be identified as follows: 8.2.9 Definition (projective incidence planes): We refer to an incidence geometry (IP, IB, I) as a projective incidence plane, if additionally The following apply: (I2) Two different straight lines each incise with exactly one point. (I3) Every straight line incised with at least three points (I4) There are at least two different straight lines. 88 § 8 on the concept of projective space 89 The simple proof that this definition is equivalent to the one given in § 1 is recommended for practice. The duality principle for projective incidence levels, which is valid as a result, is reminded. We arrive at the term projective incidence space (we use this term because the term projective space according to 7.4 has already been assigned and equivalence has yet to be ensured) if we generalize axiom (I2) in the preceding definition. 8.2.10 Definition (projective incidence space): An incidence geometry (IP, IB, I) is called projective incidence space if the Veblen-Yong axiom (VY) applies in addition to axioms (I3) and (I4). (VY) says: If A, B, C, D are four different points of the incidence geometry, so that the straight line A⊔B meets the straight line C⊔D, then the straight line A⊔C also meets the straight line B⊔D. (VY) can be pronounced a little less formally: If a straight line meets two sides of a triangle, then it also meets the third side. The meeting of the straight lines A⊔B and C⊔D clearly means that they lie in a (projective) plane. Then the straight lines A⊔C and B⊔D must also lie in this plane and therefore have an intersection point according to (I2). In this respect (VY) is a generalization of (I2). B A S D C Fig. 8.2c About the Veblen-Young axiom The thick line meets the two triangular lateral lines S⊔C and S⊔A of the triangle SAC in D and in B. According to (VY) it then also meets the lateral line A⊔C. In projective rooms cozy 7.4 In addition to points and straight lines, we have other geometric object types that we differentiate on the basis of their dimensions. In incidence spaces we gain this by reversing the considerations from 5.4. So we mark out those subsets of the point set that are closed with regard to the formation of the straight line. 89 N.Christmann: Projective Geometry WS 2007/2008 90 8.2.10 Definition (partial spaces): Let Γ = (IP, IB, I) be an incidence geometry. Then T ⊆ IP is called subspace (also linear set) of Γ, if T for two different points P and Q also contains all points which intersect with P⊔Q. We denote the set of partial spaces with TR (IP) or also TR (Γ). With & lt; A & gt; (“Shell of A”) we denote for A∈Pot (IP) the smallest subspace that encompasses A. Now we can bring the association theory into play. 8.2.11 Theorem (association of subspaces): Let Γ = (IP, IB, I) be an incidence geometry. If we explain two links ⊓ and ⊔ on TR (IP) by A ⊓ B: = A ∩ B and A ⊔ B: = & lt; A ∪ B & gt;, then (TR (IP), ⊓, ⊔) to a lattice with zero element N = {} and one element E = IP. Proof as an exercise. If P is a point then & lt; P & gt; = {P}, for the sake of simplicity we also consider P itself as a subspace, so sometimes we also set & lt; P & gt; = P. Correspondingly, we also identify a straight line g with its point set & lt; g & gt; provided that no misunderstandings are to be feared. The definition given in 8.2.10 for r & lt; A & gt; determines this envelope from above, the intersection of all subspaces is formed, which includes A (this concept formation makes sense because the subspace IP includes the set A). With the subspace definition, in addition to points and straight lines, we get further excellent objects in the form of subspaces of dimensions ≥ 1, but these do not belong to the set Ω = IP∪IB that generates the geometry, therefore they have maximum flags in this geometry still exactly two elements (a point and a straight line). Of course, you can also define flags within the set TR (IP) of the subspaces, then you get the (possibly intuitively expected) dependence of the length on the dimension of the space under consideration. For projective incidence spaces, the envelope of two sub-spaces can also be easily determined from below. The 8.2.12 lemma applies: Let Γ = (IP, IB, I) be a projective incidence space. Then for A ∈ TR (IP) and P ∈ IP: A ⊔ P = ∪a∈A & lt; a⊔P & gt ;. The connecting space of A and P is obtained by connecting P to all points of A. For proof, see [BR] p. 14f. or refer to & sect; 1 of the lecture “Introduction to Geometry”. If the point P is replaced by an arbitrary subspace, the connection space for the concept of projective space 91 consists of all straight lines which connect points in one space with points in the other. To introduce the concept of dimension, the concept of the independent set is introduced analogously to § 5. Independent generating systems are then called bases, the length of which turns out to be an invariant and - reduced by one - can be introduced as the dimension of the partial spaces. The restriction to finitely generated spaces leads to the projective incidence spaces, which are also projective spaces in the sense of 7.4.1. We will forego a proof of the equivalence of the outlined approaches to the fundamentals, so that we can develop the theory a little further. 8.3 Analytical projective geometry Already in § 1 the analytically defined projective planes PG (2, IK) were considered. In 7.4.3 it was shown that a projective space exists if the 2 is replaced by any n∈N or the vector space IK3 is replaced by any (n + 1) -dimensional IK vector space V. The resulting standard geometries PG (n, IK) and PG (V) will be examined here and in the following paragraphs. The important question of whether there are any others besides these standard geometries is touched on in Section 8.4. 8.3.1 Definition (projective spaces analytical): Let V be an (n + 1) -dimensional IK vector space (n∈N, e.g. V = IKn + 1). Then the subspace lattice (L (V), ∩, ⊔) is also called the projective standard geometry of V, in symbols L (V) = PG (V) or L (IKn + 1) = PG (n, IK). The points of PG (V) are the subspaces of the (vector space) dimension 1 (straight lines through the origin), the straight lines of this geometry are the 2-dimensional subspaces of V (planes through the origin), the incidence is given by the set-theoretical inclusion described. For a more convenient notation we write P (v) for the point assigned to a vector v ≠ 0. Ultimately, this is just another designation, reminiscent of geometry, for the subspace generated by v, which in linear algebra is called Span (v) or Lin (v) or the like. written down. In the case of sub-vector spaces, a distinction must be made between the lattice-theoretical dimension (we denote the associated dimension function with "Dim") and the dimension denoted by "dim" from the theory of vector spaces. We have Dim (U) = dim (U) - 1 (U∈L (V)). For the spaces PG (V) it has already been shown that they are projective spaces in the sense of 7.4.1. The 8.3.2 theorem also applies: If V is an (n + 1) -dimensional IK vector space (n∈N), then the geometry PG (V) with the one and two-dimensional sub-vector spaces as object types is a projective incidence space. 91 N. Christmann: Projective Geometry WS 2007/2008 92 The axioms (I1) and (I1 '), like (I3) and (I4), result from the properties of the L (V) group. The axiom (VY) results from the projective dimensional formula. One has to assume n & gt; 1. A proof without recourse to the association axioms is recommended for practice. We now want to introduce coordinates into PG (V) with the help of vector space theory. For this purpose, a basis B = (b0, b1,… .., bn) of V is given first. Then every vector v∈V can be represented uniquely in the form v = x0⋅b0 + x1⋅b1 +… .. + xn⋅bn. Because with v every λ⋅v with λ∈IK \ {0} determines the same projective point P (v), we have to allow all corresponding multiples of the (n + 1) -tuple to identify it. This is done by introducing a corresponding equivalence relation. 8.3.3 Definition (homogeneous coordinates): Let V be an (n + 1) –dimensional IK vector space and B = (b0, b1,… .., bn) a basis of V. Then we see two coordinates- (n + 1) -tuple (x0. X1, ...., Xn) and (y0, y1, ...., Yn) as & auml; equivalent to (in characters (x0. X1, ...., Xn) ~ (y0, y1, ... ., yn)), if there is a scalar λ∈IK \ {0} with (y0, y1,…., yn) = λ⋅ (x0. x1,…., xn). We denote the associated equivalence class with [x0. x1,…., xn] (= [y0, y1,…., yn]). We then call [x0, x1,…., Xn] homogeneous coordinates of the point P (x0⋅b0 + x1b1…. Xnbn) with respect to the base B. We also write briefly P [x0, x1,… .., xn] for this matter. For further investigations, the methods of linear algebra, especially matrix calculations, can now be used. As an example we consider the following 8.3.4 criterion: Let B = (b0, b1,… .., bn) be a basis of the (n + 1) -dimensional IK vector space V. In PG (V) let d + 1 points P0 = P (v0), P1 = P (v1),… .., Pd = P (vd) given. With respect to the base B, the vectors vi may have the coordinates (xi0, xi1,… .., xin) (i = 0,…., D). Then the dimension of P0 ⊔ P1 ⊔ …… ⊔ Pd agrees with the rank of the matrix (xi, k) i = 0, .., d; k = 0,…., matched. In particular, the points are independent if and only if the rank of this matrix is d + 1. The points P (v0), P (v1),… .., P (vd) are independent (free) if and only if the homogeneous coordinates of these points with respect to a base B are linearly independent in IKn + 1 . In particular, the coordinates - (n + 1) - tuples can be written as a matrix. If their rank is r, then the subspace spanned by the d + 1 points has the (projective) dimension r – 1. According to the specification of a basis of V, the homogeneous coordinates of points with regard to this basis are uniquely determined (i.e. except for a scalar not equal to 0). Conversely, one can specify n + 1 independent points B0, B1, ...., Bn, for these n + 1 vectors b0, b1, ..., bn can be determined with Bk = P (bk) (k = 0, ..., n) . Then B = (b0, b1,… .., bn) is a basis of V, with respect to which can determine the homogeneous coordinates of the points of PG (V). However, these are not uniquely determined by the specification of the points B0, B1, ..., Bn, because with 92 8 for the concept of projective space 93 the generating vector can be multiplied for each individual point with a scalar not equal to zero, with different ones Scalars can choose. If, for example, [x0, x1,… .., xn] are the homogeneous coordinates of X with respect to B and if one changes b0 to b0 ': = λ⋅b0 (λ ≠ 0, 1), then X has with respect to & uuml; if the base (b0 ', b1,… .., bn) had the homogeneous coordinates [x0 / λ, x1,… .., xn], these are obviously different from those relating to B. The deficiency is remedied by adding another appropriate point eliminated. We define 8.3.5 definition (points in general position): Let V be an (n + 1) -dimensional IK vector space. Then the n + 2 points are called P0, P1,…., Pn, E in a general position, if n + 1 of these points are independent (equivalent: no n + 1 of these points lie in a hyperplane). For a more precise justification of this definition we prove the following 8.3.6 Lemma: Let V be an (n + 1) -dimensional IK vector space. With n + 2 points P0, P1,…., Pn, E from PG (V) in general position there is a basis B = (b0, b1,… .., bn) of V with Pk = P (bk) (k = 0,… n) and E = P (b0 + b1 +… .. + bn) uniquely determined except for a common scalar factor. Proof: a) Existence: Let (b0, b1,… .., bn) be chosen such that Pk = P (bk) (k = 0,… n) holds. Then B = (b0, b1,… .., bn) and B '= (b0', b1 ',… .., bn') = (λ0⋅b0, λ1⋅b1,… .., λn⋅bn ) Bases of V, where λk ≠ 0 f & uuml; rk = 0, 1,… n. The point E then has a representation E = P (e) with e = α0''b0 '+ α1''b1' + .... + αn’⋅bn ’= α0’⋅λ0⋅b0 + α1’⋅λ1⋅b1 +…. + αn’⋅λn⋅bn. Because of the requirement "in general situation", all αk ’≠ 0 (k = 0, 1,…., N). If λk = 1 / αk 'is chosen, the desired representation for E. b) Uniqueness: Let Pk = P (bk) = P (bk') (k = 0, 1, ... , n) and E = P (b0 + b1 +… + bn) = P (b0 '+ b1' +… + bn '). Then bk '= βk ⋅bk with βk ≠ 0 (k = 0, ..., n) and b0' + b1 '+… + bn' = β0⋅b0 + β1⋅b1 +… + βn⋅bn = β⋅ ( b0 + b1 +… + bn). Because the scalars are uniquely determined with respect to a base, we get βk = β for r k = 0, 1, ...., N. Both bases therefore only differ by a common factor β. With n + 2 points in a general position, the homogeneous coordinates are clearly determined up to a scalar factor, so they are clearly defined in the sense of equivalence. This leads to the following 93 N. Christmann: Projective Geometry WS 2007/2008 94 8.3.7 Definition (projective coordinate system): Are n + 2 points in a general position according to Lemma 8.3.6 determines, we also call it the projective coordinate system, E we also call it the unit point, and the rest as the base points. 8.3.8 a) Remarks For a projective coordinate system you can choose any n + 2 points in general position, but you have to specify which point is the unit point and in which order the points are chosen as base points. b) With the choice of the points, the associated one-dimensional sub-vector spaces are fixed. For the basic points, the basic vectors are to be chosen in such a way that their sum lies in the subspace P (e) of the unit point. Except for a scalar factor ≠ 0, this is clearly possible. c) The basic points include the homogeneous coordinates [1,0,…, 0], [0, 1, 0,… .0], [0, 0,… .., 0, 1], to the unit point [1, 1,… .., 1]. With the help of the coordinate systems, partial spaces can be described by linear systems of equations. We assume that V is an (n + 1) -dimensional IK vector space, a projective coordinate system is chosen in PG (V), the homogeneous coordinates of the points of this geometry relate to this system. 8.3.9 Lemma: a) In PG (V), after defining a projective coordinate system, each hyperplane can be replaced by a linear homogeneous equation α0x0 + α1x1 +… .. + αnxn = 0 with (α0, α1,…., Αn) ≠ (0 , 0,…., 0). Conversely, every such homogeneous equation describes a hyperplane. The coefficients (α0, α1,…., Αn) are uniquely determined except for a scalar factor other than zero. We also simply denote this hyperplane with] α0, α1,…., Αn [. b) Every d-dimensional subspace can be described by a homogeneous system of n-d equations (average of n-d hyperplanes). The proof results from the corresponding theorems of vector space theory. 8.3.10 Examples: a) The three points P0: = P (0,1,4,2,3), P1: = P (2,0,1,1,0) and P (2,2,9, 5,6) from PG (4, R) lie on a (projective) straight line g, two of these points generate this straight line, so two points each are free, therefore P0, P1, P2 are in a general position with respect to g. If we set p0: = (0,1,4,2,3), p1: = (2,0,1,1,0), p2: = (2,2,9,5,6), then f & uuml applies ; r these vectors the equation 2⋅p0 + p1 - p2 = 0 ⇔ p2 = 2⋅p0 + p1. If one chooses (b0, b1) = (2⋅p0, p1), then P0, P1, P2 form a projective coordinate system of g with the base points P0, P1 and the unit point P2. The points of the straight line g are thus first given by 94 8 for the concept of the projective space 95 {P (λ⋅b0 + b1) / (λ,)) not equal to (0,0)}. For points with λ ≠ 0, P (λ⋅b0 + b1) = P (b0 + (/ λ) ⋅b1), this corresponds to a normalization of the coordinate x0 to 1, if this is possible is. This does not work for P (b1) = P1. The straight line g contains the points P (b0 + σ⋅b1) and P (b1). This corresponds to the affine parametric representation of a straight line, in addition there is only one (distant) point, which is determined by the direction vector. The equation v = λ⋅b0 + µb1 for the points P (v) is also called the parametric representation of the straight line P0 ⊔ P1, the parameters (λ, µ) are only uniquely determined up to a scalar factor. Parametric representations of higher-dimensional structures result accordingly. b) We consider a plane F in PG (4, R). Let this be determined by the three points P0: = P (1,0,1,0,0), P1: = P (1,0,2, -2, -2), P2: = P (1,1, 0.1, -1). A parametric representation for the points P (v) of F is therefore with p0: = (1,0,1,0,0); p1: = (1,0,2, -2, -2); p3: = (1,1,0,1, -1). v = λ⋅p0 + µ ⋅p1 + κ⋅p2 Here we can understand the coordinates as such with respect to the canonical coordinate system with the unit point E = P (e) with e = e0 + e1 + e2 + e3 + e4 . What does F look like as an intersection of hyperplanes? First of all, the total space has the (projective) dimension 4, F the projective dimension 2. F is to be represented as the solution space of a homogeneous system of linear equations with 5 variables, the vector space dimension of the solution space is 3, so the associated matrix must be have rank 2 (= 5 -3). If we write the solutions as column vectors, i.e. if the matrix equation A⋅x = 0 is to apply for the solution vectors x, then the row vectors a of A must have the equations a⋅piT = 0. The three vectors p0, p1, p2 result in the rows of the matrix of the corresponding homogeneous system of equations, each supplemented by a 0. This has the rank 3, 2 parameters can therefore be selected. The specializations a41 = 1 and a31 = 0 and a42 = 0 and a32 = 1 can be used to calculate the remaining solution components of two rows of A, thus obtaining the two hyperplane equations −2 x0 + x1 + 2 x2 + x3 = 0 and -2x 0 + 3 x1 + 2 x2 + x4 = 0. The plane F can therefore be obtained as the intersection of these two hyperplanes. This example also shows how parametric representations and systems of equations for projective subspaces can be converted into one another. c) We consider again a projective coordinate plane F, the points P0, P1, P2, E belonging to F are (with respect to F) in general position. Let the coordinates of the points X of F be denoted by [x0, x1, x2] with regard to this coordinate system. 95 N. Christmann: Projective Geometry WS 2007/2008 96 P2 E1 E0 E P0 E3 P1 For the coordinates of the base points and the unit point E, the conditions already shown in 8.3.8 apply: Point P0 P1 P2 E E0 E1 E2 coordinates [1.0.0] [0.1.0] [0.0.1] [1.1.1] [0.1.1] [1.0.1] [1.1.0] The Pi ⊔ E then has the equation xj - xk = 0 with j, k∈ {0,1,2} \ {i}, j ≠ k.The straight line Pi ⊔ Pk has the equation xk = 0 (i, j, k∈ {0,1,2} all three different). This results in the intersection Ei of the straight line Pi ⊔ E with the straight line Pj ⊔ Pk (again i, j, k∈ {0,1,2} all three different) the coordinates xi = 0 and xj = λ = xk with λ ≠ 0. The coordinates of the table result from the normalization λ = 1. We now want to change the coordinate system, the basic points are Q0: = E, Q1: = P1, Q2: = P2, E ': = P0 is chosen as the unit point, so compared to the old system, the Roles of P0 and E reversed. The equations q0 = λ⋅e then apply to the vectors belonging to the points (we simply denote them with the corresponding small letters, so P0 = P (p0), Q0 = P (q0) etc.) = λ⋅ (p0 + p1 + p2); q1 = λ1p1; q2 = λ2⋅p2; e ’= λ0⋅p0 = q0 + q1 + q2. Inserting the last equation yields the condition (λ - λ0) ⋅p0 + (λ + λ1) ⋅p1 + (λ + λ2) ⋅p2 = 0. The linear independence of the vectors belonging to the base points is provided for the scalars the equations λ0 = λ; λ1 = -λ; λ2 = -λ. Like the vectors belonging to the points, the transformation is only determined up to a scalar multiple. We therefore use z. B. λ = 1 and get as transformation matrix 1 0 0 T = 1 −1 0 . 1 0 −1 96 8 to the concept of projective space 97 One can also belong to v = x0p0 + x1p1 + x2p2 because p0 = e - p1 - p2 after insertion directly from x = x0 (e - p1 - p2) + x1p1 + x2p2 = x0e + (x1 - x0) p1 + (x2 - x0) p2 read off the new coordinates [x0, x1 - x0, x2 - x0]. 8.3.11 Projective and Affine Spaces In Section 1, the relationship between affine and projective incidence levels was explained. An analogous connection can be established between affine and projective spaces of any dimension. To this end, the concept of affine space must first be clarified. We limit ourselves here to the analytical version: If V is an IK vector space, then we denote the secondary classes according to sub-vector spaces as affine subspaces of V. The geometry resulting in this way is also called affine standard geometry to V (in characters: AG (V) or in the case of V = IKn also AG (n, IK)). The affine subspaces of AG (V) are therefore of the form T = a + U, where a∈V and U is a subspace of V. The subspace of T is then also referred to as the directional space of T. The dimension of a subspace is that of the directional space, the affine dimension therefore agrees with the dimension of the associated vector spaces, we therefore also write dim (T) (= dim (U)) for the dimension of T. An affine coordinate system of an n-dimensional affine space consists of n + 1 points A0, A1,… .., An, where the difference vectors A1-A0,… .., An-A0 are linearly independent. A0 is then also referred to as the origin of this coordinate system. The difference vectors A1-A0,… .., An-A0 in particular form a basis of V, each point P can therefore be assigned the scalars belonging to this basis as coordinates with regard to the coordinate system given by the points. If A is an n-dimensional IK vector space (and thus also an n-dimensional affine space over IK), further U = Span (u) a 1-dimensional IK vector space, we denote by V: = UxA = { (u, a) / u∈U and a∈A} ≅ (Ux {0A}) ⊕ ({0U} xA) The direct sum of the two vector spaces yields the mapping ϕ: A → V; a ֏ (u, a) an embedding of A in V, which is compatible with the affine structures. After choosing a basis for A we can also consider the coordinate mapping, for which we then have :K: A → IKn + 1; a ֏ (1, a1,… .., an). We can also interpret this (n + 1) -tuple as homogeneous coordinates [1, a1,… .., an] of a point from PG (V). Conversely, we can assign to all points from PG (V) with homogeneous coordinates [x0, x1,… .., xn] in the case of x0 ≠ 0 the point belonging to (x1 / x0,… .., xn / x0) assign as image in the reverse image of ϕ. We have thus embedded the n-dimensional affine space A in an n-dimensional projective space PG (V). With this embedding, the points P ((0U, a)) of V are not recorded, i.e. those for whose coordinates x0 = 0 applies. This is a hyperplane of PG (V), which is also referred to as the remote hyperplane belonging to A. The embedding of an affine space A in a projective space is also referred to as the projective closure of A. You can also look back at this process: You start from an n-dimensional projective space PG (V) (then in particular dim (V) = n + 1), selects a hyperplane H (after a suitable choice of the coordinate system to which the equation x0 = 0 belongs). For V there is then a direct decomposition V ≅ S ⊕ T into a one-dimensional subspace S and an n-dimensional subspace T such that H = {P (t) / t∈T}. 97 N. Christmann: Projective Geometry WS 2007/2008 98 After a suitable choice of the base of V, the homogeneous coordinates [1, x1,… .., xn] can be assigned to the points from P (V) \ H in a reversible and unambiguous manner, in to which one assigns the affine coordinates (x1,… .., xn) to these, one recognizes that P (V) \ H is isomorphic to a n-dimensional affine space. This process is also referred to as the slitting of PG (V) along the hyperplane H. With the help of the closure / slitting it is possible to mutually merge affine and projective questions. In particular, connections between affine and projective mappings can thus be explored. 8.3.12 Remark on the classification of geometries If IK is a field and n∈N, then with PG (n, IK) we immediately have a projective geometry of dimension n on offer. If V is an (n + 1) -dimensional IK vector space, then PG (V) is also an n-dimensional projective geometry to the coordinate field IK. We referred to both as standard geometries. This designation implies that the point set, line set and incidence “standard” are selected (points: one-dimensional sub-vector spaces, straight lines: two-dimensional sub-vector spaces, incidence: set inclusion). But there is even more to it: As soon as you have chosen n and IK, every geometry PG (V) is isomorphic to an (n + 1) -dimensional vector space V to the geometry PG (n, IK), all projective geometries of Dimension n for a given coordinate body are therefore isomorphic to one another. The admission of arbitrary vector spaces has the advantage that one is not set in advance on a possibly unfavorable standard basis, so that one can work with skilful representations of the respective vector space if necessary. The previously mentioned isomorphisms result from the isomorphism of the corresponding sub-vector space lattices. So there is, apart from isomorphism to n∈N and a body IK, exactly one projective space of dimension n over IK. Therefore one can also simply speak of the n-dimensional projective space about IK (and use PG (n, IK) as a model, for example). Corresponding considerations apply to the affine geometries AG (A) and AG (n, IK). The finite-dimensional analytically explained projective and affine geometries are thus completely known. You "only" have to get all bodies in body theory, then there is exactly one (projective or affine) geometry of the dimension n above this body according to the specification of a body and the choice of n∈N . One goal of a mathematical theory is to arrange all models belonging to the theory into classes and thus to create a "catalog" for all different models. The associated question is called a classification problem. For geometrically explained geometries, according to the preceding explanations, the classification problem is completely solved when the problem of the classification of bodies is completely solved. 8.4 The Sentences of Desargues and Pappos-Pascal In 8.3 we examined analytically defined projective spaces in the form of the standard geometries PG (V) and finally also classified them. The question of whether all projective geometries in the sense of our definitions from 7.4.1 or 8.2.7 (incidence geometrical variant) can be captured with this remains unanswered. References to geometries that cannot be represented as geometry over a vector space have already been given in § 1 for the case of projective planes. In this section, a few more sentences on this topic are to be presented. 98 § 8 on the concept of projective space 99 8.4.1 Definition (desargue spaces): A projective space (L, ⊓, ⊔) is called desargue space if the following condition (D) applies in it: For each choice of six different points A1, A2, A3, B1, B2, B3 with the properties: • The connecting lines Ai⊔Bi (i = 1, 2, 3) intersect at one of the six points Ai, Bi (i = 1, 2, 3) different point Z • The points A1, A2, A3 are not collinear, the same applies to B1, B2, B3. the three intersections P12: = (A1⊔A2) ⊓ (B1⊔B2), P23: = (A2⊔A3) ⊓ (B2⊔B3), P31: = (A3⊔A1) ⊓ (B3⊔B1) lie on one Straight lines (see Fig. 8.4a) on p. 93) Perhaps a little less technically, condition (D) can also be formulated as follows: There are two triangles A1A2A3 and B1B2B3, where the connecting straight lines Ai⊔Bi (i = 1, 2 , 3) intersect at a point Z from the corresponding pairs of corners Ai, Bi. Then the intersections of corresponding side lines lie on a straight line. Condition (D) is named after the French builder and war engineer Girard DESARGUES (1593–1640), who worked as a geometer in Paris from 1626. P13 B1 A1 P23 A2 B2 Z A3 P12 B3 Figure 8.4a Desargues' condition One can now prove that (D) holds in PG (n, IK). Furthermore one can prove that every projective space in which (D) holds can be represented as geometry PG (n, IK). Finally, it can still be shown that (D) holds in every at least three-dimensional space. 99 N. Christmann: Projective Geometry WS 2007/2008 100, all at least three-dimensional projective geometries are recorded by the standard geometries PG (n, IK). In the case of dimension 2, however, there are exceptions in the form of the non-Desarguean levels. This led to the fact that the projective (and the related affine via slotting / closing) incidence planes became a separate branch of research in mathematics in the 20th century. If one constructs the associated standard geometry PG (V) for a Desargue space, i.e. the coordinate field IK (uniquely determined except for isomorphism) and vector space V (selectable as IKn + 1 in the case of the projective dimension n), then There is one more "little thing" to consider: The field IK does not have to be commutative (cf., for example, the oblique field of the Hamiltonian quaternions). To ensure the commutativity of IK an additional geometric property is required, for this we give the following 8.4.2 definition: A projective space (L, ⊓, ⊔) of dimension ≥ 2 satisfies the condition of Pappos and Pascal ( in short (PP)), if for each choice of the six different points A1, A2, A3, B1, B2, B3 the following applies: Are A1, A2, A3 on a straight line a and B1, B2, B3 on a straight line b and if all points A1, A2, A3, B1, B2, B3 are different from the intersection point Z of the straight lines a and b, then the three intersection points S12: = (A1⊔ B2) ⊓ (B1⊔ A2), S23: = (A2⊔ B3) ⊓ (B2⊔ A3), S31: = (A3⊔ B1) ⊓ (B3⊔ A1) on a straight line (see Fig. 8.4b on p. 94). By specifying that the straight lines a and b intersect at a point Z, it is ensured that all points lie in one plane. Levels with (PP) are also called Pappos levels. Papposian spaces can then also be marked in such a way that they contain a Papposian plane, the isomorphism of all planes then ensures (PP). A3 A2 A1 a Z S12 S23 S31 b B1 B2 B3 The condition (PP) 100 & sect; 8 for the concept of the projective space 101 One can first show that the condition (D) follows from (PP). The reverse of this statement is not true. To show this, one first proves that (PP) holds in Desargue's spaces if and only if the coordinate field IK is commutative. The existence of non-commutative fields (e.g. quaternions) then ensures that (PP) does not hold in all Desarguean spaces. The evidence of the propositions addressed here will be dealt with in the lecture “Fundamentals of Geometry”. Exercise 1: Prove that in PG (n, IK) with a commutative field IK the conditions (PP) and (D) hold. Exercise 2: Show that in projective spaces cozy 7.4.1 the axiom (VY) applies (dimension at least 2). 101 N. Christmann: Projective Geometry WS 2007/2008 102 102

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