Metric absolute geometry

The metric absolute geometry is an axiomatic description of the absolute geometry that a common foundation for models of the Euclidean geometry and non-Euclidean geometry , specifically for elliptical geometries and hyperbolic geometries sets. The term and the axioms come from Friedrich Bachmann , who formulated them in his textbook "Structure of Geometry from the Reflection Concept", proves important conclusions and shows how the two-dimensional, metric absolute geometries, the metric planes , can be embedded in projective planes. Each metric level determines a certain subgroup of the projectivities group of the two-dimensional projective space in which it can be embedded through its "metric". The body is also uniquely determined by the metric level. ${\displaystyle \mathbb {P} ^{2}(K)}$ ${\displaystyle K}$

It should be noted that the term “metric” as used in this context has only distant, formal similarities with the metric of a metric space . The “metric” here determines an orthogonality between straight lines, generally no distance between points. One can describe this orthogonality in the coordinate vector space , the projective space in which the metric plane is embedded, by a symmetrical bilinear form (for this description see projective-metric geometry ). This right angle definition then corresponds formally to the usual definition for the real, Euclidean case by a scalar product , i.e. by a positive-definite symmetrical bilinear form. ${\displaystyle K^{3}}$ ${\displaystyle \mathbb {P} ^{2}(K)}$

This article mainly describes plane metric absolute geometry, its models are called metric planes .

Axioms of plane metric geometry

The axiom system formulated geometrically is equivalent to the axiom system formulated in group theory. In metric geometry, the axiom system formulated in group theory is made the basis, and conclusions are drawn from this system. The geometrical axiom system is quoted here for comparison with other axiom systems of absolute geometry:

Geometric formulation of the axioms

Is given , the elements of the set points , the elements of the set straight lines are called. , the incidence relation is a two-digit symmetric relation . If it applies , then one says “ incised with ” and uses the formulations otherwise usual in geometry. In short: is a (simple) incidence structure , whereby it is not required that the set of points and lines are disjoint. The relation , the orthogonality, can only exist between straight lines, one then says for : is perpendicular to , is a perpendicular of , etc. ${\displaystyle ({\mathfrak {P}},{\mathfrak {S}},I,\perp )}$ ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle {\mathfrak {S}}}$ ${\displaystyle I\subseteq ({\mathfrak {P}}\times {\mathfrak {S}})\cup ({\mathfrak {S}}\times {\mathfrak {P}})\;}$ ${\displaystyle PIg}$ ${\displaystyle P}$ ${\displaystyle g}$ ${\displaystyle ({\mathfrak {P}},{\mathfrak {S}},I)}$ ${\displaystyle \perp \subseteq {\mathfrak {S}}\times {\mathfrak {S}}}$ ${\displaystyle g\perp h}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle g}$ ${\displaystyle h}$

Every bijective self-mapping of the point set in which the incidence and orthogonality are preserved is called orthogonal collineation . An involutorial orthogonal collineation that leaves a straight line fixed point by point is called a reflection on the straight line . ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle g\in {\mathfrak {S}}}$ ${\displaystyle g}$

1. Axioms of incidence: There is at least one straight line and at least three points intersect with each straight line. There is exactly one straight line for two different points, which intersects with both points.
2. Axioms of orthogonality: is perpendicular to , then is perpendicular to (symmetry). Vertical straight lines have one point in common. Through every point there is a perpendicular to every straight line and, if the point intersects with the straight line, only one. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle b}$ ${\displaystyle a}$
3. At every straight line there is at least one reflection (reflection axiom). The composition of reflections on three straight lines , which have a point or a perpendicular in common, corresponds to a reflection on a straight line (theorem of the three reflections). ${\displaystyle a,b,c}$ ${\displaystyle d}$

Group theoretical concepts and agreements

The connection of a group is written multiplicatively or more often by juxtaposition , is its neutral element . The conjugation operates like the exponentiation of the right and is defined accordingly: ${\displaystyle (G,\cdot )}$ ${\displaystyle 1}$

• ${\displaystyle \sigma ^{\alpha }:=\alpha ^{-1}\sigma \alpha \;}$, so is ${\displaystyle \sigma ^{\alpha \beta }=(\sigma ^{\alpha })^{\beta }}$
• A subset of a group is called invariant if it is under the conjugation. A subset is invariant if and only if it always holds for every element and every group element . ${\displaystyle M\subseteq G}$ ${\displaystyle \mu \in M}$ ${\displaystyle \gamma \in G}$ ${\displaystyle \mu ^{\gamma }\in M}$
• In a group the elements of the order are called involutorial , so the neutral element of the group in particular is not an involution! ${\displaystyle 2}$

Basic relation

A nameless relation is defined for involutor group elements ; it turns out that this relation is incidence as well as perpendicular and describes some other relationships: If there are involutor group elements, then the following applies if and only if one of the following equivalent conditions is met: ${\displaystyle \alpha ,\beta }$ ${\displaystyle \alpha ,\beta }$ ${\displaystyle \alpha |\beta }$

• ${\displaystyle \alpha \beta }$ is involutor,
• ${\displaystyle \alpha \beta =\beta \alpha }$and , ${\displaystyle \alpha \neq \beta }$
• ${\displaystyle \alpha ^{\beta }=\alpha }$and (geometric interpretation: the movement maps the "geometric object", e.g. point or straight line, onto itself), ${\displaystyle \alpha \neq \beta }$ ${\displaystyle \beta }$ ${\displaystyle \alpha }$
• ${\displaystyle \beta ^{\alpha }=\beta }$and . ${\displaystyle \alpha \neq \beta }$

The basic relation is symmetrical and non-reflective .

Abbreviations

Instead of " and " is an abbreviation written means that every link related element associated with each right-wing in relation etc. ${\displaystyle \alpha _{1}|\beta }$ ${\displaystyle \alpha _{2}|\beta }$ ${\displaystyle \alpha _{1},\alpha _{2}|\beta }$ ${\displaystyle \alpha _{1},\alpha _{2}|\beta _{1},\beta _{2}}$ ${\displaystyle \alpha _{k}}$ ${\displaystyle \beta _{j}}$

Group-theoretical formulation of the axioms

Axiom D (axiom of the three-sided): There are two perpendicular straight lines , and one straight line that is neither to nor too perpendicular and does not intersect with the point . ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle j}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle gh}$

Basic assumption: There is an invariant generating system of a group consisting of involutor elements . ${\displaystyle {\mathfrak {S}}}$ ${\displaystyle {\mathfrak {G}}}$

The elements of are denoted by small Latin letters (they play the role of the straight line and axis mirrors), the selfreciprocal elements of the group , which as a product of two elements of the generating set are represented - that is in the form of displayable elements of the group - be with capitalized Latin letters (they play the role of points and point reflections). ${\displaystyle {\mathfrak {S}}}$ ${\displaystyle {\mathfrak {G}}}$ ${\displaystyle {\mathfrak {S}}}$ ${\displaystyle ab}$ ${\displaystyle a|b}$

Axiom 1: To there is always a with (existence of the connecting line ). ${\displaystyle P,Q}$ ${\displaystyle g}$ ${\displaystyle P,Q|g}$

Axiom 2: From it follows or (uniqueness of the connecting line, two lines have at most one point of intersection). ${\displaystyle P,Q|g,h}$ ${\displaystyle P=Q}$ ${\displaystyle g=h}$

Axiom 3: If it holds , then there is a such that is (theorem of the three reflections for three copoint lines). ${\displaystyle a,b,c|P}$ ${\displaystyle d}$ ${\displaystyle abc=d}$

Axiom 4: If it holds , then there is a such that is (theorem of the three reflections for three perpendicular lines). ${\displaystyle a,b,c|g}$ ${\displaystyle d}$ ${\displaystyle abc=d}$

Axiom D: There is such a thing that and neither nor nor holds (axiom of the three-sided). ${\displaystyle g,h,j}$ ${\displaystyle g|h}$ ${\displaystyle j|g}$ ${\displaystyle j|h}$ ${\displaystyle j|gh}$

For the geometric meaning of the axioms, compare the figure on the right for axiom D, for axioms 3 and 4 see below

More terms and relations

Points, lines, incidence and perpendicular relations

• The elements of the generating system are interpreted as straight lines. ${\displaystyle {\mathfrak {S}}}$
• The involutive elements for which a representation exists are interpreted as points. ${\displaystyle P\in {\mathfrak {G}}}$ ${\displaystyle P=ab,\;a,b\in {\mathfrak {S}}}$
• Is for a point and a straight line , the product (or equivalent ) in involution, then you say, " and incise" in short . ${\displaystyle P}$ ${\displaystyle g}$ ${\displaystyle Pg}$ ${\displaystyle gP}$ ${\displaystyle P}$ ${\displaystyle g}$ ${\displaystyle PIg:\Leftrightarrow P|g}$
• If the product is involutor for two straight lines , then one says “ is perpendicular to ”, in short . ${\displaystyle a,b}$ ${\displaystyle ab}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a\perp b:\Leftrightarrow a|b}$

Two vertical straight lines define a point, vertical straight lines always intersect, a point commutes with every straight line that cuts with it. Two straight lines commute exactly when they are perpendicular to each other.

${\displaystyle ({\mathfrak {G}},{\mathfrak {S}})}$that is, if the basic assumption and the axioms are fulfilled, the group of motions generated . The geometric structure that results from the movement group through the geometric interpretations is called the group level of . ${\displaystyle ({\mathfrak {P}},{\mathfrak {S}},I,\perp )}$ ${\displaystyle ({\mathfrak {G}},{\mathfrak {S}})}$

Transform

If there is any group element, then according to the definition of the points and straight lines, in particular due to the invariance of under conjugation: ${\displaystyle \alpha \in {\mathfrak {G}}}$ ${\displaystyle {\mathfrak {S}}}$

• for each point the transformed object is a point, ${\displaystyle P}$ ${\displaystyle P^{\alpha }}$
• for every straight line the transformed object is a straight line, ${\displaystyle g}$ ${\displaystyle g^{\alpha }}$
• the incidence relation and the perpendicular relation are retained during transformations.

By “transforming”, ie conjugating the group theory, the group operates on itself and especially on the elements that are geometrically interpreted as straight lines and points. A transformation group of points and straight lines is referred to as a group of movements of the group level of points and straight lines. ${\displaystyle {\mathfrak {G}}}$

First of all, according to the construction, it is clear that the transformation gives an epimorphism of the group to the group of its internal automorphisms . In fact: ${\displaystyle {\mathfrak {G}}}$

• The center of consists only of the one element. And therefore: ${\displaystyle {\mathfrak {G}}}$
• The axiomatically described group of movements is isomorphic to the group of movements of its group level. ${\displaystyle {\mathfrak {G}}}$

Connectivity

If the relation exists for three involutive elements , then a connection is called from . Important concrete geometric interpretations of connectivity: ${\displaystyle \rho _{1},\rho _{2},\sigma }$ ${\displaystyle \rho _{1},\rho _{2}|\sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \rho _{1},\rho _{2}}$

• ${\displaystyle A,B|g}$: The points are connected by the straight line. - The existence of this connecting line is assured by axiom 1, its uniqueness in the case by axiom 2. ${\displaystyle A,B}$ ${\displaystyle g}$ ${\displaystyle A\neq B}$
• ${\displaystyle g,h|A}$: The two straight lines are copunctal . If there is a connection point for two different straight lines, then according to axiom 2 there is no further one.
• ${\displaystyle g,h|l}$: The straight lines have a common perpendicular . They are then also called perpendicular . The geometrical statement: “Every straight line has a perpendicular” follows from the fact that every straight line contains a point (not trivial to prove from the axioms of group theory) and then simply from the definition of a point as the product of two perpendicular straight lines. ${\displaystyle g,h}$ ${\displaystyle l}$

The Lotgleichheit two lines is in the absolute geometry of an equivalence relation that the parallelism in terms of incidence can replace often.

Tufts

Straight lines are said to be “they are in a tuft”, if there is a straight line. The three-digit relation in-tuft-lying obviously has the following properties: ${\displaystyle a,b,c\in {\mathfrak {S}}}$ ${\displaystyle abc\in {\mathfrak {S}}}$ ${\displaystyle B(a,b,c)}$

• (three-digit) reflexivity : If at least two of the garades match, then applies . For example , since is involution, is straight because is invariant. ${\displaystyle a,b,c}$ ${\displaystyle B(a,b,c)}$ ${\displaystyle B(a,a,c):aac=c}$ ${\displaystyle a}$ ${\displaystyle B(a,b,a):aba=b^{a}\;}$ ${\displaystyle {\mathfrak {S}}}$
• (Three-digit) symmetry : If it applies, the relation also applies to every permutation of the straight line. For example ; etc. ${\displaystyle B(a,b,c)}$ ${\displaystyle abc\in {\mathfrak {S}}\Rightarrow bca=aa\cdot bca=a(abc)a=(abc)^{a}}$ ${\displaystyle \;abc\in {\mathfrak {S}}\Rightarrow abc=(abc)^{-1}=cba}$

Not so obvious is that:

• (three-digit) transitivity : applies and , then follows . ${\displaystyle B(a,b,c),B(a,b,d)}$ ${\displaystyle a\neq b}$ ${\displaystyle B(a,c,d)}$

The tufts relation generated as a three-digit "equivalence relation" a division of the set of lines in pencil of lines with the following properties that the equivalence class properties of a two-digit equivalence relation similar to (and purely set-theoretic inferences from said relation properties, the three-digit reflexivity, symmetry and transitivity are):

• The relation applies to three different straight lines in a tuft.
• If the relation applies to two different straight lines in a tuft and a third straight line, then this also lies in the tuft.
• Two different straight lines of a tuft clearly determine this.
• Two different tufts have at most one element in common.

Geometrically , there are three types of line tufts in absolute geometry:

1. Tufts of copoint straight lines (the classic tufts as they appear in projective space ),
2. Tufts of perpendicular straight lines, tufts of perpendicular ,
3. “Free” tufts (neither point nor plumb tufts).

An “all or nothing” principle applies to the first two types: if a point bundle is also a solder bundle , then it is all! This is exactly the case when the geometry is elliptical, in which case there are no free tufts either. In any non-elliptical geometry, the three cases are mutually exclusive. All three species are in embedding the absolute geometry in a projective ideal plane for ideal points . Exactly these ideal points are then the points of the ideal plane.

Mirror image position, angles, rotations

The fourth mirror axis required in axiom 3 for three copunctal straight lines . It then applies . When executed one after the other, the pairs result in the same transformation, a rotation around . So they represent the same "angle of rotation". ${\displaystyle d}$ ${\displaystyle a,b,c}$ ${\displaystyle ab=dc}$ ${\displaystyle (a,b),(d,c)}$ ${\displaystyle o}$

Are the lines lotgleich , in the figure at right angles to all , then the level required by Axiom 4 4. mirroring axis is likewise vertically . ${\displaystyle a,b,c}$ ${\displaystyle g}$ ${\displaystyle d}$ ${\displaystyle g}$

If this applies to 4 straight lines , then according to Hjelmslev one says: “The straight lines are mirror images of one another with respect to the straight lines .” Then the straight lines are also mirror images of one another , compare the figures. If there is , that is, and is equivalent , then you say: "The straight lines are mirror images of each other in relation to the axis ". It does not follow from the axioms that there is such a center line for two given different straight lines . ${\displaystyle a,b,c,d\;ab=dc}$ ${\displaystyle b,d}$ ${\displaystyle a,c}$ ${\displaystyle a,c}$ ${\displaystyle b,d}$ ${\displaystyle a=c}$ ${\displaystyle ab=da}$ ${\displaystyle b^{a}=d}$ ${\displaystyle b,d}$ ${\displaystyle a}$ ${\displaystyle b,d}$

For pairs of lines at a point tufts of an equivalence relation is given by the symmetrical position: . Two pairs that are in the relation to one another determine the same "rotation" around the point with which they jointly incise. Therefore, one can understand the equivalence classes as angles . The (Euclidean) angle between two axes drawn in the figure on the right (copunctal straight line) is half the Euclidean angle of rotation of the rotation , which is given by the transformation . In the second figure (perpendicular straight line) the drawn (Euclidean) shift is half of the translation that is given by transforming with (left) or  (right) in the metric-Euclidean case. ${\displaystyle (a,b),(d,c)}$ ${\displaystyle (a,b)\sim (d,c):\Leftrightarrow ab=dc}$ ${\displaystyle \sim }$ ${\displaystyle O}$ ${\displaystyle ab=dc}$ ${\displaystyle ab}$ ${\displaystyle cd}$

Distinguishing axioms

The axiom of non-connectivity in a metric-Euclidean plane: The straight lines and have a common perpendicular , i.e. they are connected perpendicular. The straight line (green) cuts in so is point bonded and therefore can neither with nor connected by a solder (since it is assumed). If there is no intersection of and , indicated by the red question mark, then the axiom applies in this metric-Euclidean plane . The axiom does not follow from the existence of the point in question ! ${\displaystyle \neg \mathbf {V} }$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle l}$ ${\displaystyle c}$ ${\displaystyle b}$ ${\displaystyle S}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \mathbf {R} }$ ${\displaystyle c}$ ${\displaystyle a}$ ${\displaystyle \neg \mathbf {V} }$ ${\displaystyle \mathbf {V} }$

The different types of absolute geometries fork at the following axioms:

• Connectivity axiom, it plays the role of a completeness axiom.
• Existence of a polar triangle (characterizes the elliptical geometry)
• Existence of a right-hand side (essentially characterizes Euclidean geometry, together with the connectivity axiom precisely this).
• Axiom limits the possibilities of non-connectivity. Together with the negation of the axiom of connectivity, it characterizes hyperbolic geometry. ${\displaystyle \mathbf {H} }$

Polar triangle and elliptical planes

A polar triangle is characteristic of the elliptical geometry. All angles marked in red are right. The point is the pole of the straight line , this is the polar of . All straight lines cut through and are plumb bobs on . Every straight line that is too perpendicular goes through the point . ${\displaystyle C}$ ${\displaystyle c}$ ${\displaystyle C}$ ${\displaystyle C}$ ${\displaystyle c}$ ${\displaystyle c}$ ${\displaystyle c}$ ${\displaystyle C}$

Geometrically, a polar triangle is a three-sided (three straight lines that are not copunctal), in which each straight line is perpendicular to the other two. In the language of the group level, there are three different ones , in which two different ones are related to one another, i.e. with the different equivalent statements and interpretations: ${\displaystyle a,b,c\in {\mathfrak {S}}}$ ${\displaystyle |}$

Each product of two of these elements

• is an involution, i.e. H. one point,
• is the same as the reverse product.

• From the second statement it follows that the three elements commute.

If one calls z. B. the product point , then are involutorial, so lies on and , so is the uniquely determined intersection of this vertical straight line. Analogously , the other two corners of the three sides. Since the three straight lines commute, is . but cannot be involutor, otherwise the straight line would also contain what would contradict the uniqueness of the perpendicular at one point of the straight line . So and the corresponding applies to every permutation of the three lines in the product. ${\displaystyle ab=:C}$ ${\displaystyle C}$ ${\displaystyle aC=aab=b,Cb=abb=a}$ ${\displaystyle C}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle B:=ac}$ ${\displaystyle A:=bc}$ ${\displaystyle (Cc)^{2}=(abc)^{2}=a^{2}b^{2}c^{2}=1}$ ${\displaystyle Cc}$ ${\displaystyle C}$ ${\displaystyle c}$ ${\displaystyle c}$ ${\displaystyle abc=1}$

If, conversely, applies to three straight lines, they must be different in pairs (otherwise the product would either be a straight line or a transformed straight line, i.e. again a straight line and in any case an involution). It follows and again one has three different paired perpendicular straight lines which, as products of straight lines, are also points. ${\displaystyle abc=1}$ ${\displaystyle ab=c,a=cb,b=ac}$

Axiom ( There is no polar triangle ) It always is . ${\displaystyle \neg \mathbf {P} }$ ${\displaystyle abc\neq 1}$

Axiom ( There is a polar triangle ) There is with . ${\displaystyle \mathbf {P} }$ ${\displaystyle a,b,c}$ ${\displaystyle abc=1}$

From the axiom it follows with the common axioms: No product of an odd number of generators is equal to a product of an even number of generators. In this case the subgroup of group elements that can be represented by an even number of generators is a subgroup of index 2 in . Then: ${\displaystyle \neg \mathbf {P} }$ ${\displaystyle {\mathfrak {U}}<{\mathfrak {G}}}$ ${\displaystyle {\mathfrak {G}}}$

• It is , ${\displaystyle {\mathfrak {U}}=\{ab:a,b\in {\mathfrak {S}}\}}$
• Each point is one of the sub-group , ${\displaystyle {\mathfrak {U}}}$
• Each element of the (real) secondary class of , the class of group elements that can be represented by an uneven number of generating axis reflections, can be represented as a sliding reflection , ${\displaystyle {\mathfrak {U}}}$ ${\displaystyle aB}$
• no product of two points is involutor,
• no point is equal to a straight line
• the involutor elements of are precisely those points ${\displaystyle {\mathfrak {U}}}$
• the involutive elements of the minor class are precisely the straight lines.

From the axiom it follows with the common axioms: Every product of an even number of generators is equal to a product of an odd number of generators, it is , and also: ${\displaystyle \mathbf {P} }$ ${\displaystyle {\mathfrak {U}}={\mathfrak {G}}}$

• Each group member is in the form represented, ${\displaystyle ab}$
• so every straight line is equal to a point ; it is called the pole of the straight line, the straight line polar of its pole ${\displaystyle c}$ ${\displaystyle C}$
• ${\displaystyle c=ab}$means at the same time , so every point as a corner belongs to a polar triangle, every straight line as a side to a polar triangle. ${\displaystyle abc=1}$

Right and metric-Euclidean planes

A right side, the angles marked in red are right. The existence of a right side characterizes the metric-Euclidean planes among the plane models of absolute geometry.

• Axiom : There is with and , that is, there is a right side, cf. the figure on the right. ${\displaystyle \mathbf {R} }$ ${\displaystyle a,b,c,d}$ ${\displaystyle a,c\perp b,d}$ ${\displaystyle a\neq c,b\neq d}$
• Axiom : From follows or . ${\displaystyle \neg \mathbf {R} }$ ${\displaystyle a,c\perp b,d}$ ${\displaystyle a=c}$ ${\displaystyle b=d}$

Geometries for which applies are called metric-Euclidean . In a metric-Euclidean plane, the perpendicular is defined as parallelism: should apply if the two straight lines have a common perpendicular. Plumb lines have none or all points in common, so they are non-intersecting or identical. Two straight lines that do not intersect, do not necessarily have to have a common perpendicular (see the models). The axiom of connectivity excludes the existence of such non-intersecting elements that are not perpendicular. ${\displaystyle \mathbf {R} }$ ${\displaystyle a\|_{R}b}$ ${\displaystyle \mathbf {V} ^{*}}$

From the axiom follows the axiom , equivalent: from the axiom follows the axiom (always on the basis of the common axioms). A polar triangle cannot coexist with a right side. Somewhat more formal: No metric-Euclidean geometry is elliptical, no elliptical metric-Euclidean. ${\displaystyle \mathbf {P} }$ ${\displaystyle \neg \mathbf {R} }$ ${\displaystyle \mathbf {R} }$ ${\displaystyle \neg \mathbf {P} }$

From the axiom that follows Rechtsseitsatz : from and follow . Compare the picture. ${\displaystyle \mathbf {R} }$ ${\displaystyle a,b\perp c}$ ${\displaystyle a\perp d}$ ${\displaystyle b\perp d}$ ${\displaystyle (\mathbf {R} )}$

Despite a certain similarity to Axiom , the right-hand proposition is a closure proposition with orthogonality - from three perpendicular relations in a four-sided follows the fourth - while the axiom is a pure existence statement. ${\displaystyle \mathbf {R} }$

Connectivity axiom and axiom H: Hyperbolic planes

Connectivity in Klein's circular disk model: two hyperbolic straight lines (green) whose projective support lines meet on the outside of the circle (at the ideal point ) have the actual part of the polar of (red), which is again inside the circle, as the only common perpendicular. Straight lines that meet inside the circle are point-connected. Two different straight lines can only be connected point or perpendicular, never both ( assuming axiom ). ${\displaystyle P}$ ${\displaystyle l}$ ${\displaystyle P}$ ${\displaystyle \neg \mathbf {P} }$

Here “connectivity” is understood in a somewhat narrower sense than above. The axioms can equivalently also be formulated in a more general way, but they should formally presuppose little here as axioms . The formulation given here is the one given by Bachmann to describe hyperbolic geometry:

• Axiom (connectivity axiom ) For two lines there is always a point with or a line with . Formulated geometrically: two straight lines have either a point or a perpendicular in common. ${\displaystyle \mathbf {V} ^{*}}$ ${\displaystyle a,b}$ ${\displaystyle C}$ ${\displaystyle a,b|C}$ ${\displaystyle g}$ ${\displaystyle a,b|g}$

If there is such an intersection or a common perpendicular for two straight lines, then they are called connectable (in the narrower sense).

• Axiom : There are those that cannot be connected. ${\displaystyle \neg \mathbf {V} ^{*}}$ ${\displaystyle a,b}$
• Axiom : If and are and and are non-binding, then or or . ${\displaystyle \mathbf {H} }$ ${\displaystyle a,b,c|P}$ ${\displaystyle a,g}$ ${\displaystyle b,g}$ ${\displaystyle c,g}$ ${\displaystyle a=b}$ ${\displaystyle a=c}$ ${\displaystyle b=c}$

In other words, the axiom says : In a cluster of copunctival straight lines there are at most two different ones that cannot be connected to a given straight line (they have neither a point nor a perpendicular in common with it). ${\displaystyle \mathbf {H} }$ ${\displaystyle g}$

The axiom of hyperbolic geometry in Klein's circular disk model: The point cluster through P contains exactly two straight lines that meet g at an “end”, ie on the peripheral circle. Only these two straight lines of the point cluster cannot be connected with g. Note that the ends of are improper ideal points of the hyperbolic plane. To the axiom : The perpendiculars through the center of the circle are Euclidean perpendiculars to the circular chords in the circular disk model. If one thinks of being drawn on this perpendicular line , then fulfilling and, together with each of the drawn straight lines, the axiom as a straight line that cannot be connected. ${\displaystyle \mathbf {H} }$ ${\displaystyle g}$ ${\displaystyle D^{*}}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle P}$ ${\displaystyle g}$ ${\displaystyle j}$

A geometry and is satisfied as hyperbolic geometry called. ${\displaystyle \neg \mathbf {V} ^{*}}$ ${\displaystyle \mathbf {H} }$

The axiom system of hyperbolic geometry expanded by the axioms of incompleteness and additional axioms can be described as equivalent, but a little more briefly, by combining the axiom with the common axiom ${\displaystyle \neg \mathbf {V} ^{*}}$ ${\displaystyle \mathbf {H} }$ ${\displaystyle \neg \mathbf {V} ^{*}}$ ${\displaystyle D}$

${\displaystyle D^{*}}$: There is , with and non-connectable (in the narrower sense described above: have neither a point nor a perpendicular in common). ${\displaystyle g,h,j}$ ${\displaystyle g\perp h}$ ${\displaystyle g,j}$ ${\displaystyle g,j}$

So one can also describe the hyperbolic groups of motion equally by the axiom system that is obtained if one additionally demands the axiom and replaces the axioms and by . ${\displaystyle \mathbf {H} }$ ${\displaystyle \neg \mathbf {V} ^{*}}$ ${\displaystyle D}$ ${\displaystyle D^{*}}$

The following also applies:

• From this follows the axiom , ("every elliptical geometry is complete"), ${\displaystyle \mathbf {P} }$ ${\displaystyle \mathbf {V} ^{*}}$
• from and the invalidity follows from ( "no incomplete metric, Euclidean plane is hyperbolic"), ${\displaystyle \neg \mathbf {V} ^{*}}$ ${\displaystyle \mathbf {R} }$ ${\displaystyle \mathbf {H} }$
• Every hyperbolic geometry is "incomplete" because of it . ${\displaystyle \neg \mathbf {V} ^{*}}$

Therefore a group level can only belong to one of the classes “elliptical”, “metric-Euclidean” or “hyperbolic” geometry.

Semi-elliptical planes

A group level which fulfills the axioms , and is called semi-elliptical ; it can also be equivalently defined by the axiom ${\displaystyle \neg \mathbf {R} }$ ${\displaystyle \mathbf {V} ^{*}}$ ${\displaystyle \neg \mathbf {P} }$

"Two different straight lines either have exactly one perpendicular or exactly one point in common, never both."

describe. When embedding in a projective-metric space, semi-elliptical groups are completed to form elliptical groups; Their movement group turns out to be a full elliptical group , but the axiomatically described generating system , that is, geometrically the straight lines of the plane, form a real subset of the projective set of straight lines and here, in group theory, the set of all involutorial group elements . ${\displaystyle {\mathfrak {G}}}$ ${\displaystyle {\mathfrak {S}}}$ ${\displaystyle {\mathfrak {S}}\subsetneq {\mathfrak {S}}'}$ ${\displaystyle {\mathfrak {S}}'}$

Embedding in the projective-metric ideal plane

Apart from the equivalence of the geometrically formulated axiom system to that formulated in terms of group theory, it is initially not clear in what sense the axiomatically described groups with a generating system made up of involutor elements are "geometrical mapping groups". The embedding in a projective-metric level (the ideal level of the group level) makes this clear and does four things:

1. For the incidence relation: The set of axis reflections is mapped injectively onto a subset of the set of ideal lines (the actual ideal lines) and the set of line bundles bijectively onto the point set of the projective ideal plane. The incidence relation for actual points (point clusters of the group level) and actual straight lines is retained. ${\displaystyle {\mathfrak {S}}}$
2. For orthogonality: The orthogonality in the group level is continued to an ideal straight line-ideal pole relationship in the projective ideal level. Two straight lines of the group level (actual ideal straight lines) are perpendicular to each other if one is contained in the ideal pole (a tuft) of the other. The continuation on any ideal straight line and its poles is clear.
3. For the group structure: The axiomatically described movement group proves to be isomorphic to a well-defined subgroup of the orthogonal group of the projective-metric plane in which it is embedded. The orthogonal group is the subgroup of the projective group that leaves the polar structure invariant. The generating system clearly defines this subgroup , but the generating system is generally not clearly defined by. ${\displaystyle {\mathfrak {G}}}$ ${\displaystyle \operatorname {PGL} (3,K)}$ ${\displaystyle {\mathfrak {S}}}$ ${\displaystyle {\mathfrak {G}}}$ ${\displaystyle {\mathfrak {S}}}$ ${\displaystyle {\mathfrak {G}}}$
4. For the algebraic representation using coordinates: The movement group determines a unique coordinate body (the coordinate body of the projective-metric plane) and, in the case of a hyperbolic movement group, its generating system also defines a unique arrangement of this body. Additional properties of the movement group (halving right angles, geometric arrangement, free mobility , ...) often correspond to a certain class of bodies. ${\displaystyle K}$ ${\displaystyle {\mathfrak {S}}}$

In addition , one observes that the half reflections, with which potential ideal straight lines are (technically) "contracted" onto actual ideal straight lines, represent injective maps on the set of actual ideal straight lines, which are only surjective if the geometry is either elliptical or Euclidean . It follows that only in these two main cases, the metric level be finally could . - In fact, finite models only exist for the Euclidean case. In the elliptical case, the required projective, elliptical polarities only exist over infinite bodies, see Correlation (Projective Geometry) # Polarities over finite spaces .

Relationships between group level and ideal level

• Every straight line at the group level of is an ideal point. The point clusters at the group level are the actual ideal points. This shows that the ideal plane is minimal in terms of the point set. ${\displaystyle ({\mathfrak {G}},{\mathfrak {S}})}$
• Each actual ideal point (i.e. each point cluster) also incises in the ideal plane exclusively with actual ideal straight lines . "In an arbitrary point absolute geometry and projective geometry are the same." This is called the group level because of this characteristic, a local full incremental level of their ideal plane.
• Each movement determines a clear projectivity in the ideal plane, which depicts the totality of the actual ideal points and straight lines in an incidence-true manner through transformation. ${\displaystyle \gamma \in {\mathfrak {G}}}$
• The reflection in the group level on a straight line , that is, the transformation with induces the involutive perspectivity in the ideal level with the actual ideal straight line as the axis and its pole as the center. ${\displaystyle c}$ ${\displaystyle c\in {\mathfrak {S}}}$ ${\displaystyle g(c)}$ ${\displaystyle G(c)}$

• Every ideal line is determined as a set of fixed points (of fixed ideal points) of an involutorial, polarity-true perspective , with an ideal point to which it does not belong as the center - the point itself is the only fixed point of which should not belong to the ideal line. In the metric-Euclidean special case, the “distance line” is also an improper ideal line, it cannot be reflected. It is the set of (ideal) poles of all actual ideal lines, in other words: the set of all solder tufts. ${\displaystyle \pi _{Z}}$ ${\displaystyle Z}$ ${\displaystyle Z}$ ${\displaystyle \pi _{Z}}$

Multi-dimensional generalization

The axioms for an absolute geometry formulated in group theory can be generalized to spaces of the dimension : Basic assumption: Be a generated group, invariant generating system of involutor elements. The basic relation for involutor group elements is defined as in the two-dimensional case. It is agreed that expressions like are used as abbreviations for conjunctions: If there is at least one symbol between two elements , then the relation should exist between them. The example says about the relationship between and against nothing. ${\displaystyle n>2}$ ${\displaystyle ({\mathfrak {G}},{\mathfrak {S}})}$ ${\displaystyle {\mathfrak {S}}}$ ${\displaystyle |}$ ${\displaystyle \alpha _{1},\alpha _{2}|\alpha _{3},\alpha _{4}|\alpha _{5}}$ ${\displaystyle \alpha _{j},\alpha _{k}}$ ${\displaystyle |}$ ${\displaystyle \alpha _{1}|\alpha _{5}}$ ${\displaystyle \alpha _{3}}$ ${\displaystyle \alpha _{4}}$

Each element of is interpreted as a reflection at a hyperplane and designated with a lowercase letter. is defined as the set of involutive products with and interpreted as the set of point reflections. The point reflections are denoted with capital letters. Common axioms: ${\displaystyle {\mathfrak {S}}}$ ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle a_{1}\cdot a_{2}\cdots a_{n}}$ ${\displaystyle a_{1}|a_{2}|\ldots |a_{n}}$

Axiom 1 _n ^* : To there is a with . ${\displaystyle a_{1},\ldots ,a_{n-1},A}$ ${\displaystyle a}$ ${\displaystyle a|a_{1},\ldots ,a_{n-1},A}$

Axiom 1 _n : For with there is a with . ${\displaystyle a_{1},\ldots ,a_{n-2},A,B}$ ${\displaystyle a_{1}|\cdots |a_{n-2}|A,B}$ ${\displaystyle a}$ ${\displaystyle a|a_{1},\ldots ,a_{n-2},A,B}$

Axiom 2 _n : From follows or . ${\displaystyle a_{1}|\cdots |a_{n-2}|a,b|A,B}$ ${\displaystyle a=b}$ ${\displaystyle A=B}$

Axiom 3 _n : Off and follows . ${\displaystyle a_{1}|\cdots |a_{n-2},A|a,b,c}$ ${\displaystyle a_{n-2}\neq A}$ ${\displaystyle abc\in {\mathfrak {S}}}$

Axiom 4 _n : Off follows . ${\displaystyle a_{1}|\cdots |a_{n-1}|a,b,c}$ ${\displaystyle abc\in {\mathfrak {S}}}$

Axiom X _n : There is with . ${\displaystyle a_{1},a_{n}}$ ${\displaystyle a_{1}|\cdots |a_{n}}$

Axiom D _n : To with there is such a thing that and neither nor holds. ${\displaystyle a_{1},a_{n}}$ ${\displaystyle a_{1}|\cdots |a_{n}}$ ${\displaystyle a}$ ${\displaystyle a|a_{1},a_{n-1}}$ ${\displaystyle a=a_{n}}$ ${\displaystyle a|a_{n}}$

The geometric interpretations:

• Existence of a perpendicular, axiom 1 _n ^* : To a point and hyperplanes there is a hyperplane through which is perpendicular to the given ones. ${\displaystyle A}$ ${\displaystyle n-1}$ ${\displaystyle A}$
• Existence of a connection, axiom 1 _n : To two points and pairwise perpendicular hyperplanes that intersect with both points, there is a hyperplane through the points that is perpendicular to the given hyperplanes. ${\displaystyle n-2}$
• Uniqueness of the connection, axiom 2 _n : If the two points from axiom 1 _{n are} different, then there is only one such hyperplane.
• Existence of the fourth reflection hyperplane , axiom 3 _n : If a point and pairwise perpendicular hyperplanes are given, of which all but at most one also incise and this is also not polar too , then there are three hyperplanes that are perpendicular to the given ones, a fourth reflection (shiper plane). ${\displaystyle A}$ ${\displaystyle n-2}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$
• Dimension, axiom X _n : There are pairwise perpendicular hyperplanes. (From this it follows with the uniqueness of the connection that the space is exactly -dimensional). ${\displaystyle n}$ ${\displaystyle n}$
• Axiom D _n : To pairwise perpendicular hyperplanes there is another one that is perpendicular to them, but different from the last given one and not perpendicular to it. ${\displaystyle n}$ ${\displaystyle n-1}$

Elliptic axiom:

Axiom (from the polar simplex): There is with . ${\displaystyle \mathbf {P} _{n}}$ ${\displaystyle a_{1},\ldots ,a_{n+1}}$ ${\displaystyle a_{1}|\cdots |a_{n+1}}$

Models of metric absolute geometry

Metric-Euclidean models with Euclidean parallelism

• All finite models of metric absolute geometry satisfy the axiom of parallels and are therefore affine spaces . Finite plane models that satisfy the common axioms 1 to 4 and D described here are precisely the finite pre-Euclidean planes . The group of motions of such a pre-Euclidean plane is the group created by the perpendicular axis reflections . ${\displaystyle {\mathfrak {G}}}$ ${\displaystyle {\mathfrak {S}}}$
• Each pre-Euclidean plane provides a model of a metric plane. This then also fulfills the axioms . ${\displaystyle \mathbf {R} ,\mathbf {V} ^{*}}$
• The pre-Euclidean planes can be extended to -dimensional affine spaces with orthogonality . These then provide models for metric-Euclidean spaces with Euclidean parallelism. ${\displaystyle n}$ ${\displaystyle (n>2)}$

Hyperbolic levels

• The Klein disc model of hyperbolic geometry, see Hyperbolic Geometry . Except for isomorphism, this is the only model of a real hyperbolic plane.

• Generally it can be on every parent body a group theoretical model for a hyperbolic plane specify: The motion group is so full Projektivitätengruppe a projective line over . The axis reflections are all involutor projectivities with a negative determinant. ${\displaystyle K}$ ${\displaystyle {\mathfrak {G}}}$ ${\displaystyle \operatorname {PGL} (2,K)}$ ${\displaystyle K}$ ${\displaystyle {\mathfrak {S}}}$ ${\displaystyle {\mathfrak {G}}=\operatorname {PGL} (2,K)}$

For the group theoretical model is isomorphic to the Klein disk model. Since the field of real numbers allows only one arrangement and every hyperbolic polarity is equivalent to the standard bilinear form with the shape matrix , it can be shown that the circular disk model is the only hyperbolic level (in the sense of the axioms formulated in this article) above the real numbers except for isomorphism is. ${\displaystyle K=\mathbb {R} }$ ${\displaystyle \operatorname {diag} (1,-1,-1)}$

The real example shows how the group theoretical model comes about: The movement group of the Klein model, that is, the group of collineations of the projective plane , which map the inner circle of the unit circle so that the hyperbolic orthogonality is preserved, consists exactly those projectivities that represent the unit circle on themselves. This subgroup of is isomorphic to for every non-degenerate conic section (for formally real fields ). ${\displaystyle \mathbb {P} ^{2}(\mathbb {R} )}$ ${\displaystyle \operatorname {PGL} (3,K)}$ ${\displaystyle \operatorname {PGL} (2,K)}$ ${\displaystyle K}$

Elliptical models

→ Main article : Elliptical geometry

The spherical geometry on the unit sphere in real three-dimensional space, in which opposing points (" antipodes ") are identified with one another if the corresponding orthogonality of great circles is also defined. See also Correlation (Projective Geometry) #An elliptical polarity . This is the (apart from isomorphism the only one) model of plane, real elliptical geometry. ${\displaystyle S}$ ${\displaystyle S}$

In general, for every elliptical projective-metric space, the full group of motions with the set of all involutive elements of this group as the generating system is a model of a -dimensional elliptical geometry. The existence of a projective elliptical polarity in this space is necessary and sufficient for an elliptical “metric” to be explained , with which it becomes an elliptical projective-metric space. ${\displaystyle {\mathfrak {G}}:=O_{n+1}(K,B)}$ ${\displaystyle {\mathfrak {S}}=\{a\in {\mathfrak {G}}\setminus \{1\}:a^{2}=1\}}$ ${\displaystyle n}$ ${\displaystyle \mathbb {P} ^{n}(K)}$

In general, two elliptical planes need not be isomorphic over the same body. They are then (sufficient condition) if the elliptical polarity of one plane can be brought to the form it has on the other plane by choosing a suitable basis for the coordinate vector space . ${\displaystyle K^{n+1}}$

literature

Original literature

• Arnold Schmidt : The duality of incidence and perpendicularity in absolute geometry . In: Math. Ann. tape 118 , 1943, pp. 609-625 .
• Johannes Hjelmslev : New justification of the plane geometry . In: Math. Ann. tape 64 , 1907, pp. 449-474 .

To the multidimensional (n> 2) generalization

• H. Kinder: Elliptic geometry of finite dimension . In: Arch. Math. Band 21 , 1970, pp. 515-527 .
• Gerhard Huebner: Classification of n-dimensional absolute geometries . In: Treatises from the Mathematical Seminar of the University of Hamburg . tape 33 , 1969, p. 165-182 .

Textbook (main source)

• Friedrich Bachmann: Structure of geometry from the concept of reflection . 2nd supplemented edition. Springer, Berlin / Heidelberg / New York 1973, ISBN 3-540-06136-3 .

References and comments

1. ↑ The basic idea for a “reflection geometry” comes from Hjelmslev, z. B. Hjelmslev (1907), Bachman, in his words, “reduced” a system of axioms by Schmidt (Schmidt, 1943).
2. ↑ Bachmann (1973)
3. ↑ Bachmann (1973), p. 32.
4. ↑ If one formulates the geometry in a group-theoretical way, as here, it results that points and straight lines cannot be distinguished as group elements in elliptical geometry. Nevertheless, with many formulations in which perpendicular or incidence relations occur, one is formally on the safe side if one thinks the set of points and lines to be “made disjoint”.
5. ↑ From this statement of uniqueness follows the (much weaker) “simplicity” of the plane as an incidence structure: A straight line is completely determined by the set of points which it incurs. It is therefore sufficient to define the morphisms , here the orthogonal collineations, as point mappings.
6. ↑ Bachmann (1973), § 3.7 sentence 18
7. ↑ Bachmann (1973), § 3.7 sentence 19
8. ↑ Bachmann (1973), §4.4 sentence 6 (transitivity theorem)
9. ↑ Hjelmslev (1907)
10. ↑ Please note here: pole and polar are identical elements in the group, but never incident as geometric objects, because this means that there is no involution. ${\displaystyle C}$ ${\displaystyle c}$ ${\displaystyle cC=1}$ ${\displaystyle cC}$
11. ↑ Bachmann (1973), §6,8, sentence 13 (right end sentence)
12. ↑ ^{a b c d} Bachmann (1973) §14.1 The axioms of hyperbolic groups of motion
13. ↑ Bachmann (1973) §6.12 Justification of Absolute Geometry
14. ↑ In fact, the embedding image has to be composed of several half-reflections that act on straight lines and their "mirrored inverses" that act on points, and the polarity and generating system must be reconstructed from the actual objects. The basic idea goes back to Hjelmslev. Bachmann (1973) §6.10 Justification of Metric Geometry
15. ↑ Since the actual ideal straight line never incurs with its pole, there is only an involutorial perspective with these specifications.
16. ↑ Bachmann (1973), §6.5 sentence 10
17. ↑ Projective is every involutorial perspective with the distant line as the axis and an actual point as the center (an actual point reflection on ) such an axis reflection. But there is no pole to the distant line. ${\displaystyle Z}$ ${\displaystyle Z}$
18. ↑ Bachmann (1973), §20.9 -dimensional absolute geometry. ${\displaystyle n}$
19. ↑ Children (1970)