Editing Projective line over a ring

{{short description|Projective construction in ring theory}}
[[File:Projectivisation_F7P%5E1.svg|thumb|200px|Eight colors illustrate the projective line over Galois field GF(7)]]

In [[mathematics]], the '''projective line over a ring''' is an extension of the concept of [[projective line]] over a [[field (mathematics)|field]]. Given a [[ring (mathematics)|ring]] ''A'' (with 1), the projective line P<sup>1</sup>(''A'') over ''A'' consists of points identified by [[projective coordinates]].  Let ''A''<sup>×</sup> be the [[group of units]] of ''A''; pairs {{nowrap|(''a'', ''b'')}} and {{nowrap|(''c'', ''d'')}} from {{nowrap|''A'' × ''A''}} are related when there is a ''u'' in ''A''<sup>×</sup> such that {{nowrap|1=''ua'' = ''c''}} and {{nowrap|1=''ub'' = ''d''}}. This relation is an [[equivalence relation]]. A typical [[equivalence class]] is written {{nowrap|''U''[''a'', ''b'']}}.

{{nowrap|1=P<sup>1</sup>(''A'') = {{mset| ''U''[''a'', ''b''] | ''aA'' + ''bA'' {{=}} ''A'' }}}}, that is, {{nowrap|''U''[''a'', ''b'']}} is in the projective line if the [[right ideal|one-sided ideal]] generated by ''a'' and ''b'' is all of&nbsp;''A''.

The projective line P<sup>1</sup>(''A'') is equipped with a [[homography#Homography groups|group of homographies]]. The homographies are expressed through use of the [[matrix ring]] over ''A'' and its group of units ''V'' as follows: If ''c'' is in Z(''A''<sup>×</sup>), the [[center (group theory)|center]] of ''A''<sup>×</sup>, then the [[Group action (mathematics)|group action]] of matrix <math>\left(\begin{smallmatrix}c & 0 \\ 0 & c \end{smallmatrix}\right)</math> on P<sup>1</sup>(''A'') is the same as the action of the identity matrix. Such matrices represent a [[normal subgroup]] ''N'' of ''V''. The homographies of P<sup>1</sup>(''A'') correspond to elements of the [[quotient group]] {{nowrap|''V''{{hsp}}/{{hsp}}''N''}}.

P<sup>1</sup>(''A'') is considered an extension of the ring ''A'' since it contains a copy of ''A'' due to the embedding 
{{nowrap|''E'' : ''a'' → ''U''[''a'', 1]}}.  The [[multiplicative inverse]] mapping {{nowrap|''u'' → 1/''u''}}, ordinarily restricted to ''A''<sup>×</sup>, is expressed by a homography on P<sup>1</sup>(''A''):
: <math>U[a,1]\begin{pmatrix}0&1\\1&0\end{pmatrix} = U[1, a] \thicksim U[a^{-1}, 1].</math>
Furthermore, for {{nowrap|''u'',''v'' ∈ ''A''<sup>×</sup>}}, the mapping {{nowrap|''a'' → ''uav''}} can be extended to a homography:
: <math>\begin{pmatrix}u & 0 \\0 & 1 \end{pmatrix}\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} v & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} u & 0 \\ 0 & v \end{pmatrix}. </math>
: <math>U[a,1]\begin{pmatrix}v&0\\0&u\end{pmatrix} = U[av,u] \thicksim U[u^{-1}av,1].</math>
Since ''u'' is arbitrary, it may be substituted for ''u''<sup>−1</sup>.
Homographies on P<sup>1</sup>(''A'') are called '''linear-fractional transformations''' since
: <math>U[z,1] \begin{pmatrix}a&c\\b&d\end{pmatrix} = U[za+b,zc+d] \thicksim  U[(zc+d)^{-1}(za+b),1].</math>

== Instances ==
[[File:Projectivisation_F5P%5E1.svg|thumb|200px|Six colors illustrate the projective line over Galois field GF(5)]]

[[File:Lines through origin finite field 25.jpg|thumb|Six lines through the origin in F_25, each corresponding to a point in the projective line P(F_5).]]

Rings that are [[field (mathematics)|field]]s are most familiar: The projective line over [[GF(2)]] has three elements: {{nowrap|''U''[0, 1]}}, {{nowrap|''U''[1, 0]}}, and {{nowrap|''U''[1, 1]}}. Its homography group is the [[permutation group]] on these three.<ref name=Rankin>{{citation |last=Rankin |first=R.A. |authorlink=Robert Alexander Rankin |date=1977 |title=Modular forms and functions |publisher=[[Cambridge University Press]] |isbn=0-521-21212-X }}</ref>{{rp|29}}

The ring [[Modular arithmetic#Integers modulo m|'''Z'''{{hsp}}/{{hsp}}3'''Z''']], or GF(3), has the elements 1, 0, and −1; its projective line has the four elements {{nowrap|''U''[1, 0]}}, {{nowrap|''U''[1, 1]}}, {{nowrap|''U''[0, 1]}}, {{nowrap|''U''[1, −1]}} since both 1 and −1 are [[unit (ring theory)|unit]]s. The homography group on this projective line has 12 elements, also described with matrices or as permutations.<ref name=Rankin/>{{rp|31}} For a [[finite field]] GF(''q''), the projective line is the [[Galois geometry]] {{nowrap|PG(1, ''q'')}}. [[J. W. P. Hirschfeld]] has described the [[projective harmonic conjugate#Galois tetrads|harmonic tetrads]] in the projective lines for ''q'' = 4, 5, 7, 8, 9.<ref>{{cite book |title=Projective Geometries Over Finite Fields |first1=J. W. P. |last1=Hirschfeld |authorlink=J. W. P. Hirschfeld |publisher=[[Oxford University Press]] |year=1979 |page=129 |isbn=978-0-19-850295-1 }}</ref>

=== Over discrete rings ===
Consider {{nowrap|P<sup>1</sup>('''Z'''{{hsp}}/{{hsp}}''n'''''Z''')}} when ''n'' is a [[composite number]]. If ''p'' and ''q'' are distinct primes dividing ''n'', then {{angle bracket|''p''}} and {{angle bracket|''q''}} are [[maximal ideal]]s in {{nowrap|'''Z'''{{hsp}}/{{hsp}}''n'''''Z'''}} and by [[Bézout's identity]] there are ''a'' and ''b'' in '''Z''' such that {{nowrap|1=''ap'' + ''bq'' = ''1''}}, so that {{nowrap|''U''[''p'', ''q'']}} is in {{nowrap|P<sup>1</sup>('''Z'''{{hsp}}/{{hsp}}''n'''''Z''')}} but it is not an image of an element under the canonical embedding. The whole of {{nowrap|P<sup>1</sup>('''Z'''{{hsp}}/{{hsp}}''n'''''Z''')}} is filled out by elements {{nowrap|''U''[''up'', ''vq'']}}, where {{nowrap|''u'' ≠ ''v''}} and {{nowrap|''u'', ''v'' ∈ ''A''<sup>×</sup>}}, ''A''<sup>×</sup> being the units of {{nowrap|'''Z'''{{hsp}}/{{hsp}}''n'''''Z'''}}. The instances {{nowrap|'''Z'''{{hsp}}/{{hsp}}''n'''''Z'''}} are given here for ''n'' = 6, 10, and 12, where according to [[modular arithmetic]] the group of units of the ring is {{nowrap|1=('''Z'''{{hsp}}/{{hsp}}6'''Z''')<sup>×</sup> = {{mset|1, 5}}}}, {{nowrap|1=('''Z'''{{hsp}}/{{hsp}}10'''Z''')<sup>×</sup> = {{mset|1, 3, 7, 9}}}}, and {{nowrap|1=('''Z'''{{hsp}}/{{hsp}}12'''Z''')<sup>×</sup> = {{mset|1, 5, 7, 11}}}} respectively. Modular arithmetic will confirm that, in each table, a given letter represents multiple points. In these tables a point {{nowrap|''U''[''m'', ''n'']}} is labeled by ''m'' in the row at the table bottom and ''n'' in the column at the left of the table. For instance, the [[point at infinity]] {{nowrap|1=A = ''U''[''v'', 0]}}, where ''v'' is a unit of the ring.

{|
|style="width: 20em;"|
{| class="wikitable" style="text-align: center;"
|+ Projective line over the ring {{nowrap|'''Z'''{{hsp}}/{{hsp}}6'''Z'''}}
 ! 5
 | B || G || F || E || D || C
|-
 ! 4
 |   || J ||   || K ||   || H
|-
 ! 3
 |   || I || L ||   || L || I
|-
 ! 2
 |   || H ||   || K ||   || J
|-
 ! 1
 | B || C || D || E || F || G
|-
 ! 0
 |   || A ||   ||   ||   || A
|-
 ! 
 ! 0 !! 1 !! 2 !! 3 !! 4 !! 5
|}
|style="width: 30em;"| 
{| class="wikitable" style="text-align: center;"
|+ Projective line over the ring {{nowrap|'''Z'''{{hsp}}/{{hsp}}10'''Z'''}}
 ! 9
 | B || K || J || I || H || G || F || E || D || C
|-
 ! 8
 |   || P ||   || O ||   || Q ||   || M ||   || L
|-
 ! 7
 | B || E || H || K || D || G || J || C || F || I
|-
 ! 6
 |   || O ||   || L ||   || Q ||   || P ||   || M
|-
 ! 5
 |   || N || R || N || R ||   || R || N || R || N
|-
 ! 4
 |   || M ||   || P ||   || Q ||   || L ||   || O
|-
 ! 3
 | B || I || F || C || J || G || D || K || H || E
|-
 ! 2
 |   || L ||   || M ||   || Q ||   || O ||   || P
|-
 ! 1
 | B || C || D || E || F || G || H || I || J || K
|-
 ! 0
 |   || A ||   || A ||   ||   ||   || A ||   || A
|-
 ! 
 ! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9
|}
| 
{| class="wikitable" style="text-align: center;"
|+ Projective line over the ring {{nowrap|'''Z'''{{hsp}}/{{hsp}}12'''Z'''}}
 ! 11
 | B || M || L || K || J || I || H || G || F || E || D || C
|-
 ! 10
 |   || T ||   || U ||   || N ||   || T ||   || U ||   || N
|-
 ! 9
 |   || S || V ||   || W || S ||   || O || W ||   || V || O
|-
 ! 8
 |   || R ||   || X ||   || P ||   || R ||   || X ||   || P
|-
 ! 7
 | B || I || D || K || F || M || H || C || J || E || L || G
|-
 ! 6
 |   || Q ||   ||   ||   || Q ||   || Q ||   ||   ||   || Q
|-
 ! 5
 | B || G || L || E || J || C || H || M || F || K || D || I
|-
 ! 4
 |   || P ||   || X ||   || R ||   || P ||   || X ||   || R
|-
 ! 3
 |   || O || V ||   || W || O ||   || S || W ||   || V || S
|-
 ! 2
 |   || N ||   || U ||   || T ||   || N ||   || U ||   || T
|-
 ! 1
 | B || C || D || E || F || G || H || I || J || K || L || M
|-
 ! 0
 |   || A ||   ||   ||   || A ||   || A ||   ||   ||   || A
|-
 !
 ! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !!10 !!11
|}
|+ Tables showing the projective lines over rings {{nowrap|'''Z'''{{hsp}}/{{hsp}}''n'''''Z'''}} for ''n'' = 6, 10, 12. Ordered pairs marked with the same letter belong to the same point.
|}

The extra points can be associated with {{nowrap|'''Q''' ⊂ '''R''' ⊂ '''C'''}}, the rationals in the [[extended complex upper-half plane]]. The group of homographies on {{nowrap|P<sup>1</sup>('''Z'''{{hsp}}/{{hsp}}''n'''''Z''')}} is called a [[congruence subgroup|principal congruence subgroup]].<ref>{{citation |first1=Metod |last1=Saniga |first2=Michel |last2=Planat |first3=Maurice R. |last3=Kibler |first4=Petr |last4=Pracna |date=2007 |title=A classification of the projective lines over small rings |journal=[[Chaos, Solitons & Fractals]] |volume=33 |issue=4 |pages=1095–1102 |doi=10.1016/j.chaos.2007.01.008 |arxiv=math/0605301 |bibcode=2007CSF....33.1095S |mr=2318902 }}</ref>

For the [[rational number]]s '''Q''', homogeneity of coordinates means that every element of P<sup>1</sup>('''Q''') may be represented by an element of P<sup>1</sup>('''Z'''). Similarly, a homography of P<sup>1</sup>('''Q''') corresponds to an element of the [[modular group]], the automorphisms of P<sup>1</sup>('''Z''').

=== Over continuous rings ===
The projective line over a [[division ring]] results in a single auxiliary point {{nowrap|1=∞ = ''U''[1, 0]}}. Examples include the [[real projective line]], the [[complex projective line]], and the projective line over [[quaternion]]s. These examples of [[topological ring]]s have the projective line as their [[one-point compactification]]s. The case of the [[complex number]] field '''C''' has the [[Möbius group]] as its homography group.

The projective line over the [[dual number]]s was described by Josef Grünwald in 1906.<ref name="Grünwald">{{citation |first=Josef |last=Grünwald |date=1906 |title=Über duale Zahlen und ihre Anwendung in der Geometrie |journal=[[Monatshefte für Mathematik]] |volume=17 |pages=81–136 |doi=10.1007/BF01697639 }}</ref> This ring includes a nonzero [[nilpotent]] ''n'' satisfying {{nowrap|1=''nn'' = 0}}. The plane {{nowrap|{{mset| ''z'' {{=}} ''x'' + ''yn'' | ''x'', ''y'' ∈ '''R''' }}}} of dual numbers has a projective line including a line of points {{nowrap|''U''[1, ''xn''], ''x'' ∈ '''R'''}}.<ref name=CS>[[Corrado Segre]] (1912) "Le geometrie proiettive nei campi di numeri duali", Paper XL of ''Opere'', also ''Atti della R. Academia della Scienze di Torino'', vol XLVII.</ref> [[Isaak Yaglom]] has described it as an "inversive Galilean plane" that has the [[topology]] of a [[cylinder (geometry)|cylinder]] when the supplementary line is included.<ref name=Yaglom79>{{citation |last=Yaglom |first=Isaak |authorlink=Isaak Yaglom |date=1979 |title=A Simple Non-Euclidean Geometry and its Physical Basis |publisher=Springer |isbn=0387-90332-1 |mr=520230 }}</ref>{{rp|149–153}} Similarly, if ''A'' is a [[local ring]], then P<sup>1</sup>(''A'') is formed  by adjoining points corresponding to the elements of the [[maximal ideal]] of&nbsp;''A''.

The projective line over the ring ''M'' of [[split-complex number]]s introduces auxiliary lines {{nowrap|{{mset| ''U''[1, ''x''(1 + j)] | ''x'' ∈ '''R''' }}}} and {{nowrap|{{mset| ''U''[1, ''x''(1 − j)] | ''x'' ∈ '''R''' }}}} Using [[stereographic projection]] the plane of split-complex numbers is [[motor variable#Compactification|closed up]] with these lines to a [[hyperboloid]] of one sheet.<ref name=Yaglom79/>{{rp| 174–200}}<ref name=Benz73>[[Walter Benz]] (1973) ''Vorlesungen über Geometrie der Algebren'', §2.1 Projective Gerade über einem Ring, §2.1.2 Die projective Gruppe, §2.1.3 Transitivitätseigenschaften, §2.1.4 Doppelverhaltnisse, Springer {{isbn|0-387-05786-2}} {{MathSciNet|id=353137}}</ref> The projective line over ''M'' may be called the [[Minkowski plane]] when characterized by behaviour of hyperbolas under homographic mapping.

== Modules ==
The projective line P<sup>1</sup>(''A'') over a ring ''A'' can also be identified as the space of [[projective module]]s in the [[module (mathematics)|module]] {{nowrap|''A'' ⊕ ''A''}}. An element of P<sup>1</sup>(''A'') is then a [[direct sum of modules|direct summand]] of {{nowrap|''A'' ⊕ ''A''}}. This more abstract approach follows the view of [[projective geometry]] as the geometry of [[linear subspace|subspaces]] of a [[vector space]], sometimes associated with the [[lattice theory]] of [[Garrett Birkhoff]]<ref>Birkhoff and Maclane (1953) ''Survey of modern algebra'', pp. 293–298, or 1997 AKP Classics edition, pp. 312–317</ref> or the book ''Linear Algebra and Projective Geometry'' by [[Reinhold Baer]]. In the case of the ring of rational [[integer]]s '''Z''', the module summand definition of P<sup>1</sup>('''Z''') narrows attention to the {{nowrap|''U''[''m'', ''n'']}}, ''m'' [[coprime]] to ''n'', and sheds the embeddings that are a principal feature of P<sup>1</sup>(''A'') when ''A'' is topological. The 1981 article by W. Benz, Hans-Joachim Samaga, & Helmut Scheaffer mentions the direct summand definition.

In an article "Projective representations: projective lines over rings"<ref>{{citation |first1=A. |last1=Blunck |first2=H. |last2=Havlicek |date=2000 |title=Projective representations: projective lines over rings |journal=[[Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg]] |volume=70 |pages=287–299 |doi=10.1007/BF02940921 |mr=1809553 |arxiv=1304.0098 }}. This article uses an alternative definition of projective line over a ring that restricts elements of the projective line over '''Z''' to those of the form {{nowrap|''U''[''m'', ''n'')}} where ''m'' and ''n'' are coprime.</ref> the [[group of units]] of a [[matrix ring]] M<sub>2</sub>(''R'') and the concepts of module and [[bimodule]] are used to define a projective line over a ring. The group of units is denoted by {{nowrap|GL(2, ''R'')}}, adopting notation from the [[general linear group]], where ''R'' is usually taken to be a field.

The projective line is the set of orbits under {{nowrap|GL(2, ''R'')}} of the free cyclic [[module (mathematics)#Submodules and homomorphisms|submodule]] {{nowrap|''R''(1, 0)}} of {{nowrap|''R'' × ''R''}}. Extending the commutative theory of Benz, the existence of a right or left [[multiplicative inverse]] of a ring element is related to P<sup>1</sup>(''R'') and {{nowrap|GL(2, ''R'')}}. The [[Dedekind-infinite set#Generalizations|Dedekind-finite]] property is characterized. Most significantly, [[representation theory|representation]] of P<sup>1</sup>(''R'') in a projective space over a division ring ''K'' is accomplished with a {{nowrap|(''K'', ''R'')}}-bimodule ''U'' that is a left ''K''-vector space and a right ''R''-module. The points of P<sup>1</sup>(''R'') are subspaces of {{nowrap|P<sup>1</sup>(''K'', ''U'' × ''U'')}} isomorphic to their complements.

== Cross-ratio ==
A homography ''h'' that takes three particular ring elements ''a'', ''b'', ''c'' to the projective line points {{nowrap|''U''[0, 1]}}, {{nowrap|''U''[1, 1]}}, {{nowrap|''U''[1, 0]}} is called the '''cross-ratio homography'''. Sometimes<ref>{{citation |first1=Gareth |last1=Jones |first2=David |last2=Singerman |date=1987 |title=Complex Functions |pages=23, 24 |publisher=[[Cambridge University Press]] }}</ref><ref>[[Joseph A. Thas]] (1968/9) "Cross ratio of an ordered point quadruple on the projective line over an associative algebra with at unity element" (in Dutch) [[Simon Stevin (journal)|Simon Stevin]] 42:97–111 {{MathSciNet|id=0266032}}</ref> the [[cross-ratio]] is taken as the value of ''h'' on a fourth point {{nowrap|1=''x'' : (''x'', ''a'', ''b'', ''c'') = ''h''(''x'')}}.

To build ''h'' from ''a'', ''b'', ''c'' the generator homographies
: <math>\begin{pmatrix}0 & 1\\1 & 0 \end{pmatrix}, \begin{pmatrix}1 & 0\\t & 1 \end{pmatrix}, \begin{pmatrix}u & 0\\0 & 1 \end{pmatrix}</math>
are used, with attention to [[fixed point (mathematics)|fixed point]]s: +1 and −1 are fixed under inversion, {{nowrap|''U''[1, 0]}} is fixed under translation, and the "rotation" with ''u'' leaves {{nowrap|''U''[0, 1]}} and {{nowrap|''U''[1, 0]}} fixed. The instructions are to place ''c'' first, then bring ''a'' to {{nowrap|''U''[0, 1]}} with translation, and finally to use rotation to move ''b'' to {{nowrap|''U''[1, 1]}}.

Lemma: If ''A'' is a [[commutative ring]] and {{nowrap|''b'' − ''a''}}, {{nowrap|''c'' − ''b''}}, {{nowrap|''c'' − ''a''}} are all units, then {{nowrap|(''b'' − ''c'')<sup>−1</sup> + (''c'' − ''a'')<sup>−1</sup>}} is a unit.

Proof: Evidently <math>\frac{b-a}{(b-c)(c-a)} = \frac{(b-c)+(c-a)}{(b-c)(c-a)}</math> is a unit, as required.

Theorem: If {{nowrap|(''b'' − ''c'')<sup>−1</sup> + (''c'' − ''a'')<sup>−1</sup>}} is a unit, then there is a homography ''h'' in G(''A'') such that 
: {{nowrap|1=''h''(''a'') = ''U''[0, 1]}}, {{nowrap|1=''h''(''b'') = ''U''[1, 1]}}, and {{nowrap|1=''h''(''c'') = ''U''[1, 0]}}.

Proof: The point {{nowrap|1=''p'' = (''b'' − ''c'')<sup>−1</sup> + (''c'' − ''a'')<sup>−1</sup>}} is the image of ''b'' after ''a'' was put to 0 and then inverted to {{nowrap|''U''[1, 0]}}, and the image of ''c'' is brought to {{nowrap|''U''[0, 1]}}. As ''p'' is a unit, its inverse used in a rotation will move ''p'' to {{nowrap|''U''[1, 1]}}, resulting in ''a'', ''b'', ''c'' being all properly placed. The lemma refers to sufficient conditions for the existence of ''h''.

One application of cross ratio defines the [[projective harmonic conjugate]] of a triple ''a'', ''b'', ''c'', as the element ''x'' satisfying {{nowrap|1=(''x'', ''a'', ''b'', ''c'') = −1}}. Such a quadruple is a [[projective harmonic conjugate#Galois tetrads|harmonic tetrad]]. Harmonic tetrads on the projective line over a [[finite field]] GF(''q'') were used in 1954 to delimit the projective linear groups {{nowrap|PGL(2, ''q'')}} for ''q'' =  5, 7, and 9, and demonstrate [[accidental isomorphism]]s.<ref>{{citation |first1=Jean |last1=Dieudonné |authorlink=Jean Dieudonné |date=1954 |title=Les Isomorphisms exceptionnals entre les groups classiques finis |journal=[[Canadian Journal of Mathematics]] |volume=6 |pages=305–315 |doi=10.4153/CJM-1954-029-0 }}</ref>

== Chains ==
The [[real line#In real algebras|real line]] in the [[complex plane]] gets permuted with circles and other real lines under [[Möbius transformation]]s, which actually permute the canonical embedding of the [[real projective line]] in the [[complex projective line]]. Suppose ''A'' is an [[algebra over a field]] ''F'', generalizing the case where ''F'' is the real number field and ''A'' is the field of complex numbers. The canonical embedding of P<sup>1</sup>(''F'') into P<sup>1</sup>(''A'') is
: <math>U_F[x, 1] \mapsto U_A[x, 1] , \quad U_F[1, 0] \mapsto U_A[1, 0].</math>
A '''chain''' is the image of P<sup>1</sup>(''F'') under a homography on P<sup>1</sup>(''A''). Four points lie on a chain [[if and only if]] their cross-ratio is in ''F''. [[Karl von Staudt]] exploited this property in his theory of "real strokes" [reeler Zug].<ref>{{citation |last=von Staudt |first=Karl |authorlink=Karl von Staudt |date=1856 |title=Beträge zur Geometrie der Lage }}</ref>

=== Point-parallelism ===
Two points of P<sup>1</sup>(''A'') are '''parallel''' if there is ''no'' chain connecting them. The convention has been adopted that points are parallel to themselves. This relation is [[invariant (mathematics)|invariant]] under the action of a homography on the projective line. Given three pair-wise non-parallel points, there is a unique chain that connects the three.<ref>[[Walter Benz]], Hans-Joachim Samaga, & Helmut Scheaffer (1981) "Cross Ratios and a Unifying Treatment of von Staudt's Notion of Reeller Zug", pp. 127–150 in ''Geometry – von Staudt's Point of View'', Peter Plaumann & Karl Strambach editors, Proceedings of NATO Advanced Study Institute, Bad Windsheim, July/August 1980, [[D. Reidel]], {{isbn|90-277-1283-2}}, {{MathSciNet|id=0621313}}</ref>

== History ==
[[August Ferdinand Möbius]] investigated the [[Möbius transformation]]s between his book ''Barycentric Calculus'' (1827) and his 1855 paper "Theorie der Kreisverwandtschaft in rein geometrischer Darstellung". [[Karl Wilhelm Feuerbach]] and [[Julius Plücker]] are also credited with originating the use of homogeneous coordinates. [[Eduard Study]] in 1898, and [[Élie Cartan]] in 1908, wrote articles on [[hypercomplex numbers]] for German and French ''Encyclopedias of Mathematics'', respectively, where they use these arithmetics with [[linear fractional transformation]]s in imitation of those of Möbius. In 1902 [[Theodore Vahlen]] contributed a short but well-referenced paper exploring some linear fractional transformations of a [[Clifford algebra]].<ref>{{citation |last=Vahlen |first=Theodore |authorlink=Theodore Vahlen |date=1902 |title=Über Bewegungen und complexe Zahlen |journal=[[Mathematische Annalen]] |volume=55 |issue=4 |pages=585–593 |doi=10.1007/BF01450354 }}</ref> The ring of [[dual numbers]] ''D'' gave Josef Grünwald opportunity to exhibit P<sup>1</sup>(''D'') in 1906.<ref name="Grünwald"/> [[Corrado Segre]] (1912) continued the development with that ring.<ref name=CS/>

[[Arthur W. Conway|Arthur Conway]], one of the early adopters of relativity via [[biquaternion]] transformations, considered the quaternion-multiplicative-inverse transformation in his 1911 relativity study.<ref>{{citation |last=Conway |first=Arthur |authorlink=Arthur W. Conway |date=1911 |title=On the application of quaternions to some recent developments of electrical theory |journal=[[Proceedings of the Royal Irish Academy]] |volume=29 |pages=1–9, particularly page 9 }}</ref> In 1947 some elements of inversive quaternion geometry were described by P.G. Gormley in Ireland.<ref>{{citation |first=P.G. |last=Gormley |date=1947 |title=Stereographic projection and the linear fractional group of transformations of quaternions |journal=[[Proceedings of the Royal Irish Academy]], Section A |volume=51 |pages=67–85 }}</ref> In 1968 [[Isaak Yaglom]]'s ''Complex Numbers in Geometry'' appeared in English, translated from Russian. There he uses P<sup>1</sup>(''D'') to describe [[line coordinates#With complex numbers|line geometry]] in the Euclidean plane and P<sup>1</sup>(''M'') to describe it for Lobachevski's plane. Yaglom's text ''A Simple Non-Euclidean Geometry'' appeared in English in 1979. There in pages 174 to 200 he develops ''Minkowskian geometry'' and describes P<sup>1</sup>(''M'') as the "inversive Minkowski plane". The Russian original of Yaglom's text was published in 1969. Between the two editions, [[Walter Benz]] (1973) published his book,<ref name=Benz73/> which included the homogeneous coordinates taken from&nbsp;''M''.

== See also ==
* [[Euclid's orchard]]

== Notes and references ==
{{reflist}}
{{refbegin}}
* {{citation |first=Sky |last=Brewer |date=2012 |title=Projective Cross-ratio on Hypercomplex Numbers |journal=[[Advances in Applied Clifford Algebras]] |doi=10.1007/s00006-12-0335-7 |doi-broken-date=1 November 2024 }}
* {{citation |first=I. M. |last=Yaglom |date=1968 |title=Complex Numbers in Geometry }}
{{refend}}

== Further reading ==
{{refbegin}}
* {{citation |first=G. |last=Ancochea |date=1942 |title=Le théorèm de von Staudt en géométrie projective quaternionienne |journal=Journal für die reine und angewandte Mathematik |volume=184 |pages=193–198 |doi=10.1515/crll.1942.184.193}}
* {{citation |first=N. B. |last=Limaye |date=1972 |title=Cross-ratios and Projectivities of a line |journal=[[Mathematische Zeitschrift]] |volume=129 |pages=49–53 |doi=10.1007/BF01229540 |mr=0314823 }}
* {{citation |first1=B.V. |last1=Limaye |first2=N.B. |last2=Limaye |date=1977 |title=The Fundamental Theorem for the Projective Line over Commutative Rings |journal=Aequationes Mathematica |volume=16 |issue=3 |pages=275–281 |doi=10.1007/BF01836039 |mr=0513873 }}
* {{citation |first1=B.V. |last1=Limaye |first2=N.B. |last2=Limaye |date=1977 |title=The Fundamental Theorem for the Projective Line over Non-Commutative Local Rings |journal=[[Archiv der Mathematik]] |volume=28 |issue=1 |pages=102–109 |doi=10.1007/BF01223897 |mr=0480495 }}
* {{citation |first=Marcel |last=Wild |date=2006 |title=The Fundamental Theorem of Projective Geometry for an Arbitrary Length Two Module |journal=[[Rocky Mountain Journal of Mathematics]] |volume=36 |issue=6 |pages=2075–2080 |doi=10.1216/rmjm/1181069362 }}
{{refend}}

== External links ==
* Mitod Saniga (2006) [http://www.ta3.sk/~msaniga/pub/ftp/RCQI_06.pdf Projective Lines over Finite Rings] (pdf) from [http://www.ta3.sk Astronomical Institute of the Slovak Academy of Sciences]

[[Category:Algebraic geometry]]
[[Category:Ring theory]]
[[Category:Projective geometry]]