Editing Dual number (section)

==Projective line==
The idea of a projective line over dual numbers was advanced by Grünwald<ref>{{cite journal|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|s2cid=119840611}}</ref> and [[Corrado Segre]].<ref>{{cite book|first=Corrado|last=Segre|author-link=Corrado Segre|date=1912|chapter=XL. Le geometrie proiettive nei campi di numeri duali|title=Opere}} Also in ''Atti della Reale Accademia della Scienze di Torino'' '''47'''.</ref>

Just as the [[Riemann sphere]] needs a north pole [[point at infinity]] to close up the [[complex projective line]], so a [[line at infinity]] succeeds in closing up the plane of dual numbers to a [[cylinder (geometry)|cylinder]].<ref name="yaglom">{{cite book|first=I. M.|last=Yaglom|date=1979|title=A Simple Non-Euclidean Geometry and its Physical Basis|publisher=Springer|isbn=0-387-90332-1|mr=520230|url-access=registration|url=https://archive.org/details/simplenoneuclide0000iagl}}</ref>{{rp|pp=149–153}}

Suppose {{mvar|D}} is the ring of dual numbers {{math|''x'' + ''yε''}} and {{mvar|U}} is the subset with {{math|''x'' ≠ 0}}. Then {{mvar|U}} is the [[group of units]] of {{mvar|D}}. Let {{math|''B'' {{=}} {(''a'', ''b'') ∈ ''D'' × ''D'' : ''a'' ∈ U or ''b'' ∈ U}<nowiki/>}}. A [[relation (mathematics)|relation]]  is defined on B as follows: {{math|(''a'', ''b'') ~ (''c'', ''d'')}} when there is a {{mvar|u}} in {{mvar|U}} such that {{math|''ua'' {{=}} ''c''}} and {{math|''ub'' {{=}} ''d''}}. This relation is in fact an [[equivalence relation]]. The points of the projective line over {{mvar|D}} are [[equivalence class]]es in {{mvar|B}} under this relation: {{math|''P''(''D'') {{=}} ''B''/~}}. They are represented with [[projective coordinates]] {{math|[''a'', ''b'']}}.

Consider the [[embedding]] {{math|''D'' → ''P''(''D'')}} by {{math|''z'' → [''z'', 1]}}. Then points {{math|[1, ''n'']}}, for {{math|''n''<sup>2</sup> {{=}} 0}}, are in {{math|''P''(''D'')}} but are not the image of any point under the embedding. {{math|''P''(''D'')}} is mapped onto a [[cylinder (geometry)|cylinder]] by  [[projection (mathematics)|projection]]: Take a cylinder tangent to the double number plane on the line {{math|{''yε'' : ''y'' ∈ '''R'''}<nowiki/>}}, {{math|''ε''<sup>2</sup> {{=}} 0}}. Now take the opposite line on the cylinder for the axis of a [[pencil (mathematics)|pencil]] of planes. The planes intersecting the dual number plane and cylinder provide a correspondence of points between these surfaces. The plane parallel to the dual number plane corresponds to points {{math|[1, ''n'']}}, {{math|''n''<sup>2</sup> {{=}} 0}} in the projective line over dual numbers.