View source for Von Staudt conic

In [[projective geometry]], a '''von Staudt conic''' is the point set defined by all the absolute points of a polarity that has absolute points. In the [[real projective plane]] a von Staudt conic is a [[conic section]] in the usual sense. In more general [[projective plane]]s this is not always the case. [[Karl Georg Christian von Staudt]] introduced this definition in ''Geometrie der Lage'' (1847) as part of his attempt to remove all metrical concepts from projective geometry.

==Polarities==
A '''polarity''', {{pi}}, of a projective plane, {{mvar|''P''}}, is an involutory (i.e., of order two) [[bijection]] between the points and the lines of {{mvar|''P''}} that preserves the [[Incidence (geometry)|incidence relation]]. Thus, a polarity relates a point {{mvar|''Q''}} with a line {{mvar|''q''}} and, following [[Gergonne]], {{mvar|''q''}} is called the '''polar''' of {{mvar|''Q''}} and {{mvar|''Q''}} the '''pole''' of {{mvar|''q''}}.<ref>{{harvnb|Coxeter|1964|page=60}}</ref> An '''absolute point''' ('''line''') of a polarity is one which is incident with its polar (pole).<ref>{{harvnb|Garner|1979|page=132}}</ref><ref>Coxeter and several other authors use the term ''self-conjugate'' instead of absolute.</ref>

A polarity may or may not have absolute points. A polarity with absolute points is called a '''hyperbolic polarity''' and one without absolute points is called an '''elliptic polarity'''.<ref name=Coxeter72>{{harvnb|Coxeter|1964|page=72}}</ref> In the [[complex projective plane]] all polarities are hyperbolic but in the [[real projective plane]] only some are.<ref name=Coxeter72 />

A classification of polarities over arbitrary fields follows from the classification of sesquilinear forms given by Birkhoff and von Neumann.<ref>{{citation|first1=G.|last1=Birkhoff|first2=J.|last2=von Neumann|title=The logic of quantum mechanics|journal=Ann. Math.|volume=37|year=1936|pages=823–843}}</ref> Orthogonal polarities, corresponding to symmetric bilinear forms, are also called ''ordinary polarities'' and the locus of absolute points forms a non-degenerate conic (set of points whose coordinates satisfy an irreducible homogeneous quadratic equation) if the field does not have [[characteristic (field)|characteristic]] two. In characteristic two the orthogonal polarities are called ''pseudopolarities'' and in a plane the absolute points form a line.<ref>{{citation|first1=Susan|last1=Barwick|first2=Gary|last2=Ebert|title=Unitals in Projective Planes|publisher=Springer|year=2008|isbn=978-0-387-76364-4|pages=16–18}}</ref>

==Finite projective planes==
If {{pi}} is a polarity of a finite projective plane (which need not be desarguesian), {{mvar|''P''}}, of order {{mvar|''n''}} then the number of its absolute points (or absolute lines), {{mvar|''a''({{pi}})}} is given by:
: {{math|1=''a''({{pi}}) = ''n'' + 2''r''{{sqrt|''n''}} + 1}},
where {{mvar|''r''}} is a non-negative integer.<ref>{{citation|first=R.W.|last=Ball|title=Dualities of Finite Projective Planes|journal=Duke Mathematical Journal|volume=15|year=1948|pages=929–940|doi=10.1215/s0012-7094-48-01581-6}}</ref>
Since {{mvar|''a''({{pi}})}} is an integer, {{math|1=''a''({{pi}}) = ''n'' + 1}} if {{mvar|''n''}} is not a square, and in this case, {{pi}} is called an ''orthogonal polarity''.

R. Baer has shown that if {{mvar|''n''}} is odd, the absolute points of an orthogonal polarity form an [[oval (projective plane)|oval]] (that is, {{math|1=''n'' + 1}} points, no three [[collinear]]), while if {{mvar|''n''}} is even, the absolute points lie on a non-absolute line.<ref>{{citation|first=Reinhold|last=Baer|title=Polarities in Finite Projective Planes|journal=Bulletin of the American Mathematical Society|volume=52|year=1946|pages=77–93|doi=10.1090/s0002-9904-1946-08506-7|doi-access=free}}</ref>

In summary, von Staudt conics are not ovals in finite projective planes (desarguesian or not) of even order.<ref>{{harvnb|Garner|1979|page=133}}</ref><ref>{{harvnb|Dembowski|1968|pages=154–155}}</ref>

==Relation to other types of conics==
{{main|Non-Desarguesian plane#Conics}}

In a [[pappian plane]] (i.e., a projective plane coordinatized by a [[field (algebra)|field]]), if the field does not have [[characteristic (field)|characteristic]] two, a von Staudt conic is equivalent to a [[Steiner conic]].<ref>{{harvnb|Coxeter|1964|page=80}}</ref> However, R. Artzy has shown that these two definitions of conics can produce non-isomorphic objects in (infinite) [[Moufang plane]]s.<ref>{{citation|first=R.|last=Artzy|title=The Conic y = x<sup>2</sup> in Moufang Planes|journal=[[Aequationes Mathematicae]]|volume=6|year=1971|pages=30–35|doi=10.1007/bf01833234}}</ref>

==Notes==
{{reflist}}

==References==
* {{citation|first=H. S. M.|last=Coxeter|title=Projective Geometry|year=1964|publisher=Blaisdell}}
* {{Citation | last1=Dembowski | first1=Peter | title=Finite geometries | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]], Band 44 | mr=0233275 | year=1968 | isbn=3-540-61786-8 | url-access=registration | url=https://archive.org/details/finitegeometries0000demb }}
* {{citation|first=Cyril W L.|last=Garner|title=Conics in Finite Projective Planes|journal=Journal of Geometry|year=1979|volume=12|issue=2|pages=132–138|doi=10.1007/bf01918221}}

==Further reading==
* {{citation|first=T.G.|last=Ostrom|chapter=Conicoids: Conic-like figures in Non-Pappian planes|editor1-first=Peter|editor1-last=Plaumann|editor2-first=Karl|editor2-last=Strambach|title=Geometry - von Staudt's Point of View|publication-date=1981|publisher=D. Reidel|pages=175–196|isbn=90-277-1283-2}}

[[Category:Conic sections]]
[[Category:Projective geometry]]