projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...

, 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 In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has bas ...

a von Staudt conic is a

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...

in the usual sense. In more general

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...

s this is not always the case.

Karl Georg Christian von Staudt Karl Georg Christian von Staudt (24 January 1798 – 1 June 1867) was a German mathematician who used synthetic geometry to provide a foundation for arithmetic. Life and influence Karl was born in the Free Imperial City of Rothenburg, which is n ...

introduced this definition in ''Geometrie der Lage'' (1847) as part of his attempt to remove all metrical concepts from projective geometry.

Polarities

A polarity, , of a projective plane, , is an involutory (i.e., of order two)

bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...

between the points and the lines of that preserves the

incidence relation In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is ''X'' and the second is ''Y'', the matrix has one row for each element o ...

. Thus, a polarity relates a point with a line and, following

Gergonne Joseph Diez Gergonne (19 June 1771 at Nancy, France – 4 May 1859 at Montpellier, France) was a French mathematician and logician. Life In 1791, Gergonne enlisted in the French army as a captain. That army was undergoing rapid expansion becau ...

, is called the polar of and the pole of . An absolute point (line) of a polarity is one which is incident with its polar (pole). 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. In the

complex projective plane In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1, ...

all polarities are hyperbolic but in the

only some are. A classification of polarities over arbitrary fields follows from the classification of sesquilinear forms given by Birkhoff and von Neumann. 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 two. In characteristic two the orthogonal polarities are called ''pseudopolarities'' and in a plane the absolute points form a line.

Finite projective planes

If is a polarity of a finite projective plane (which need not be desarguesian), , of order then the number of its absolute points (or absolute lines), is given by: : , where is a non-negative integer. Since is an integer, if is not a square, and in this case, is called an ''orthogonal polarity''. R. Baer has shown that if is odd, the absolute points of an orthogonal polarity form an

oval An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one or ...

(that is, points, no three

collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...

), while if is even, the absolute points lie on a non-absolute line. In summary, von Staudt conics are not ovals in finite projective planes (desarguesian or not) of even order.

Relation to other types of conics

In a

pappian plane In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that *given one set of Collinearity, collinear points A, B, C, and another set of collinear points a,b,c, then the intersection points X,Y,Z of line (mathemati ...

(i.e., a projective plane coordinatized by a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

), if the field does not have characteristic two, a von Staudt conic is equivalent to a

Steiner conic The Steiner conic or more precisely Steiner's generation of a conic, named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non-degenerate projective conic section in a projective plane over a field. The usual d ...

. However, R. Artzy has shown that these two definitions of conics can produce non-isomorphic objects in (infinite)

Moufang plane In geometry, a Moufang plane, named for Ruth Moufang, is a type of projective plane, more specifically a special type of translation plane. A translation plane is a projective plane that has a ''translation line'', that is, a line with the property ...

Notes

References

* * *

Polarities

Finite projective planes

Relation to other types of conics

Notes

References

Further reading