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 (''p ...

, 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, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...

a von Staudt conic is a

conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...

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 (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...

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

bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

between the points and the lines of that preserves the

incidence relation In geometry, an incidence relation is a heterogeneous relation that captures the idea being expressed when phrases such as "a point ''lies on'' a line" or "a line is ''contained in'' a plane" are used. The most basic incidence relation is that betw ...

. Thus, a polarity relates a point with a line and, following Gergonne, 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 or 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 \C^3, \qquad (Z_1,Z_2, ...

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 of mathematics (projective geometry, technical drawing, etc.), it is given a more precise definition, which may inc ...

(that is, points, no three

collinear In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...

), 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 (i.e., a projective plane coordinatized by a field), if the field does not have characteristic two, a von Staudt conic is equivalent to a Steiner conic. However, R. Artzy has shown that these two definitions of conics can produce non-isomorphic objects in (infinite) Moufang planes.

Notes

References

* * *

Polarities

Finite projective planes

Relation to other types of conics

Notes

References

Further reading