Von Staudt conic

In projective geometry, a von Staudt conic is a special set of points connected to a concept called polarity. This idea was introduced by the mathematician Karl Georg Christian von Staudt in his book Geometrie der Lage in 1847. He wanted to study shapes without using measurements like distance, focusing only on how points and lines relate to each other.

In the real projective plane, a von Staudt conic looks like a regular conic section, which includes shapes like circles, ellipses, parabolas, and hyperbolas. But in more general settings called projective planes, these shapes might not always look the same. This concept helps mathematicians understand the deep relationships between points and lines in space, showing how geometry can be studied in very abstract ways.

Polarities

A polarity in a projective plane is a special matching system between points and lines. It pairs each point with a line, calling the line the polar of the point and the point the pole of the line. An absolute point is a point that lies on its matching line.

Some polarities have absolute points, called hyperbolic polarities, while others do not, called elliptic polarities. In certain types of projective planes, like the real projective plane, only some polarities have absolute points. These special points can form shapes called conics, which are curves defined by particular mathematical rules.

Finite projective planes

In a finite projective plane of order n, the number of absolute points for a polarity is given by a(π) = n + 2^r√n + 1, where r is a non-negative integer. If n is not a square, this number simplifies to n + 1, and the polarity is called an orthogonal polarity.

When n is odd, these absolute points form an oval, which is a set of n + 1 points with no three lying on a straight line. However, when n is even, the absolute points do not form an oval but instead lie on a non-absolute line. Thus, von Staudt conics are not ovals in finite projective planes of even order.

Relation to other types of conics

Main article: Non-Desarguesian plane § Conics

In certain types of geometric spaces, called pappian planes, a von Staudt conic behaves the same as another kind of shape known as a Steiner conic. This similarity only holds when the number system used does not have a specific property called "characteristic two". However, a mathematician named R. Artzy discovered that in other special spaces called Moufang planes, these two types of conics can look different from each other.