Incidence (geometry)

Incidence is in the geometry of the simplest relationship that can occur between geometric elements such as point, line, circle, plane, etc.. Incidence to consist, for example, if a point lies on a straight line, a plane containing a straight line or in each case vice versa. So mathematically speaking it is a relation, that is, a subset of the union of Cartesian products of the set of points with the set of lines, the set of lines with the amount of points, the amount of levels with the amount of straight, the set of points with the set of levels, etc.

Definition

A geometric structure with incidence relation is a mathematical structure

Consisting of sets of points, lines, planes, etc., together with a relation

Which defines the incidence. ( In the right standing association of Cartesian products, the products of all pairs of sets, with belonging to the structure are formed. ) The relation is also called flags amount of structure.

History and Significance

The incidence term plays at least since David Hilbert's axiomatic foundation in geometry a role, since it is no longer tries to Hilbert's approach to give descriptions of the "nature" of geometric objects, but these objects are defined solely by their mathematically tangible relationships. Hilbert calls its incidence axioms " axioms of logic " and combines them in Group I of his axiom system together. The parallel postulate, which also belongs formally to the incidence axioms, forms in Hilbert own group (IV). When not using the axiom of parallels and Hilbert's axiom weakens group III ( axioms of congruence ), we reach the absolute geometry, a generalization for non-Euclidean geometries.

Lower incidence geometry are understood in synthetic geometry, more generally, a geometric structure that is solely based on incidence axioms ( and possibly other Reichhaltigkeitsaxiomen ).

In the more recent, especially the Anglo-American literature on the concept of incidence (as separately defined relation ) is often omitted and the relation of content largely through the " is member of " relation or general relation and their inversions " subset of " shall be replaced. Then the incidence is a generic term for this set-theoretically defined relations. The advantage of the classical incidence ratio is that this relationship can be defined, thus allowing symmetric elegant formulations dualisierbare statements of projective geometry. In addition, one can principally be described in this way also a geometry in which there are varying blank objects, such as lines, the incident with no point. Such applications have proved to be very fruitful and barely survived.

The original, historical purpose of defining a " Contain or Embrace " relation, which is not based on the element relation and the subset relation, it was probably to use as few axioms of set theory in the construction of the geometry. The standing in relation objects are from today's perspective (for example, straight lines are at no point sets, but can be incident with points ) even with a formulation of the axioms of geometry with a non- set-theoretic incidence relation be regarded as quantities in the sense of Zermelo -Fraenkel set theory.

Ways of speaking

Besides the known ways of speaking "a point p lies on a straight G" or " a plane contains a line G" for " p is incident with G " or " G is incident with " are also common ways of speaking:

• Incise two straight lines with the same point, this is the point of intersection of the straight line.
• Incise two different points with the same straight line, this is the straight line connecting the points.
• Incise several items of the same line, they are called collinear.
• Resecting multiple lines with the same point, they are called kopunktal.

Examples of structures with an incidence relation

• Affine and projective planes
• Polygons and polyhedra
• Euclidean space
• Möbius plane
• With a suitable definition of the incidence but also tables, chairs, and beer mugs.