Arguesian geometry

In synthetic Geometry, the Arguesian geometry is a " construction" simple (due to Desargues), based on the introduction of unsuitable elements, to make enter the Géométrie refines (and the parallelism) in the mussel of the projective Géométrie.

Introduction

The first axiom of the projective geometry states (inter alia):

On the other hand, the axiom of the parallelism of the geometry closely connected (a simplified formulation of fifth postulate of the geometry of Euclide) is:

It thus seems that projective geometry and geometry refine are irreconcilable since by definition.

Actually, it of it is nothing.

Description

The Arguesian geometry is a means of reconciling geometry closely connected and projective geometry:

Parallels

Desargues redefined the concept of parallelism by introducing the unsuitable elements : not unsuitable (comparable to the break point ), unsuitable right-hand side or plan. It goes without saying the elements of an unsuitable form are unsuitable. The Arguesian geometry is thus characterized by the distinction of unsuitable elements. The definition of parallelism becomes:

In projective geometry (in elliptic Geometry also), it there not of unsuitable points thus not of parallelism. On the other hand, one built there new geometries and first of all the geometry closely connected in two extremely simple stages:

one defines unsuitable points
one removes them

The characterization of the unsuitable elements in geometry closely connected is:

The elimination of the points consists in saying: “ one transforms a projective line into a line closely connected by removing its unsuitable point to him. ”. One then finds immediately the axiom of the parallelism of the geometry refines. Moreover, the removed unsuitable point is comparable to the direction of its lines.

Let us note that one can also resort to the unsuitable elements to characterize the parallelism of the hyperbolic Géométrie; but the latter is not entirely compatible with the projective geometry.

Segment

In projective geometry, two points define two segments known as additional But in geometry, two points define nothing any more but one segment: it is that of both which does not contain the unsuitable point. The order closely connected thus becomes binary more familiar.

Conclusion

The concept of element unsuitable is not necessary to the projective geometry; but is used as " passerelle" between this geometry and geometry closely connected. The removal of the unsuitable elements is comparable with an opening (with the topological direction) of space. Conversely the projective geometry is connected with a closing of the geometry closely connected.

Moreover Arguesian construction allows a rewriting, a fast transposition of the theorems of the projective geometry (the Théorème of Desargues initially). Paradoxically, one quickly realizes simplifying character of the projective geometry which removes us from the singularities of parallelism (called “decays”).

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