First Steps in Non-standard Projective Geometry

TL;DR

This paper introduces non-standard analysis into projective geometry, analyzing properties over non-Archimedean fields and exploring how infinitesimal deviations affect geometric objects and transformations.

Contribution

It pioneers the application of non-standard analysis to projective geometry, examining infinitesimal deviations and their effects on classical concepts like transformations and conics.

Findings

Infinitesimal deviations often lead to only infinitesimal changes in projective objects.

Discontinuity and degeneration occur when infinitesimal properties do not hold.

Non-standard analysis provides new insights into projective concepts over non-Archimedean fields.

Abstract

In this article, we will introduce methods of non-standard analysis into projective geometry. Especially, we will analyze the properties of a projective space over a non-Archimedean field. Non-Archimedean fields contain numbers that are smaller than every real number: the so called "infinitesimal" numbers. The theory is well known from non-standard analysis. This enables us to define projective objects that deviate only infinitesimally (in an appropriate metric) from each other. And we show that in most cases operations involving such objects that deviate infinitesimally also experience only infinitesimal change. Another focus will be where this property does not hold true and show that this usually involves discontinuity or degeneration. Furthermore, we will explore common projective concepts like projective transformations, cross-ratios and conics in a non-standard setting.