Affine Geometry

Abstract

An affine space is a set of points; it contains lines, etc. and affine geometry⁽¹⁾ deals, for instance, with the relations between these points and these lines (collinear points, parallel or concurrent lines…). To define these objects and describe their relations, one can:

• Either state a list of axioms, describing incidence properties, like “through two points passes a unique line”. This is the way followed by Euclid (and more recently by Hilbert). Even if the process and a fortiori the axioms themselves are not explicitly stated, this is the way used in secondary schools.

• Or decide that the essential thing is that two points define a vector and define everything starting from linear algebra, namely from the axioms defining the vector spaces.