The stratification of 3D geometry

Usually the world is perceived as a Euclidean 3D space. In some cases (e.g. starting from images) it is not possible or desirable to use the full Euclidean structure of 3D space. It can be interesting to only deal with the less structured and thus simpler projective geometry. Intermediate layers are formed by the affine and metric geometry. These structures can be thought of as different geometric strata which can be overlaid on the world. The simplest being projective, then affine, next metric and finally Euclidean structure.

This concept of stratification is closely related to the groups of transformations acting on geometric entities and leaving invariant some properties of configurations of these elements. Attached to the projective stratum is the group of projective transformations, attached to the affine stratum is the group of affine transformations, attached to the metric stratum is the group of similarities and attached to the Euclidean stratum is the group of Euclidean transformations. It is important to notice that these groups are subgroups of each other, e.g. the metric group is a subgroup of the affine group and both are subgroups of the projective group.

An important aspect related to these groups are their invariants. An *invariant* is a property of a configuration of geometric entities that is not altered by any transformation belonging to a specific group. Invariants therefore correspond to the measurements that one can do considering a specific stratum of geometry. These invariants are often related to geometric entities which stay unchanged - at least as a whole - under the transformations of a specific group. These entities play an important role in part of this text. Recovering them allows to upgrade the structure of the geometry to a higher level of the stratification.

In the following paragraphs the different strata of geometry are discussed. The associated groups of transformations, their invariants and the corresponding invariant structures are presented. This idea of stratification can be found back in [132] and [30].