Unterebene

Unterebene is a term used in geometry to denote a subplane of a given plane. It consists of a subset of points together with a subset of lines of the parent plane that forms a plane again when equipped with the same incidence relations. In other words, the Unterebene inherits the geometric structure of the parent plane.

For a subset to be a Unterebene, it must be closed under incidence: for any two points in the Unterebene, the unique line determined by them in the parent plane lies entirely in the Unterebene. Equivalently, the lines of the Unterebene are exactly those lines of the parent plane that lie completely within the chosen point-set. The resulting structure must satisfy the axioms of a plane, including that any two points determine a unique line and that there exist at least three non-collinear points.

The concept appears in different geometric contexts. In projective and affine planes, especially those constructed from finite fields (Desarguesian planes), proper Unterebenen can exist and correspond to subplanes. In many finite examples, a subplane can be isomorphic to a projective plane of smaller order, such as PG(2, q^m) inside PG(2, q^n) when m divides n. By contrast, in the ordinary Euclidean plane, nontrivial Unterebenen do not exist; the only subplane is the entire plane.

Unterebenen are studied to understand the substructures and symmetries of geometric systems. They connect to notions of subfields in algebraic constructions of planes and to broader questions about how complex geometries decompose into simpler, self-contained parts. Related concepts include projective planes, affine planes, and subspaces in linear geometry.