Gegenfläche

Gegenfläche, literally “opposite surface” in German, is a geometric term used to describe a surface that is opposite to another surface within a given object or configuration. The precise meaning depends on context, and there is no single universal definition.

In solid geometry, Gegenfläche commonly denotes the surface on the far side of a body relative to a chosen direction or point. This can include opposite faces of a polyhedron or pairs of parallel supporting planes that bound a convex body from opposite sides. In these settings the concept helps describe antipodal relations and symmetry within the object.

In differential and projective geometry, the term is sometimes used to refer to a dual or polar surface. Such a Gegenfläche encodes tangential information of the original surface, for example as the envelope of tangent planes, and may be considered within dual space in projective geometry. The idea is to pair the given surface with a complementary construction that reflects its orientation, curvature, or contact with other geometric objects.

Overall, Gegenfläche denotes a surface associated with another by a form of opposition or duality. Its exact construction—whether as an opposite side, a dual surface, or an offset/parallel variant—is dependent on the mathematical framework and authors' conventions.

See also: opposite face, antipodal surface, dual surface, polar dual, parallel surface.