Gequivalence

Gequivalence is a term used in mathematics and related fields to denote a type of equivalence that identifies geometric objects up to a specified set of transformations. In its common formalization, a group G of geometric transformations acts on a set X of objects. Two objects x and y in X are considered gequivalent if there exists a transformation g in G such that y = g·x. The resulting gequivalence relation is the orbit relation of the G-action, and gequivalence classes are the orbits of this action.

Formally, if X is a geometric space and G a group of transformations acting on X, then gequivalence partitions X into equivalence classes given by [x] = {g·x : g ∈ G}. This relation is reflexive, symmetric, and transitive when G contains the identity and is closed under composition, yielding well-defined orbits as equivalence classes. The choice of G determines what is considered "the same" for the objects in X, effectively encoding invariants preserved by the transformations in G.

Examples and applications illustrate the concept. In plane geometry, two figures are gequivalent under the Euclidean group if one can be translated, rotated, or reflected to coincide with the other. In image analysis, two images may be gequivalent under a specified set of transforms (rotations, flips, affine changes). In graph theory, two labeled graphs can be gequivalent under a group of relabelings representing automorphisms.

Notes: gequivalence is a general formulation that encompasses notions such as congruence and similarity, with the specific meaning depending on the chosen transformation group. The term is not universally standardized across disciplines. See also orbit, group action, symmetry, and equivalence relation.