an orbit

Stand at one spot and let a group of transformations carry you everywhere it possibly can. The complete set of places you could be taken to is your orbit. If the group is all rotations about the origin and you start at the point (1, 0), every rotation sends you to some point one unit from the origin, and together they sweep out the whole unit circle — so the orbit of (1, 0) is that circle.

Precisely, for a group G acting on a space X and a point x, the orbit of x is the set of all points g . x as g ranges over the entire group. To picture it concretely: list every transformation in the group, apply each to your starting point, and collect all the landing spots; that collection is the orbit. A fundamental fact is that orbits never partly overlap — two orbits are either identical or completely disjoint — so the orbits carve the whole space into separate pieces, a partition. This is because 'can be carried to' is an equivalence relation, courtesy of the group laws (you can stay put, undo, and chain moves). When the group can carry any point to any other, there is just one orbit, the whole space; such an action is called transitive, and the space is then called homogeneous.

Orbits are the natural 'types' a geometry can distinguish. In the Erlangen view, two configurations are geometrically the same exactly when they lie in the same orbit of the transformation group, so classifying figures up to the geometry's notion of sameness is literally listing the orbits. Under the Euclidean group acting on pairs of points, the orbit of a pair is determined by the distance between them — which is why distance is a Euclidean invariant. Under the larger affine group, all pairs of distinct points form a single orbit, so distance is no longer something affine geometry can see.