Circlegroup

Circlegroup is a term used in geometry and group theory to describe a class of algebraic structures that arise from the study of circles and circle symmetries. In different contexts, circlegroup may refer to a group acting on the set of circles in the plane, to a Lie group constructed from the geometry of the circle, or to transformation groups that preserve circle structure. The concept is closely related to the circle group, Möbius transformations, and diffeomorphisms of the circle.

One standard realization views circlegroup as the group of Möbius transformations that preserve the unit circle in the complex plane. Under this realization, elements are orientation-preserving automorphisms of the circle, and composition of transformations corresponds to the group operation. This realization connects circlegroup to PSL(2,R) and to the action on the boundary of the hyperbolic plane.

Another approach treats circlegroup as the group Diff(S^1) of diffeomorphisms of the circle, an infinite-dimensional Lie group under composition. This viewpoint emphasizes smooth deformations of circle parametrizations and has applications in dynamical systems and conformal geometry. Special cases, such as restricting to rotations, recover the classical circle group S^1.

Applications include modeling rotational symmetries in computer graphics, analysis of circle packings and Apollonian gaskets, and the study of geometric structures on surfaces. Though the details vary by construction, circlegroup broadly captures the idea of symmetry and transformation related to circles.

See also circle group, Möbius transformation, Diff(S^1), PSL(2,R), SO(2).