axisymmetry

Axisymmetry, or axial symmetry, is a property of an object or system that remains unchanged under rotations about a fixed line called the axis. In practical terms, if you rotate the object around that axis by any angle, its shape and features map onto themselves. In mathematical terms, with a chosen axis, the object or field is invariant under all rotations around that axis. In cylindrical coordinates, taking the axis as the z-axis means the object or field is independent of the azimuthal angle φ.

Axisymmetry is related to cylindrical symmetry and is distinct from spherical symmetry. An axisymmetric body is not necessarily invariant under rotations about every axis, only about the designated fixed axis. A sphere, by contrast, has full rotational symmetry about any axis.

Examples of axisymmetric objects include a right circular cylinder, a torus with a circular cross-section, and any solid of revolution generated by rotating a plane curve about an axis. A vase formed by revolving a profile around a central axis is axisymmetric. In general, solids of revolution exhibit axisymmetry.

In physics and engineering, axisymmetry simplifies analyses by reducing dependence on the azimuthal coordinate. In fluid dynamics, axisymmetric flows have velocity fields that do not depend on φ. In electromagnetism and gravitation, axisymmetric problems often leverage symmetry to solve field equations more readily. In general relativity, axisymmetric spacetimes describe rotating bodies, such as the Kerr solution for rotating black holes.

Limitations include the requirement of a fixed axis; tilting or redefining the axis generally destroys axisymmetry.