Spherical

Spherical is an adjective describing things related to a sphere, a perfectly round three-dimensional object where every point on the surface is equidistant from a central point. In three dimensions, the sphere is the boundary of a ball, consisting of all points at a fixed radius.

In mathematics, spherical geometry studies properties of figures on the surface of a sphere. The shortest path between two points on the sphere is a great circle, not a straight line in Euclidean space. Spherical trigonometry deals with triangles drawn on a sphere and uses relationships among angles and side lengths; the spherical excess equals the sum of the triangle's angles minus π.

Coordinate systems: Spherical coordinates express points by radius r, inclination θ, and azimuth φ, with Cartesian coordinates x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. Note conventions vary, especially for θ and φ.

In physics and engineering: Spherical symmetry describes problems that are invariant under rotations about a center. Spherical harmonics Y_lm provide a basis for functions on the sphere and are used to solve Laplace's equation in spherical coordinates and in quantum mechanics to describe electron orbitals.

Other uses and related terms: Spherical aberration in optics arises when light rays refract at different distances from the axis. The term also appears in everyday language as a simplified thought experiment. Spherical is contrasted with spheroidal or ellipsoidal shapes.