superficie

Superficie, in mathematics and related fields, refers to a two-dimensional geometric object that locally resembles a plane. In a precise sense, a surface is a two-dimensional manifold embedded in three-dimensional space, though in everyday use it can be the boundary or outer shell of a three-dimensional object. Surfaces may be flat or curved and may have boundaries or be closed and boundaryless.

Surfaces can be described in several ways. A common approach is parametric, where a map r(u, v) from a domain in the plane into three-dimensional space defines the surface. An implicit form uses a function F(x, y, z) = 0. At each point, the surface has a tangent plane and a normal vector obtained from the partial derivatives. Curvature measures include Gaussian curvature K, which is intrinsic to the surface, and mean curvature H, which reflects how the surface bends in ambient space.

Topology and classification play a central role. Surfaces are characterized by orientability and genus. An orientable closed surface with genus g has Euler characteristic χ = 2 − 2g. Non-orientable surfaces include the projective plane and the Klein bottle. Surfaces can be combined through connected sums to produce a wide range of topological types.

Applications and examples span geometry, physics, and computer science. Plain surfaces like planes, spheres, cylinders, and tori arise in geometry; minimal surfaces have zero mean curvature and appear in materials science and architecture. In geography, the Earth’s surface is modeled as a two-dimensional manifold on a curved manifold. In computer graphics, surface models animate and render three-dimensional shapes, while in physics and chemistry surface properties influence phenomena such as surface tension and roughness.