piiripinna

Piiripinna, translated as boundary surface, is a concept used in geometry, topology, and related fields to describe the two‑dimensional surface that forms the boundary of a three‑dimensional region. In formal terms, if S is a subset of three‑dimensional space, the piiripinna (boundary) consists of all points where every neighborhood intersects both S and its complement. In Euclidean space, the boundary of a solid is typically a two‑dimensional manifold, though it may have singularities or edges.

A common way to think about piiripinna is through familiar solids. The boundary of a solid ball is the unit sphere, a smooth two‑dimensional surface. The boundary of a cube is made up of its six square faces, joined along edges. Some solids have more complex boundaries, such as a torus, whose boundary is a toroidal surface. If a region has a smooth surface, the boundary can be described by a smooth parameterization; if the region is polyhedral, the boundary is a union of polygonal faces.

Mathematically, piiripinna can be represented in several ways, including parametric surfaces r(u,v), implicit surfaces defined by F(x,y,z)=0, or as a polygonal mesh in computer graphics. Properties of the boundary include orientability, curvature, and topology (for example, genus). In computational contexts, the boundary is often approximated by meshes or level‑set methods for simulations and rendering.

Applications of piiripinna appear in computer graphics, CAD, finite element methods, geospatial modeling, and physical simulations, where the boundary delineates interface conditions and governs how objects interact with their environment.