Perimeteren

Perimeteren is a term used in geometry to denote the family of perimeters associated with a given region under a specified mode of approximation or discretization. The concept is not part of a formal, standardized terminology, but it appears in discussions of digital geometry, computational geometry, and tiling theory as a way to compare how boundary length changes when a shape is represented in discrete form.

In its basic form, a perimeteren can be defined for a planar region R together with a discretization parameter t. The perimeteren P(t, R) is the length of the boundary of the polygonal approximation of R obtained at scale t, for example by representing R on a grid with cell size t or by selecting a family of polygonal outlines with vertices constrained to a lattice. More generally, the perimeteren may be treated as the set of perimeters of all allowable approximations of R under a given representation. Related concepts include the isoperimetric profile, which maps area to the minimal possible perimeter for shapes of that area, and the digital perimeter in grid-based models.

Examples include approximating a circle by inscribed or circumscribed lattice polygons, where the computed perimeter approaches 2πr as the resolution increases. In tiling and mesh generation, perimeteren analysis helps assess boundary quality and convergence properties.

Notes: Perimeteren is a descriptive term rather than a standard mathematical object; precise definitions vary by author.

See also: Isoperimetric inequality, Isoperimetric profile, Digital geometry, Boundary length, Discrete geometry.