Isometrinensanaa

Isometrinensanaa is a fictional construct in metric geometry used to describe a class of distance-preserving transformations that are piecewise isometric. In this framework, a metric space X is partitioned into a collection of regions called cells. A map f: X -> Y is called an isometrinensanaa if its restriction to each cell is an isometry onto its image, while the distortion of distances across boundaries between cells is restricted by a global tolerance epsilon. The notion serves as an intermediate concept between strict isometries and more flexible piecewise or quasi-isometries.

Formally, let P = {C_i} be a partition of X. For each i, there exists an isometry f_i: C_i -> f(C_i) with d_Y(f_i(x), f_i(y)) = d_X(x,y) for all x,y in C_i. The map f is an isometrinensanaa if there exists epsilon ≥ 0 such that for all x in C_i, y in C_j with i and j adjacent in P, |d_Y(f(x), f(y)) - d_X(x,y)| ≤ epsilon.

Properties include: the concept is local-to-global; composition is defined when partitions align; the class is not as rigid as isometries but respects uniform control.

Examples: a grid partition of the plane where each cell is mapped by a rotation or reflection; a tiling-based map that preserves distances within tiles but allows small boundary distortions.

Relation to other concepts: it sits between isometries, piecewise isometries, and quasi-isometries; related to bi-Lipschitz mappings but with a focus on local exactness.

See also: Isometry, Piecewise isometry, Quasi-isometry, Bi-Lipschitz map.