Parametrisointina

Parametrisointina is a theoretical construct in geometry and computational modeling describing how complex objects can be represented through parameterized mappings. In its broad sense, it envisions describing a shape S by a collection of local parameterizations, each defined on a parameter domain U_i, together with transition rules that ensure coherence on overlaps. This patchwork approach mirrors the classical notion of an atlas in differential geometry, but parametrisointina emphasizes practical assembly rules suitable for computational workflows.

Definition and structure: A parametrisointina representation of S consists of a finite or countable collection {f_i: U_i -> S} of maps, together with transition functions φ_ij: U_i ∩ U_j -> U_j that satisfy compatibility on triple overlaps. The advantage is modularity: one can refine or replace individual patches without rewriting the entire representation, facilitating local adaptation, anisotropic resolution, and robust handling of singularities.

Origins and usage: The term appears in theoretical discussions and exploratory papers within geometry and computer graphics, often as an umbrella concept for patch-based parameterizations. It is not yet a standard formalism with universal axioms, and different authors may formalize it with varying emphases on continuity, differentiability, or computational implementability.

Applications and examples: In practice, parametrisointina aligns with common CAD practices that use multiple charts to model surfaces, such as polygonal meshes with texture coordinates or NURBS surface patches. An example is a torus represented by several local maps, each covering a portion of the surface, whose overlaps are reconciled by smooth transition functions.

See also: Parameterization, Atlas (mathematics), Patch-based modeling, CAD/CAM, Mesh parameterization.