2Dparametreja

2Dparametreja is a Finnish term used to refer to parameters that depend on two variables, commonly in mathematics and computer graphics. A 2D parameterization is a mapping from a two-dimensional parameter space to a geometric object. The domain is typically a subset of R^2 with coordinates (u, v). The image is a point X(u, v) in the target space, for example in R^2 for curves or in R^3 for surfaces. In 3D modeling, 2D parameterizations are used to describe surfaces by a pair of parameters, often called texture coordinates (u, v) or parameterization coordinates.

Parameterizations can be used to define families of shapes, to describe surfaces, or to map textures. For a surface S, S(u, v) = (x(u, v), y(u, v), z(u, v)). The choice of parameterization affects geometric properties such as distortion, area, and locality. The Jacobian matrix J = dX/d(u,v) encodes local behavior; its determinant yields local area scaling. Reparameterization, i.e., composing with a smooth function φ: (u, v) -> (ξ, η), changes parameter coordinates but not the underlying surface.

Applications include computer graphics texture mapping, surface modeling in CAD, finite element mesh generation, and scientific visualization. In practice, common parameterizations arise from coordinate systems (rectangular, cylindrical, spherical) or from intrinsic surface parameterizations optimized to reduce distortion.

Related terms include parameterization, parameter space, parametric surface, and texture coordinates. The concept of 2D parameterization is foundational in fields dealing with surfaces and mappings from a two-dimensional domain.