Parameterization

Parameterization is the representation of a geometric object or a model using one or more parameters. In geometry, a curve is described by a parameterization, a map gamma from an interval of real numbers into the ambient space, where t is the parameter and gamma(t) gives the point on the curve. A regular (or smooth) parameterization has a nonzero derivative, ensuring that the curve is traced without cusps as t varies. Surfaces are parameterized by two parameters, typically denoted (u, v), via a map from a subset of R^2 to R^3. This provides coordinate charts in differential geometry and a practical way to compute geometric quantities through partial derivatives and the Jacobian matrix.

Common examples include the circle x = cos t, y = sin t, the sphere x = sin phi cos theta, y = sin phi sin theta, z = cos phi, and the torus with x = (R + r cos v) cos u, y = (R + r cos v) sin u, z = r sin v. In each case, the parameters control position on the object.

In statistics and modeling, parameterization expresses a family of distributions or models in terms of parameters that govern behavior, while reparameterization changes the parameterization to improve estimation, interpretation, or numerical stability.

In computer graphics and geometric modeling, parameterizations are used to map surfaces to a parameter domain (for texture coordinates, shading, or remeshing). Parameterization also contrasts with implicit descriptions, where the object is defined by an equation F(x) = 0 rather than an explicit map from parameters.