Parametrize

Parametrize, or parameterize (British English parametrise), is the act of expressing an object, function, or model in terms of parameters. A parameterization is a specific representation obtained by a mapping from a parameter domain to points of the object. The term is widely used in mathematics, computer science, statistics, and engineering.

In geometry, a curve or surface is described parametrically by coordinate functions such as x(t), y(t), and z(t). The parameter t ranges over an interval, and the functions give the coordinates of points on the object as t varies. Classic examples include a circle x = r cos t, y = r sin t; a line x = x0 + a t, y = y0 + b t; a helix uses z = c t.

Parametrization reduces complex problems to analysis of a parameter set. It enables integration and differentiation along curves, simplifies the description of motion, and underpins texture mapping and surface modeling in computer graphics. Parameterizations can also reveal structure not obvious from the implicit definition of an object.

In software development and testing, parametrization refers to running code with multiple input values. A parameterized test, for example, executes the same test logic across different data sets. In statistics and machine learning, reparameterization changes how a model or probability distribution is described, often to improve identifiability, convergence, or numerical stability.

Related terms include parameterization and reparameterization.