parameterizes

Parameterizes is the third-person singular present tense of parameterize, meaning to describe or represent something in terms of parameters. A parameterization introduces one or more independent variables—the parameters—that generate the object as the parameter varies over a specified domain. Different parameterizations may describe the same object, and reparameterization refers to changing the parameter variables, often to improve properties such as simplicity or numerical behavior.

In geometry and calculus, a parameterization expresses curves and surfaces as functions of one or two variables. For a plane curve, a parameterization is a map t -> (x(t), y(t)); for a surface, (u, v) -> (x(u, v), y(u, v), z(u, v)). Classic examples include the unit circle parameterized by (cos t, sin t) and the sphere by (sin φ cos θ, sin φ sin θ, cos φ). Reparameterization replaces t with a different parameter s = g(t), often to regularize speed or domain.

In computing, to parameterize a function or procedure means to generalize its behavior by using inputs as parameters. Parameterized queries in databases separate code from data to prevent injection. Parameterized tests run the same test logic with different inputs, improving coverage and reducing duplication.

In statistics and modeling, parameterization refers to choosing a specific set of parameters to describe a model, such as expressing a probability distribution in terms of location and scale. Reparameterization can improve identifiability or numerical stability, for example transforming a model to constrain parameters to be positive.

Across fields, parameterizes describes the act of expressing an object or problem with parameters, enabling flexible representation and analysis.