parameterizing

Parameterizing is the process of expressing a mathematical object or a family of objects in terms of parameters. In mathematics and related fields, a parameterization is a map from a parameter domain, often a subset of Euclidean space, into the object of study. By varying the parameters one can generate the entire object or a family of objects, and many properties become easier to analyze or compute.

Geometric parameterization: For a plane curve, a parameterization gives coordinates as functions of a parameter t: (x(t), y(t)). A classic example is the circle x = r cos t, y = r sin t, t in [0, 2π). For a surface, two parameters (u, v) define a vector-valued map X(u, v) = (x(u, v), y(u, v), z(u, v)), which traces the surface. Regularity requires the partial derivatives to be linearly independent. The choice of parameterization is not unique; different parameterizations describe the same object up to reparameterization. Arc-length parameterization uses the curve length s as the parameter and yields unit-speed traversals.

Beyond geometry, parameterization appears in modeling and statistics as a way to express a model in terms of parameters to be estimated from data. This includes parametric families of distributions, regression models, and structural models in physics and engineering. Parameter estimation, identifiability, and reparameterization are common concerns: reparameterization can simplify optimization or improve numerical stability. An example is the parametric equation of an ellipse, x = a cos t, y = b sin t, where a and b control axes and t is the parameter.