Parametrization

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Parametrization (or parameterization; also parameterisation, parametrisation) is the process of deciding and defining the parameters necessary for a complete or relevant specification of a model or geometric object.[citation needed]

Parametrization is also the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitization.

Sometimes, this may only involve identifying certain parameters or variables. If, for example, the model is of a wind turbine with a particular interest in the efficiency of power generation, then the parameters of interest will probably include the number, length and pitch of the blades.

Most often, parametrization is a mathematical process involving the identification of a complete set of effective coordinates or degrees of freedom of the system, process or model, without regard to their utility in some design. Parametrization of a line, surface or volume, for example, implies identification of a set of coordinates that allows one to uniquely identify any point (on the line, surface, or volume) with an ordered list of numbers. Each of the coordinates can be defined parametrically in the form of a parametric curve (one-dimensional) or a parametric equation (2+ dimensions).

Non-uniqueness[edit]

Parametrizations are not generally unique. The ordinary three-dimensional object can be parametrized (or 'coordinatized') equally efficiently with Cartesian coordinates (x,y,z), cylindrical polar coordinates (ρ, φ, z), spherical coordinates (r,φ,θ) or other coordinate systems.

Similarly, the color space of human trichromatic color vision can be parametrized in terms of the three colors red, green and blue, RGB, or alternatively with cyan, magenta, yellow and black, CMYK.

Dimensionality[edit]

Generally, the minimum number of parameters required to describe a model or geometric object is equal to its dimension, and the scope of the parameters—within their allowed ranges—is the parameter space. Though a good set of parameters permits identification of every point in the parameter space, it may be that, for a given parametrization, different parameter values can refer to the same 'physical' point. Such mappings are surjective but not injective. An example is the pair of cylindrical polar coordinates (ρ,φ,z) and (ρ,φ + 2π,z).

Parametrization invariance[edit]

As indicated above, there is arbitrariness in the choice of parameters of a given model, geometric object, etc. Often, one wishes to determine intrinsic properties of an object that do not depend on this arbitrariness, which are therefore independent of any particular choice of parameters. This is particularly the case in physics, wherein parametrization invariance (or 're-parametrization invariance') is a guiding principle in the search for physically acceptable theories (particularly in general relativity).

For example, whilst the location of a fixed point on some curved line may be given by different numbers depending on how the line is parametrized, the length of the line between two such fixed points will be independent of the choice of parametrization, even though it might have been computed using other coordinate systems.

Parametrization invariance implies that either the dimensionality or the volume of the parameter space is larger than that which is necessary to describe the physics in question (as exemplified in scale invariance).

Examples of parametrized models/objects[edit]

Parametrization techniques[edit]

See also[edit]