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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.
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).
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.
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).
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 'reparametrization 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 a set of numbers whose values depend on how the curve is parametrized, the length (appropriately defined) of the curve between two such fixed points will be independent of the particular choice of parametrization (in this case: the method by which an arbitrary point on the line is uniquely indexed). The length of the curve is a parameterization-invariant quantity therefore. In such cases parameterization is a mathematical tool employed to extract a result whose value does not depend on, or make reference to, the details of the parameterization. More generally, parametrization invariance of a physical theory implies that either the dimensionality or the volume of the parameter space is larger than is necessary to describe the physics (the quantities of physical significance) in question.
Though the theory of general relativity can be expressed without reference to a coordinate system, calculations of physical (i.e. observable) quantities such as the curvature of spacetime invariably involve the introduction of a particular coordinate system in order to refer to spacetime points involved in the calculation. In the context of general relativity then, the choice of coordinate system may be regarded as a method of 'parameterizing' the spacetime, and the insensitivity of the result of a calculation of a physically-significant quantity to that choice can be regarded as an example of parameterization invariance.
As another example, physical theories whose observable quantities depend only on the relative distances (the ratio of distances) between pairs of objects are said to be scale invariant. In such theories any reference in the course of a calculation to an absolute distance would imply the introduction of a parameter to which the theory is invariant.
Examples of parametrized models/objects
- Boy's surface
- McCullagh's parametrization of the Cauchy distributions
- Parametrization (climate), the parametric representation of general circulation models and numerical weather prediction
- Singular isothermal sphere profile
- Lambda-CDM model, the standard model of Big Bang cosmology
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