In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation.
If x, y, z are the coordinates of the point, the movement is thus described by a parametric equation [1] where t is the parameter and denotes the time.
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.
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).
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.