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Such a parametric equation completely determines the curve, without the need of any interpretation of t as time, and is thus called a parametric equation of the curve (this is sometimes abbreviated by saying that one has a parametric curve). One similarly gets the parametric equation of a surface by considering functions of two parameters t and u.
In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point, in which case, the parameter is often, but not necessarily, time, and the point describes a curve, called a parametric curve. In the case of two parameters, the point describes a surface, called a parametric surface.
Parametrization (geometry), the process of finding parametric equations of a curve, surface, etc. Parametrization by arc length, a natural parametrization of a curve; Parameterization theorem or s mn theorem, a result in computability theory; Parametrization (atmospheric modeling), a method of approximating complex processes
A parametric C r-curve or a C r-parametrization is a vector-valued function: that is r-times continuously differentiable (that is, the component functions of γ are continuously differentiable), where , {}, and I is a non-empty interval of real numbers.
Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus , Stokes' theorem and the divergence theorem , are frequently given in a parametric form.
This definition of a curve has been formalized in modern mathematics as: A curve is the image of an interval to a topological space by a continuous function. In some contexts, the function that defines the curve is called a parametrization, and the curve is a parametric curve.
The curve is thus parametrized in a preferred manner by its arc length. With a non-degenerate curve r(s), parameterized by its arc length, it is now possible to define the Frenet–Serret frame (or TNB frame): The tangent unit vector T is defined as :=.
Fitting of a noisy curve by an asymmetrical peak model, with an iterative process (Gauss–Newton algorithm with variable damping factor α).Curve fitting [1] [2] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, [3] possibly subject to constraints.