Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay

In mathematics, parametric equations are a bit like functions: they allow someone to fill in some variables, called parameters or independent variables, with any values they wish. When this is done, the equations then tell the values of dependent variables. A simple example is, in kinematics, filling in a time parameter to get the position, velocity, and other facts about a moving object. Image File history File links Graph of butterfly curve, a parametric equation discovered by Temple H. Fay: File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Graph of butterfly curve, a parametric equation discovered by Temple H. Fay: File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A sextic plane curve given by the equation: Categories: Curves ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... An independent variable is presumed to cause or determine a dependent variable. ... In experimental design, a dependent variable is a variable whose values in different treatment conditions are compared. ... In physics, kinematics is the branch of mechanics concerned with the motions of objects without being concerned with the forces that cause the motion. ...

Abstractly, a relation is given in the form of an equation, and it is shown also to be the image of a functions from, say, Rⁿ. It is therefore somewhat more accurately defined as a parametric representation. See also parameter, parametrization, regular parametric representation. In mathematics, a relation is a generalization of arithmetic relations, such as = and <, which occur in statements, such as 5 < 6 or 2 + 2 = 4. See relation (mathematics), binary relation and relational algebra. ... In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ... A parameter is a measurement or value on which something else depends. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ...

For example, the simplest equation for a parabola, A parabola The parabola (from the Greek: παραβολή) is a conic section generated by the intersection of a cone and a plane tangent to the cone or parallel to some plane tangent to the cone. ...

can be parametrized by using a free parameter t, and setting

Although the preceding example appears somewhat trivial, consider the following parametrization of a circle of radius a: In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). ... RADIUS (Remote Authentication Dial In User Service) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ...

Finally, there are certain geometric forms that are nearly impossible to describe as a single equation but have very elegant expressions in parametric form:

which describe a three-dimensional curve, the helix, which has a radius of a and rises by 2πb units per turn. (Note that the equations are identical in the plane to those for a circle; in fact, a helix is just "a circle whose ends don't have the same z-value".) A helix (pl: helices), from the Greek word έλικας/έλιξ, is a twisted shape like a spring, screw or a spiral staircase. ... This article is in need of attention from an expert on the subject. ...

Such expressions as the one above are commonly written as

This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves termwise. Thus, one can describe the velocity of a particle following such a parametrized path as: Integration may be any of the following: Usually integration is the construction of an object, a theory, etc. ... Differentiation can mean the following: In biology: cellular differentiation; evolutionary differentiation; In mathematics: see: derivative In cosmogony: planetary differentiation Differentiation (geology); Differentiation (logic); Differentiation (marketing). ... This article is about velocity in physics. ...

and the acceleration as: Acceleration is the time rate of change of velocity, and at any point on a v-t graph, it is given by the gradient of the tangent to that point In physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. ...

In general, a parametric curve is a function of one independent parameter (usually denoted t). Parametrized surfaces, of great use in such vector calculus applications as Stokes' theorem, are functions of two parameters, most commonly (s, t) or (u,v). Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ... The Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

An example of a parametrized surface is the (capless) cylinder given by A right circular cylinder In mathematics, a cylinder is a quadric, i. ...

Considering the equation as representing a circle in the plane, it is evident that this represents a cylinder. It is then allowed to take on arbitrary values of z.