In mathematics, a **parametric equation** defines a group of quantities as functions of one or more independent variables called parameters.^{ [1] } Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a **parametric representation** or **parameterization** (alternatively spelled as **parametrisation**) of the object.^{ [1] }^{ [2] }^{ [3] }

- Applications
- Kinematics
- Computer-aided design
- Integer geometry
- Implicitization
- Examples in two dimensions
- Parabola
- Explicit equations
- Circle
- Ellipse
- Lissajous Curve
- Hyperbola
- Hypotrochoid
- Some sophisticated functions
- Examples in three dimensions
- Helix
- Parametric surfaces
- Examples with vectors
- See also
- Notes
- External links

For example, the equations

form a parametric representation of the unit circle, where *t* is the parameter: A point (*x*, *y*) is on the unit circle if and only if there is a value of *t* such that these two equations generate that point. Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors:

Parametric representations are generally nonunique (see the "Examples in two dimensions" section below), so the same quantities may be expressed by a number of different parameterizations.^{ [1] }

In addition to curves and surfaces, parametric equations can describe manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is *one* and *one* parameter is used, for surfaces dimension *two* and *two* parameters, etc.).

Parametric equations are commonly used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a single parameter is often labeled *t*; however, parameters can represent other physical quantities (such as geometric variables) or can be selected arbitrarily for convenience. Parameterizations are non-unique; more than one set of parametric equations can specify the same curve.^{ [4] }

In kinematics, objects' paths through space are commonly described as parametric curves, with each spatial coordinate depending explicitly on an independent parameter (usually time). Used in this way, the set of parametric equations for the object's coordinates collectively constitute a vector-valued function for position. Such parametric curves can then be integrated and differentiated termwise. Thus, if a particle's position is described parametrically as

then its velocity can be found as

and its acceleration as

- .

Another important use of parametric equations is in the field of computer-aided design (CAD).^{ [5] } For example, consider the following three representations, all of which are commonly used to describe planar curves.

Type | Form | Example | Description |
---|---|---|---|

1. Explicit | Line | ||

2. Implicit | Circle | ||

3. Parametric | ; | Line Circle | |

Each representation has advantages and drawbacks for CAD applications. The explicit representation may be very complicated, or even may not exist. Moreover, it does not behave well under geometric transformations, and in particular under rotations. On the other hand, as a parametric equation and an implicit equation may easily be deduced from an explicit representation, when a simple explicit representation exists, it has the advantages of both other representations. Implicit representations may make it difficult to generate points of the curve, and even to decide whether there are real points. On the other hand, they are well suited for deciding whether a given point is on a curve, or whether it is inside or outside of a closed curve. Such decisions may be difficult with a parametric representation, but parametric representations are best suited for generating points on a curve, and for plotting it.^{ [6] }

Numerous problems in integer geometry can be solved using parametric equations. A classical such solution is Euclid's parametrization of right triangles such that the lengths of their sides *a*, *b* and their hypotenuse *c* are coprime integers. As *a* and *b* are not both even (otherwise *a*, *b* and *c* would not be coprime), one may exchange them to have *a* even, and the parameterization is then

where the parameters *m* and *n* are positive coprime integers that are not both odd.

By multiplying *a*, *b* and *c* by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths.

Converting a set of parametric equations to a single implicit equation involves eliminating the variable from the simultaneous equations This process is called **implicitization**. If one of these equations can be solved for *t*, the expression obtained can be substituted into the other equation to obtain an equation involving *x* and *y* only: Solving to obtain and using this in gives the explicit equation while more complicated cases will give an implicit equation of the form

If the parametrization is given by rational functions

where *p*, *q*, *r* are set-wise coprime polynomials, a resultant computation allows one to implicitize. More precisely, the implicit equation is the resultant with respect to *t* of *xr*(*t*) – *p*(*t*) and *yr*(*t*) – *q*(*t*)

In higher dimensions (either more than two coordinates or more than one parameter), the implicitization of rational parametric equations may by done with Gröbner basis computation; see Gröbner basis § Implicitization in higher dimension.

To take the example of the circle of radius *a*, the parametric equations

can be implicitized in terms of *x* and *y* by way of the Pythagorean trigonometric identity:

As

and

we get

and thus

which is the standard equation of a circle centered at the origin.

The simplest equation for a parabola,

can be (trivially) parameterized by using a free parameter *t*, and setting

More generally, any curve given by an explicit equation

can be (trivially) parameterized by using a free parameter *t*, and setting

A more sophisticated example is the following. Consider the unit circle which is described by the ordinary (Cartesian) equation

This equation can be parameterized as follows:

With the Cartesian equation it is easier to check whether a point lies on the circle or not. With the parametric version it is easier to obtain points on a plot.

In some contexts, parametric equations involving only rational functions (that is fractions of two polynomials) are preferred, if they exist. In the case of the circle, such a *rational parameterization* is

With this pair of parametric equations, the point (-1, 0) is not represented by a real value of *t*, but by the limit of *x* and *y* when *t* tends to infinity.

An ellipse in canonical position (center at origin, major axis along the *X*-axis) with semi-axes *a* and *b* can be represented parametrically as

An ellipse in general position can be expressed as

as the parameter *t* varies from 0 to 2*π*. Here is the center of the ellipse, and is the angle between the -axis and the major axis of the ellipse.

Both parameterizations may be made rational by using the tangent half-angle formula and setting

A Lissajous curve is similar to an ellipse, but the *x* and *y* sinusoids are not in phase. In canonical position, a Lissajous curve is given by

where and are constants describing the number of lobes of the figure.

An east-west opening hyperbola can be represented parametrically by

- or, rationally

A north-south opening hyperbola can be represented parametrically as

- or, rationally

In all these formulae (*h*,*k*) are the center coordinates of the hyperbola, *a* is the length of the semi-major axis, and *b* is the length of the semi-minor axis.

A hypotrochoid is a curve traced by a point attached to a circle of radius *r* rolling around the inside of a fixed circle of radius *R*, where the point is at a distance *d* from the center of the interior circle.

A hypotrochoid for which *r*=*d*A hypotrochoid for which *R*= 5,*r*= 3,*d*= 5

The parametric equations for the hypotrochoids are:

Other examples are shown:

**j=3 k=3****j=3 k=3****j=3 k=4****j=3 k=4****j=3 k=4**

**i=1 j=2**

Parametric equations are convenient for describing curves in higher-dimensional spaces. For example:

describes a three-dimensional curve, the helix, with a radius of *a* and rising by 2π*b* units per turn. The equations are identical in the plane to those for a circle. Such expressions as the one above are commonly written as

where **r** is a three-dimensional vector.

A torus with major radius *R* and minor radius *r* may be defined parametrically as

where the two parameters *t* and *u* both vary between 0 and 2π.

*R*=2,*r*=1/2

As *u* varies from 0 to 2π the point on the surface moves about a short circle passing through the hole in the torus. As *t* varies from 0 to 2π the point on the surface moves about a long circle around the hole in the torus.

The parametric equation of the line through the point and parallel to the vector is^{ [7] }

- 1 2 3 Weisstein, Eric W. "Parametric Equations".
*MathWorld*. - ↑ Thomas, George B.; Finney, Ross L. (1979).
*Calculus and Analytic Geometry*(fifth ed.). Addison-Wesley. p. 91. - ↑ Nykamp, Duane. "Plane parametrization example".
*mathinsight.org*. Retrieved 2017-04-14. - ↑ Spitzbart, Abraham (1975).
*Calculus with Analytic Geometry*. Gleview, IL: Scott, Foresman and Company. ISBN 0-673-07907-4 . Retrieved August 30, 2015. - ↑ Stewart, James (2003).
*Calculus*(5th ed.). Belmont, CA: Thomson Learning, Inc. pp. 687–689. ISBN 0-534-39339-X. - ↑ Shah, Jami J.; Martti Mantyla (1995).
*Parametric and feature-based CAD/CAM: concepts, techniques, and applications*. New York, NY: John Wiley & Sons, Inc. pp. 29–31. ISBN 0-471-00214-3. - ↑
*Calculus: Single and Multivariable*. John Wiley. 2012-10-29. p. 919. ISBN 9780470888612. OCLC 828768012.

In mathematics, an **ellipse** is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity , a number ranging from to .

In mathematics, the **polar coordinate system** is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the *pole*, and the ray from the pole in the reference direction is the *polar axis*. The distance from the pole is called the *radial coordinate*, *radial distance* or simply *radius*, and the angle is called the *angular coordinate*, *polar angle*, or *azimuth*. The radial coordinate is often denoted by *r* or *ρ*, and the angular coordinate by *φ*, *θ*, or *t*. Angles in polar notation are generally expressed in either degrees or radians.

A **sphere** is a geometrical object in three-dimensional space that is the surface of a ball.

In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the *radial distance* of that point from a fixed origin, its *polar angle* measured from a fixed zenith direction, and the *azimuthal angle* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In mathematics, **curvature** is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

In geometry, a **torus** is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.

An **ellipsoid** is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In geometry, a **hyperboloid of revolution**, sometimes called a **circular hyperboloid**, is the surface generated by rotating a hyperbola around one of its principal axes. A **hyperboloid** is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.

Unit quaternions, known as *versors*, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites, and crystallographic texture analysis.

In the mathematical field of differential geometry, one definition of a **metric tensor** is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface and produces a real number scalar *g*(*v*, *w*) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.

A **nonholonomic system** in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints, such that when the system evolves along a path in its parameter space but finally returns to the original set of parameter values at the start of the path, the system itself may not have returned to its original state.

A **cardioid** is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion. It's also the set of points of reflections of a fixed point on a circle through all tangents to the circle.

In mathematics, a **Dupin cyclide** or **cyclide of Dupin** is any geometric inversion of a standard torus, cylinder or double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered by Charles Dupin in his 1803 dissertation under Gaspard Monge. The key property of a Dupin cyclide is that it is a channel surface in two different ways. This property means that Dupin cyclides are natural objects in Lie sphere geometry.

**Spirograph** is a geometric drawing device that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids. The well known toy version was developed by British engineer Denys Fisher and first sold in 1965.

In geometry, the area enclosed by a circle of radius r is π*r*^{2}. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.1416.

A **parametric surface** is a surface in the Euclidean space which is defined by a parametric equation with two parameters . 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. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

In mathematics, a **surface** is a generalization of a plane, which is not necessarily flat – that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line. There are many more precise definitions, depending on the context and the mathematical tools that are used to analyze the surface.

In geometry, various **formalisms** exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

In geometry, a **centered trochoid** is the roulette formed by a circle rolling along another circle. That is, it is the path traced by a point attached to a circle as the circle rolls without slipping along a fixed circle. The term encompasses both epitrochoid and hypotrochoid. The **center** of this curve is defined to be the center of the fixed circle.

In the geometry of curves, an **orthoptic** is the set of points for which two tangents of a given curve meet at a right angle.

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