Parametric equation

# Parametric equation

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, parametric equation is a method of defining a relation
Relation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...

using parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

s. A simple kinematic
Kinematics
Kinematics is the branch of classical mechanics that describes the motion of bodies and systems without consideration of the forces that cause the motion....

example is when one uses a time parameter to determine the position, velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

, and other information about a body in motion.

Abstractly, a parametric equation defines a relation as a set of equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

s. Therefore, it is somewhat more accurately defined as a parametric representation. It is part of regular parametric representation.

### Parabola

For example, the simplest equation for a parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

,

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

### Circle

A more sophisticated example might be the following. Consider the unit circle which is described by the ordinary equation

This equation can be parametrized as well, giving

With the normal 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.

### Helix

Parametric equations are convenient for describing curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

s in higher-dimensional spaces. For example:

describes a three-dimensional curve, the helix
Helix
A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...

, which has a radius of a and rises by 2πb units per turn. (Note that the equations are identical in the plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

to those for a circle; in fact, a helix is sometimes humorously described as just "a circle whose ends don't have the same z-value". (This is not exactly true, as a circle is by definition a two dimensional curve and a helix is by definition a three dimensional curve.)

Such expressions as the one above are commonly written as

## Parametric Surfaces

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.

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.

## Usefulness

This way of expressing curves is practical as well as efficient; for example, one can integrate
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

and differentiate
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

such curves termwise. Thus, one can describe the velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

of a particle following such a parametrized path as:

and the acceleration
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...

as:

In general, a parametric curve is a function of one independent parameter (usually denoted t). For the corresponding concept with two (or more) independent parameters, see Parametric surface
Parametric surface
A parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters. Parametric representation is the most general way to specify a surface. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence...

.

Another important use of parametric equations is in the field of computer aided design (CAD). For example, consider the following three representations, all of which are commonly used to describe planar curves
Plane curve
In mathematics, a plane curve is a curve in a Euclidean plane . The most frequently studied cases are smooth plane curves , and algebraic plane curves....

.
Type Form Example Description
1. Explicit Line
2. Implicit Circle
3. Parametric ;

Line

Circle

The first two types are known as analytical or nonparametric representations of curves, and, in general, tend to be unsuitable for use in CAD applications. For instance, both are dependent upon the choice of coordinate system and do not lend themselves well to geometric transformations, such as rotations, translations, and scaling. In addition, the implicit representation is awkward for generating points on a curve because x values may be chosen which do not actually lie on the curve. These problems are eliminated by rewriting the equations in parametric form.

## Conversion from two parametric equations to a single equation

Converting a set of parametric equations to a single equation involves eliminating the variable from the simultaneous equations . 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. If and are rational functions then the techniques of the theory of equations
Theory of equations
In mathematics, the theory of equations comprises a major part of traditional algebra. Topics include polynomials, algebraic equations, separation of roots including Sturm's theorem, approximation of roots, and the application of matrices and determinants to the solving of equations.From the point...

such as resultants can be used to eliminate t. In some cases there is no single equation in closed form that is equivalent to the parametric equations.

To take the example of the circle of radius a above, the parametric equations
can be simply expressed in terms of x and y by way of the Pythagorean trigonometric identity
Pythagorean trigonometric identity
The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one the basic relations between the sine and cosine functions, from which all others may be derived.-Statement of...

:
which is easily identifiable as a type of conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

(in this case, a circle).