Parametric equation

One example of a sketch defined by parametric equations is the butterfly curve.

In mathematics, parametric equations of a curve express the coordinates of the points of the curve as functions of a variable, called a parameter.[1][2] For example,

\begin{align} x &= \cos t \\ y &= \sin t \end{align}

are parametric equations for the unit circle, where t is the parameter. Together, these equations are called a parametric representation of the curve.

A common example occurs in kinematics, where the trajectory of a point is usually represented by a parametric equation with time as the parameter.

The notion of parametric equation has been generalized to surfaces, 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.).

2D examples

Parabola

The simplest equation for a parabola,

$y = x^2\,$

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

$x = t, y = t^2 \quad \mathrm{for} -\infty < t < \infty.\,$

Circle

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

$x^2 + y^2 = 1.\,$

This equation can be parameterized as follows:

$(\cos(t),\; \sin(t))\quad\mathrm{for}\ 0\leq t < 2\pi.\,$

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

\begin{align} x &= \frac{1 - t^2}{1 + t^2} \\ y &= \frac{2t}{1 + t^2} \end{align}.

With this parametric equation, 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.

Ellipse

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

\begin{align} x &= a\,\cos t y &= b\,\sin t. \end{align}

An ellipse in general position can be expressed as

\begin{align} x &= X_c + a\,\cos t\,\cos \varphi - b\,\sin t\,\sin\varphi \\ y &= Y_c + a\,\cos t\,\sin \varphi + b\,\sin t\,\cos\varphi \end{align}

as the parameter t varies from 0 to 2π. Here $(X_c,Y_c)$ is the center of the ellipse, and $\varphi$ is the angle between the $X$-axis and the major axis of the ellipse.

Both parametrizations may be made rational by using tangent half-angle formula and setting $\tan\frac{t}{2} = u.$

Hyperbola

An east-west opening hyperbola can be represented parametrically by

\begin{align} x &= a\sec t + h \\ y &= b\tan t + k \end{align} or, rationally \begin{align} x &= a\frac{1 + t^2}{1 - t^2} + h \\ y &= b\frac{2t}{1 - t^2} + k \end{align}

A north-south opening hyperbola can be represented parametrically as

$\begin{matrix} x = b\tan t + h \\ y = a\sec t + k \\ \end{matrix} \qquad \mathrm{or, rationally,} \qquad\begin{matrix} x = b\frac{2t}{1 - t^2} + h \\ y = a\frac{1 + t^2}{1 - t^2} + k \\ \end{matrix}$

In all 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.

Hypotrochoid

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.

The parametric equations for the hypotrochoids are:

\begin{align} x (\theta) &= (R - r)\cos\theta + d\cos\left({R - r \over r}\theta\right) \\ y (\theta) &= (R - r)\sin\theta - d\sin\left({R - r \over r}\theta\right) \end{align}

Some sophisticated functions

Other examples are shown:

\begin{align} x &= [a - b] \cos(t)\ + b \cos \left[t \left(\frac{a}{b} - 1\right)\right] \\ y &= [a - b] \sin(t)\ - b \sin \left[t \left(\frac{a}{b} - 1\right)\right], k = \frac{a}{b} \end{align}
Several graphs by variation of k
\begin{align} x &= \cos(a t) - \cos(b t)^j \\ y &= \sin(c t) - \sin(d t)^k \end{align}
\begin{align} x &= i \cos(a t) - \cos(b t) \sin(c t) \\ y &= j \sin(d t) - \sin(e t) \end{align}

3D examples

Helix

Parametric helix

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

\begin{align} x &= a \cos(t) \\ y &= a \sin(t) \\ z &= bt\, \end{align}

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

$\mathbf{r}(t) = (x(t), y(t), z(t)) = (a \cos(t), a \sin(t), b t),$

where r is a three-dimensional vector.

Parametric surfaces

Main article: Parametric surface

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

\begin{align} x &= \cos[t]\left[R + r \cos(u)\right], \\ y &= \sin[t]\left[R + r \cos(u)\right], \\ z &= r \sin[u]. \end{align}

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;[opinion] for example, one can integrate and differentiate such curves termwise. Thus, one can describe the velocity of a particle following the parametrized path of a helix as:

$v(t) = r'(t) = (x'(t), y'(t), z'(t)) = (-a \sin(t), a \cos(t), b)\,$

and the acceleration as:

$a(t) = r''(t) = (x''(t), y''(t), z''(t)) = (-a \cos(t), -a \sin(t), 0)\,$

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.

Another important use of parametric equations is in the field of computer aided design (CAD).[3] For example, consider the following three representations, all of which are commonly used to describe planar curves.

Type Form Example Description
1. Explicit $y = f(x) \,\!$ $y = mx + b \,\!$ Line
2. Implicit $f(x,y) = 0 \,\!$ $\left(x - a \right)^2 + \left( y - b \right)^2=r^2$ Circle
3. Parametric $x = \frac{x(t)}{w(t)}$; $y = \frac{y(t)}{w(t)}$ $x = a_0 + a_1t; \,\!$ $y = b_0 + b_1t\,\!$

$x = a+r\,\cos t; \,\!$ $y = b+r\,\sin t\,\!$

Line

Circle

The first two types are known as analytical or non-parametric representations of curves, and, in general tend to be unsuitable for use in CAD applications.[opinion] For instance, the first one is dependent upon the choice of a coordinate system and does not lend itself well to geometric transformations, such as rotations, translations, and scaling. In addition, with the implicit representation, it is more difficult of generating points on a curve. These problems are made easier by rewriting the equations in parametric form.[4]

Implicitization

Converting a set of parametric equations to a single equation involves eliminating the variable $t$ from the simultaneous equations $x=x(t),\ y=y(t).$ 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.

If the parametrization is given by rational functions

$x=\frac{p(t)}{r(t)},\qquad y=\frac{q(t)}{r(t)},$

where p, q, r are set-wize coprime polynomials, a resultant computation allows 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 dimension (either more than two coordinates of more than one parameter), the implicitization of rational parametric equations may by done with Gröbner basis computation; see Gröbner basis § Implicitization in higher dimension.

In some cases there is no single equation in closed form that is equivalent to the parametric equations.[5]

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

\begin{align} x &= a \cos(t) \\ y &= a \sin(t) \end{align}

can be simply expressed in terms of x and y by way of the Pythagorean trigonometric identity:

\begin{align} \frac{x}{a} &= \cos(t) \\ \frac{y}{a} &= \sin(t) \\ \cos(t)^2 + \sin(t)^2 &= 1 \\ \therefore \left(\frac{x}{a}\right)^2 + \left(\frac{y}{a}\right)^2 &= 1, \end{align}

which is easily identifiable as a type of conic section (in this case, a circle).

In integer geometry

Numerous problems in integer geometry can be solved using parametric equations. The most widely known such solution is Euclid's solution in integers for the legs a, b and the hypotenuse c of a primitive right triangle:

$a = 2mn, \ \ b = m^2 - n^2, \ \ c = m^2 + n^2,$

which is parametric on the coprime integers m and n of opposite parity.