Parametric equation

Converting a set of parametric equations to a single implicit equation involves eliminating the variable t from the simultaneous equations ${\displaystyle x=f(t),\ y=g(t).}$ 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: Solving ${\displaystyle y=g(t)}$ to obtain ${\displaystyle t=g^{-1}(y)}$ and using this in ${\displaystyle x=f(t)}$ gives the explicit equation ${\displaystyle x=f(g^{-1}(y)),}$ while more complicated cases will give an implicit equation of the form ${\displaystyle h(x,y)=0.}$

If the parametrization is given by rational functions $${\displaystyle x={\frac {p(t)}{r(t)}},\qquad y={\frac {q(t)}{r(t)}},}$$

where p, q, and 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 $${\displaystyle {\begin{aligned}x&=a\cos(t)\\y&=a\sin(t)\end{aligned}}}$$

can be implicitized in terms of x and y by way of the Pythagorean trigonometric identity. With

$${\displaystyle {\begin{aligned}{\frac {x}{a}}&=\cos(t)\\{\frac {y}{a}}&=\sin(t)\\\end{aligned}}}$$ and $${\displaystyle \cos(t)^{2}+\sin(t)^{2}=1,}$$ we get $${\displaystyle \left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{a}}\right)^{2}=1,}$$ and thus $${\displaystyle x^{2}+y^{2}=a^{2},}$$

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

The simplest equation for a parabola, $${\displaystyle y=x^{2}}$$

can be (trivially) parameterized by using a free parameter t, and setting $${\displaystyle x=t,y=t^{2}\quad \mathrm {for} -\infty <t<\infty .}$$

More generally, any curve given by an explicit equation $${\displaystyle y=f(x)}$$

can be (trivially) parameterized by using a free parameter t, and setting $${\displaystyle x=t,y=f(t)\quad \mathrm {for} -\infty <t<\infty .}$$

A more sophisticated example is the following. Consider the unit circle which is described by the ordinary (Cartesian) equation $${\displaystyle x^{2}+y^{2}=1.}$$

This equation can be parameterized as follows: $${\displaystyle (x,y)=(\cos(t),\;\sin(t))\quad {\text{ for }}0\leq t<2\pi .}$$

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 $${\displaystyle {\begin{aligned}x&={\frac {1-t^{2}}{1+t^{2}}}\\[6pt]y&={\frac {2t}{1+t^{2}}}\,.\end{aligned}}}$$

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

An ellipse in general position can be expressed as $${\displaystyle {\begin{alignedat}{4}x={}&&X_{\mathrm {c} }&+a\,\cos t\,\cos \varphi {}&&-b\,\sin t\,\sin \varphi \\y={}&&Y_{\mathrm {c} }&+a\,\cos t\,\sin \varphi {}&&+b\,\sin t\,\cos \varphi \end{alignedat}}}$$

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

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

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 $${\displaystyle {\begin{aligned}x&=a\,\cos(k_{x}t)\\y&=b\,\sin(k_{y}t)\end{aligned}}}$$ where k_x and k_y are constants describing the number of lobes of the figure.

An east-west opening hyperbola can be represented parametrically by

$${\displaystyle {\begin{aligned}x&=a\sec t+h\\y&=b\tan t+k\,,\end{aligned}}}$$

or, rationally

$${\displaystyle {\begin{aligned}x&=a{\frac {1+t^{2}}{1-t^{2}}}+h\\y&=b{\frac {2t}{1-t^{2}}}+k\,.\end{aligned}}}$$

A north-south opening hyperbola can be represented parametrically as

$${\displaystyle {\begin{aligned}x&=b\tan t+h\\y&=a\sec t+k\,,\end{aligned}}}$$

or, rationally

$${\displaystyle {\begin{aligned}x&=b{\frac {2t}{1-t^{2}}}+h\\y&=a{\frac {1+t^{2}}{1-t^{2}}}+k\,.\end{aligned}}}$$

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. Note that in the rational forms of these formulae, the points (−a , 0) and (0 , −a), respectively, are not represented by a real value of t, but are the limit of x and y as t tends to infinity.

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:

$${\displaystyle {\begin{aligned}x(\theta )&=(R-r)\cos \theta +d\cos \left({R-r \over r}\theta \right)\\y(\theta )&=(R-r)\sin \theta -d\sin \left({R-r \over r}\theta \right)\,.\end{aligned}}}$$

Some examples:

• 
  R = 6 r = 4 d = 1
• 
  R = 7 r = 4 d = 1
• 
  R = 8 r = 3 d = 2
• 
  R = 7 r = 4 d = 2
• 
  R = 15 r = 14 d = 1

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

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

describes a three-dimensional curve, the helix, with a radius of a and rising by 2πb 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

$${\displaystyle {\begin{aligned}\mathbf {r} (t)&=(x(t),y(t),z(t))\\&=(a\cos(t),a\sin(t),bt)\,,\end{aligned}}}$$

where r is a three-dimensional vector.

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

$${\displaystyle {\begin{aligned}x&=\cos(t)\left(R+r\cos(u)\right),\\y&=\sin(t)\left(R+r\cos(u)\right),\\z&=r\sin(u)\,.\end{aligned}}}$$

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 ${\displaystyle \left(x_{0},y_{0},z_{0}\right)}$ and parallel to the vector ${\displaystyle a{\hat {\mathbf {i} }}+b{\hat {\mathbf {j} }}+c{\hat {\mathbf {k} }}}$ is^[7]

$${\displaystyle {\begin{aligned}x&=x_{0}+at\\y&=y_{0}+bt\\z&=z_{0}+ct\end{aligned}}}$$

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 $${\displaystyle \mathbf {r} (t)=(x(t),y(t),z(t))\,,}$$

then its velocity can be found as $${\displaystyle {\begin{aligned}\mathbf {v} (t)&=\mathbf {r} '(t)\\&=(x'(t),y'(t),z'(t))\,,\end{aligned}}}$$

and its acceleration as $${\displaystyle {\begin{aligned}\mathbf {a} (t)&=\mathbf {v} '(t)=\mathbf {r} ''(t)\\&=(x''(t),y''(t),z''(t))\,.\end{aligned}}}$$

Another important use of parametric equations is in the field of computer-aided design (CAD).^[8] For example, consider the following three representations, all of which are commonly used to describe planar curves.

Type	Form	Example	Description
Explicit	${\displaystyle y=f(x)\,\!}$	${\displaystyle y=mx+b\,\!}$	Line
Implicit	${\displaystyle f(x,y)=0\,\!}$	${\displaystyle \left(x-a\right)^{2}+\left(y-b\right)^{2}=r^{2}}$	Circle
Parametric	${\displaystyle x={\frac {g(t)}{w(t)}};\,\!}$ ${\displaystyle y={\frac {h(t)}{w(t)}}}$	${\displaystyle x=a_{0}+a_{1}t;\,\!}$ ${\displaystyle y=b_{0}+b_{1}t\,\!}$	Line
		${\displaystyle x=a+r\,\cos t;\,\!}$ ${\displaystyle y=b+r\,\sin t\,\!}$	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 on 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.^[9]

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

$${\displaystyle {\begin{aligned}a&=2mn\\b&=m^{2}-n^{2}\\c&=m^{2}+n^{2}\,,\end{aligned}}}$$

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.

A system of m linear equations in n unknowns is underdetermined if it has more than one solution. This occurs when the matrix of the system and its augmented matrix have the same rank r and r < n. In this case, one can select n − r unknowns as parameters and represent all solutions as a parametric equation where all unknowns are expressed as linear combinations of the selected ones. That is, if the unknowns are ${\displaystyle x_{1},\ldots ,x_{n},}$ one can reorder them for expressing the solutions as^[10]

$${\displaystyle {\begin{aligned}x_{1}&=\beta _{1}+\sum _{j=r+1}^{n}\alpha _{1,j}x_{j}\\\vdots \\x_{r}&=\beta _{r}+\sum _{j=r+1}^{n}\alpha _{r,j}x_{j}\\x_{r+1}&=x_{r+1}\\\vdots \\x_{n}&=x_{n}.\end{aligned}}}$$

Such a parametric equation is called a parametric form of the solution of the system.^[10]

The standard method for computing a parametric form of the solution is to use Gaussian elimination for computing a reduced row echelon form of the augmented matrix. Then the unknowns that can be used as parameters are the ones that correspond to columns not containing any leading entry (that is the left most non zero entry in a row or the matrix), and the parametric form can be straightforwardly deduced.^[10]

• Curve
• Parametric estimating
• Position vector
• Vector-valued function
• Parametrization by arc length
• Parametric derivative