Editing Parametric equation (section)

==Parametric plane curves==
{{further|Plane curve}}

===Parabola===

The simplest equation for a [[parabola]],
<math display="block">y = x^2</math>

can be (trivially) parameterized by using a free parameter {{mvar|t}}, and setting
<math display="block">x = t, y = t^2 \quad \mathrm{for} -\infty < t < \infty.</math>

===Explicit equations===
More generally, any curve given by an explicit equation
<math display="block">y = f(x)</math>

can be (trivially) parameterized by using a free parameter {{mvar|t}}, and setting
<math display="block">x = t, y = f(t) \quad \mathrm{for} -\infty < t < \infty.</math>

===Circle===

A more sophisticated example is the following. Consider the unit circle which is described by the ordinary (Cartesian) equation
<math display="block"> x^2 + y^2 = 1.</math>

This equation can be parameterized as follows: 
<math display="block">(x,y)=(\cos(t),\; \sin(t))\quad\mathrm{for}\ 0\leq t < 2\pi.</math>

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 function]]s (that is fractions of two [[polynomial]]s) are preferred, if they exist. In the case of the circle, such a ''{{dfn|rational parameterization}}'' is 
<math display="block">\begin{align}
  x &= \frac{1 - t^2}{1 + t^2} \\
  y &= \frac{2t}{1 + t^2}\,.
\end{align}</math>

With this pair of parametric equations, the point {{math|(−1, 0)}} is not represented by a [[real number|real]] value of {{mvar|t}}, but by the [[limit (mathematics)|limit]] of {{mvar|x}} and {{mvar|y}} when {{mvar|t}} tends to [[infinity]].

===Ellipse===

An [[ellipse]] in canonical position (center at origin, major axis along the {{mvar|x}}-axis) with semi-axes {{mvar|a}} and {{mvar|b}} can be represented parametrically as
<math display="block">\begin{align}
  x &= a\,\cos t \\
  y &= b\,\sin t\,.
\end{align}</math>

An ellipse in general position can be expressed as
<math display="block">\begin{alignat}{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{alignat}</math>

as the parameter {{mvar|t}} varies from {{math|0}} to {{math|2''{{pi}}''}}. Here {{math|(''X''{{sub|c}} , ''Y''{{sub|c}})}} is the center of the ellipse, and {{mvar|φ}} is the angle between the {{mvar|x}}-axis and the major axis of the ellipse.

Both parameterizations may be made [[rational function|rational]] by using the [[tangent half-angle formula]] and setting <math display="inline">\tan\frac{t}{2} = u\,.</math>

=== Lissajous curve ===
[[File:Lissajous curve 3by2.svg|thumbnail|A Lissajous curve where {{math|1=''k{{sub|x}}'' = 3}} and {{math|1=''k{{sub|y}}'' = 2}}.]]
A [[Lissajous curve]] is similar to an ellipse, but the {{mvar|x}} and {{mvar|y}} [[sinusoid]]s are not in phase. In canonical position, a Lissajous curve is given by
<math display="block">\begin{align}
  x &=  a\,\cos(k_xt) \\
  y &=  b\,\sin(k_yt)
\end{align}</math>
where {{mvar|k{{sub|x}}}} and {{mvar|k{{sub|y}}}} are constants describing the number of lobes of the figure.

===Hyperbola===
An east-west opening [[hyperbola]] can be represented parametrically by

<math display="block">\begin{align}
  x &= a\sec t + h \\
  y &= b\tan t + k\,,
\end{align}</math>

or, [[rational function|rationally]]

<math display="block">\begin{align}
  x &= a\frac{1 + t^2}{1 - t^2} + h \\
  y &= b\frac{2t}{1 - t^2}  + k\,.
\end{align}</math>

A north-south opening hyperbola can be represented parametrically as

<math display="block">\begin{align}
  x &= b\tan t + h \\
  y &= a\sec t + k\,,
\end{align}</math>

or, rationally

<math display="block">\begin{align}
  x &= b\frac{2t}{1 - t^2} + h \\
  y &= a\frac{1 + t^2}{1 - t^2} + k\,.
\end{align}</math>

In all these formulae {{math|(''h'' , ''k'')}} are the center coordinates of the hyperbola, {{mvar|a}} is the length of the semi-major axis, and {{mvar|b}} is the length of the semi-minor axis. Note that in the rational forms of these formulae, the points {{math|(''−a'' , 0)}} and {{math|(0 , ''−a'')}}, respectively, are not represented by a real value of {{mvar|t}}, but are the limit of {{mvar|x}} and {{mvar|y}} as {{mvar|t}} tends to infinity.

===Hypotrochoid===
A [[hypotrochoid]] is a curve traced by a point attached to a circle of radius {{mvar|r}} rolling around the inside of a fixed circle of radius {{mvar|R}}, where the point is at a distance {{mvar|d}} from the center of the interior circle. 
<gallery>
Image:2-circles hypotrochoid.gif|<div class="center">A hypotrochoid for which {{math|1= ''r'' = ''d''}}</div>
Image:HypotrochoidOutThreeFifths.gif|<div class="center">A hypotrochoid for which {{math|1= ''R'' = 5}}, {{math|1= ''r'' = 3}}, {{math|1= ''d'' = 5}}</div>
</gallery>
The parametric equations for the hypotrochoids are:

<math display="block">\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}</math>

Some examples:
<gallery>
Image: Param1a 6 4 1 a2.jpg|<div class="center">{{math|1= ''R'' = 6}} {{math|1= ''r'' = 4}} {{math|1= ''d'' = 1}}</div>
Image: Param1a 7 4 1 a4.jpg|<div class="center">{{math|1= ''R'' = 7}} {{math|1= ''r'' = 4}} {{math|1= ''d'' = 1}}</div>
Image: Param1a 8 3 2 a3.jpg|<div class="center">{{math|1= ''R'' = 8}} {{math|1= ''r'' = 3}} {{math|1= ''d'' = 2}}</div>
Image: Param1a 7 4 2 a4.jpg|<div class="center">{{math|1= ''R'' = 7}} {{math|1= ''r'' = 4}} {{math|1= ''d'' = 2}}</div>
Image: Param1a 15 14 1 a14.jpg|<div class="center">{{math|1= ''R'' = 15}} {{math|1= ''r'' = 14}} {{math|1= ''d'' = 1}}</div>
</gallery>