Editing Parametric equation (section)

==Parametric space curves==
{{further|Space curve}}
[[File:Animated_Parametric_Function.webm|thumb|440x440px|Animated Parametric helix]]

===Helix===
[[File:Parametric Helix.png|thumb|300px|right|Parametric helix]]

Parametric equations are convenient for describing [[curve]]s in higher-dimensional spaces. For example:

<math display="block">\begin{align}
  x &= a \cos(t) \\
  y &= a \sin(t) \\
  z &= bt\,
\end{align}</math>

describes a three-dimensional curve, the [[helix]], with a radius of {{mvar|a}} and rising by {{math|2''{{pi}}b''}} units per turn. The equations are identical in the [[plane (mathematics)|plane]] to those for a circle.
Such expressions as the one above are commonly written as

<math display="block">\begin{align}
\mathbf{r}(t) &= (x(t), y(t), z(t)) \\
  &= (a \cos(t), a \sin(t), b t)\,,
\end{align}</math>

where {{math|'''r'''}} is a three-dimensional vector.

===Parametric surfaces===
{{main| Parametric surface}}
A [[torus]] with major radius {{mvar|R}} and minor radius {{mvar|r}} may be defined parametrically as

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

where the two parameters {{mvar|t}} and {{mvar|u}} both vary between {{math|0}} and {{math|2''{{pi}}''}}.

<gallery>
File:Torus.png|{{math|1=''R'' = 2}}, {{math|1=''r'' = 1/2}}
</gallery>

As {{mvar|u}} varies from {{math|0}} to {{math|2''{{pi}}''}} the point on the surface moves about a short circle passing through the hole in the torus. As {{mvar|t}} varies from {{math|0}} to {{math|2''{{pi}}''}} the point on the surface moves about a long circle around the hole in the torus.

===Straight line===
{{further|Linear equation}}

The parametric equation of the line through the point <math>\left( x_0, y_0, z_0 \right)</math> and parallel to the vector <math> a \hat\mathbf{i} + b \hat\mathbf{j} + c \hat\mathbf{k}</math> is<ref>{{Cite book|title=Calculus: Single and Multivariable.|date=2012-10-29|publisher=John Wiley|isbn=9780470888612|oclc=828768012|pages=919}}</ref>

<math display="block">\begin{align}
x & = x_0 +a t \\
y & = y_0 +b t \\
z & = z_0 +c t
\end{align}</math>