Definition:Parametric Equation

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Definition

Let $\map \RR {x_1, x_2, \ldots, x_n}$ be a relation on the variables $x_1, x_2, \ldots, x_n$.

Let the truth set of $\RR$ be definable as:

$\forall k \in \N: 1 \le k \le n: x_k = \map {\phi_k} t$

where:

$t$ is a variable whose domain is to be defined
each of $\phi_k$ is a mapping whose domain is the domain of $t$ and whose codomain is the domain of $x_k$.


Then each of:

$x_k = \map {\phi_k} t$

is a parametric equation.


The set:

$\set {\phi_k: 1 \le k \le n}$

is a set of parametric equations specifying $\RR$.


Parameter

$t$ is referred to as the (independent) parameter of $\set {\phi_k: 1 \le k \le n}$.


Analytic Geometry

Let $\CC$ be a curve.

A set of parametric equations for $\CC$ is a set of equations that determine the locus of $\CC$ in terms of a single parameter.


Examples

Plane Curve

Let $\CC$ be a plane curve embedded in a Cartesian plane.

Then a set of parametric equations for $\CC$ can be expressed in the form:

\(\ds x\) \(=\) \(\ds \map f p\)
\(\ds y\) \(=\) \(\ds \map g p\)

where $p$ is the parameter.


Circle

Let $\CC$ be the circle embedded in a Cartesian plane with the equation:

$x^2 y^2 = 16$

This can be expressed in parametric equations as:

\(\ds x\) \(=\) \(\ds 4 \cos \theta\)
\(\ds y\) \(=\) \(\ds 4 \sin \theta\)

where $\theta$ is the parameter representing the angle between the $x$-axis and the point $\paren {x, y}$ on $\CC$.

Each point on $\CC$ corresponds exactly to a value of $\theta$ such that $\theta \in \hointr 0 {2 \pi}$.


Ellipse

Let $\EE$ be the ellipse embedded in a Cartesian plane with the equation:

$\dfrac {x^2} {a^2} \dfrac {y^2} {b^2} = 1$

This can be expressed in parametric equations as:

\(\ds x\) \(=\) \(\ds a \cos \phi\)
\(\ds y\) \(=\) \(\ds b \sin \phi\)

where $\phi$ is the parameter representing the eccentric angle of the point $\paren {x, y}$ on $\EE$.

Each point on $\CC$ corresponds exactly to a value of $\phi$ such that $\phi \in \hointr 0 {2 \pi}$.


Also known as

Some older texts, particularly in the context of analytic geometry, refer to parametric equations as freedom-equations, as they express the freedom of the movement of the tuple $\tuple {x_1, x_2, \ldots, x_n}$ as $t$ changes.


Also see

  • Results about parametric equations can be found here.


Sources

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