Parametric Equation/Examples
Examples of Parametric Equations
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 $\phi \in \hointr 0 {2 \pi}$.