Parametric Equations of an Ellipse
Parametric Equations of an Ellipse
Definition:
Parametric equations are a way to represent the coordinates of points on a curve using a third variable, often denoted as a parameter. For an ellipse, parametric equations express the $x$ and $y$ coordinates of points on the ellipse in terms of a parameter.
Parametric Equations for an Ellipse:
The parametric equations for an ellipse centered at $(h,k)$ with semimajor axis $a$along the xaxis and semiminor axis $b$ along the yaxis are:
$x=h+a\mathrm{cos}(\theta )$ $y=k+b\mathrm{sin}(\theta )$
Where:
 $(h,k)$ represents the center of the ellipse.
 $a$ is the length of the semimajor axis.
 $b$ is the length of the semiminor axis.
 $\theta $ is the parameter that ranges from $0$ to $2\pi $ to trace the entire ellipse.
Interpretation and Properties:

Parameter $\theta $:
 Varies from $0$to $2\pi $ to sweep the ellipse's full path.
 Each value of $\theta $corresponds to a specific point on the ellipse.

Ellipse Points:
 As $\theta $ varies, $x$ and $y$ values change, generating a series of points on the ellipse.

Direction of Tracing:
 As $\theta $ increases, the parametric equations trace the ellipse counterclockwise from $0$ to $2\pi $.
Special Cases:

Circle Parametric Equations:
 For a circle ($a=b$), the parametric equations become: $x=h+r\mathrm{cos}(\theta )$ $y=k+r\mathrm{sin}(\theta )$ Where $r$ is the radius of the circle.

Changing Parameter Limits:
 Changing the limits of $\theta $ affects how much of the ellipse is traced. For instance, $\theta $ ranging from $0$ to $\pi $ traces only half of the ellipse.
Example:
Given: An ellipse has a center at $(3,2)$, a semimajor axis of length $6$, and a semiminor axis of length $4$. Find the parametric equations of this ellipse.
Solution:
Given: Center of the ellipse: $(h,k)=(3,2)$ Length of the semimajor axis: $a=6$ Length of the semiminor axis: $b=4$
The parametric equations for an ellipse centered at $(h,k)$ with semimajor axis $a$ and semiminor axis $b$ are: $x=h+a\mathrm{cos}(\theta )$ $y=k+b\mathrm{sin}(\theta )$
Substitute the given values: $x=3+6\mathrm{cos}(\theta )$ $y=2+4\mathrm{sin}(\theta )$
These equations represent the points on the ellipse in terms of a parameter $\theta $ranging from $0$ to $2\pi $.
Conclusion:
The parametric equations for the given ellipse centered at $(3,2)$ with a semimajor axis of length $6$ and a semiminor axis of length $4$ are $x=3+6\mathrm{cos}(\theta )$ and $y=2+4\mathrm{sin}(\theta )$. These equations define the coordinates of points on the ellipse as $\theta $ varies from $0$ to $2\pi $.