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 semi-major axis a along the x-axis and semi-minor axis b along the y-axis are:

x=h+acos(θ)  y=k+bsin(θ)

Where:

  • (h,k) represents the center of the ellipse.
  • a is the length of the semi-major axis.
  • b is the length of the semi-minor axis.
  • θ is the parameter that ranges from 0 to 2π to trace the entire ellipse.

Interpretation and Properties:

  1. Parameter θ:

    • Varies from 0 to 2π to sweep the ellipse's full path.
    • Each value of θ corresponds to a specific point on the ellipse.
  2. Ellipse Points:

    • As θ varies, x and y values change, generating a series of points on the ellipse.
  3. Direction of Tracing:

    • As θ increases, the parametric equations trace the ellipse counterclockwise from 0 to 2π.

Special Cases:

  1. Circle Parametric Equations:

    • For a circle (a=b), the parametric equations become: x=h+rcos(θ) y=k+rsin(θ) Where r is the radius of the circle.
  2. Changing Parameter Limits:

  • Changing the limits of θ affects how much of the ellipse is traced. For instance, θ ranging from 0 to π traces only half of the ellipse.

Example:

Given: An ellipse has a center at (3,2), a semi-major axis of length 6, and a semi-minor axis of length 4. Find the parametric equations of this ellipse.

Solution:

Given: Center of the ellipse: (h,k)=(3,2) Length of the semi-major axis: a=6 Length of the semi-minor axis: b=4

The parametric equations for an ellipse centered at (h,k) with semi-major axis a and semi-minor axis b are: x=h+acos(θ) y=k+bsin(θ)

Substitute the given values: x=3+6cos(θ) y=2+4sin(θ)

These equations represent the points on the ellipse in terms of a parameter θ ranging from 0 to 2π.

Conclusion:

The parametric equations for the given ellipse centered at (3,2) with a semi-major axis of length 6 and a semi-minor axis of length 4 are x=3+6cos(θ) and y=2+4sin(θ). These equations define the coordinates of points on the ellipse as θ varies from 0 to 2π.