How to Write Equations for Ellipses

Comets follow ellipsoidal orbits.

When Halley's comet comes rocketing past the Earth every 76 years on its way to perihelion, its nearest approach to the sun, it is following an elliptical orbit. An ellipse or elliptical orbit is like an elongated circle. Scientists describe ellipses using parametric equations. Parametric equations define the variables "x" and "y" in terms of a third variable, often called "t." A single x-value and a single y-value still define a single point on the ellipse, however.

Instructions

1. Write the equation "x = a * cos (t) + b." Write the equation "y = c * sin (t) + d."

2. Decide how many units you would like the horizontal radius of your ellipse to be. Replace "a" in your equation with this number. For example, you decide your ellipse will have a horizontal radius of 7 units. You write, "x = 7cos (t) + b."

3. Decide how many units to the left or the right of the center of the graph you want the center point of your ellipse to be. If the center point will be to the left, replace "b" in your equation with the negative of that number. If the center point of the ellipse will be to the right of the center of the graph, replace "b" in your equation with the number. For example, you decide the center point of your ellipse will be 3 units to the left of the center of the graph. You write, "x = 7cos (t) - 3."

4. Decide how many units you would like the vertical radius of your ellipse to be. Replace c in your equation with this number. For example, you decide your ellipse will have a horizontal radius of 5 units. You write, "y = 5sin (t) + d."

5. Decide how many units above or below the center of the graph you want the center point of your ellipse to be. If the center point will be below, replace "d" in your equation with the negative of that number. If the center point of the ellipse will be above the center of the graph, replace "d" in your equation with the number. For example, you decide the center point of your ellipse will be 3 units above the center of the graph. You write, "y = 5sin (t) + 3."

6. Write the completed parametric equations for your ellipse. For example, you write, "x = 7cos (t) - 3 and y = 5sin (t) + 3."

References

• University of California, Berkeley; The Comet's Tale: Comet Orbits
• Paul's Online Math Notes; Parametric Equations and Curves; Paul Dawkins; 2011
• University of Denver; Ellipse: Definition of the Ellipse
• University of Georgia; Parametric Curves; J. Wilson; 2010
• "Calculus: Concepts and Contexts"; James Stewart; 2006