Parametric Equation of an Ellipse

An ellipse can be defined as the locus of all points that satisfy the equations

x = a cos t
y = b sin t

where:
x,y are the coordinates of any point on the ellipse,
a, b are the radius on the x and y axes respectively, ( * See radii notes below )
t is the parameter, which ranges from 0 to 2π radians.

This equation is very similar to the one used to define a circle, and much of the discussion is omitted here to avoid duplication. See Parametric equation of a circle as an introduction to this topic.

The only difference between the circle and the ellipse is that in a circle there is one radius, but an ellipse has two:

One radius is measured along the x-axis and is usually called a.
The other is measured along the y-axis and is usually called b.

For a circle both these radii have the same value.

Ellipses centered at the origin

If the ellipse is centered on the origin (0,0) the equations are

where
a is the radius along the x-axis ( * See radii notes below )
b is the radius along the y-axis

Note that the equations on this page are true only for ellipses that are aligned with the coordinate plane, that is, where the major and minor axes are parallel to the coordinate system.

In the applet above, drag one of the four orange dots around the ellipse to resize it, and note how the equations change to match.

Ellipses not centered at the origin

Just as with the circle equations, we add offsets to the x and y terms to translate (or "move") the ellipse to the correct location. So the full form of the equations are

where, as before
a is the radius along the x-axis ( * See radii note below )
b is the radius along the y-axis

(h,k) are the x and y coordinates of the ellipse's center.

In the applet above, drag the orange dot at the center to move the ellipse, and note how the equations change to match. Also, adjust the ellipse so that a and b are the same length, and convince yourself that in this case, these are the same equations as for a circle.

A circle is just a particular ellipse

In the applet above, drag the right orange dot left until the two radii are the same. This is a circle, and the equations for it look just like the parametric equations for a circle. This demonstrates that a circle is just a special case of an ellipse.

The parameter t

The parameter t can be a little confusing with ellipses. For any value of t, there will be a corresponding point on the ellipse. But t is not the angle subtended by that point at the center. To see why this is so, consider an ellipse as a circle that has been stretched or squashed along each axis. In the figure below we start with a circle, and for simplicity give it a radius of one (a "unit circle").

The angle t defines a point on the circle which has the coordinates

The radius is one, so it is omitted. The blue ellipse is defined by the equations

So to get the corresponding point on the ellipse, the x coordinate is multiplied by two, thus moving it to the right. This causes the ellipse to be wider than the circle by a factor of two, whereas the height remains the same, as directed by the values 2 and 1 in the ellipse's equations.

So as you can see, the angle t is not the same as the angle that the point on the ellipse subtends at the center.

However, when you graph the ellipse using the parametric equations, simply allow t to range from 0 to 2π radians to find the (x, y) coordinates for each value of t.

Other forms of the equation

Using the Pythagorean Theorem to find the points on the ellipse, we get the more common form of the equation. For more see General equation of an ellipse

Algorithm for drawing ellipses

This form of defining an ellipse is very useful in computer algorithms that draw circles and ellipses. In fact, all the circles and ellipses in the applets on this site are drawn using this equation form. For more on this see An Algorithm for Drawing Circles.

Things to try

In the above applet click 'reset', and 'hide details'.
Drag the five orange dots to create a new ellipse at a new center point.
Write the equations of the ellipse in parametric form.
Click "show details" to check your answers.

* Note on radii

In many textbooks, the two radii are specified as being the semi-major and semi-minor axes. Recall that these are the longest and shortest radii of the ellipse respectively. The trouble with this is that if the ellipse is tall and narrow, they have to be reversed, so you wind up with two forms of the equations, one for tall thin ellipses and another for short wide ones.

Regardless of what you call these radii, remember that the x equation must use the radius along the x-axis, and the y equation must use the radius along the y-axis: