## The Ellipse

An ellipse is a curve that generalizes a circle. A circle is defined to be the set of points at a constant distance from a single fixed point, the center. An ellipse, on the other hand, is defined using two fixed points, called the *foci* (the plural of focus). For any point on the ellipse, the sum of the distances from the point to the two foci is a constant. As Kepler discovered, the orbit of the Earth is an ellipse with the Sun at one of the foci.

One equation of an ellipse is

^{x2}⁄_{a2} + ^{y2}⁄_{b2} = 1

where a and b are constants. The figure below shows the graph of this equation for the values of a and b in the boxes. The red points are the foci. Try changing the values of a and b in the boxes. Then click **Draw ellipse**.

Notice that if a is greater than b, the foci lie on the x-axis and have coordinates (c, 0) and (-c, 0), where c^{2} = a^{2} - b^{2}. In this case, the ellipse is longer in the horizontal direction than the vertical. If b is greater than a, the foci lie on the y-axis and the ellipse is longer in the vertical direction.

What curve do you get if a equals b?

Paul Trow's math software.

Copyright 2009 by Paul Trow