## 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

x2a2  +  y2b2  =  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.

 a: b: Notice that if a is greater than b, the foci lie on the x-axis and have coordinates (c, 0) and (-c, 0), where c2 = a2 - b2. 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