More About Ellipses

Find the Center of an Ellipse

Sometimes you have an ellipse but don't know the center. Finding the center is easy.

1. Draw two arbitrary parallel lines cutting chords across the ellipse (in red).
2. Bisect the chords and draw a line through the midpoints of the chords (in blue).
3. Construct a second pair of arbitrary parallel lines cutting chords across the ellipse (in green).
4. Bisect the chords and draw a line through the midpoints of the chords (in purple).

The intersection of the lines is the center of the ellipse. The simplest proof is to imagine doing this construction on a circle, the shearing the circle out of shape into an ellipse.

Find the Axes of an Ellipse

Since you can easily find the center of an ellipse, finding the axes is just as simple.

1. Given an ellipse with unknown axes and center, find the center as above.
2. Construct a circle with center at the center of the ellipse and intersecting the ellipse at four points.
3. Either:
   1. Bisect the arcs of the circle (not shown), or
   2. Construct the rectangle joining the points where the ellipse and circle intersect (green).
4. Construct the perpendicular bisectors of the sides of the rectangle (purple), or connect opposing pairs of arc bisectors.

Find the Foci of an Ellipse

Given the major and minor axes of an ellipse, you can always find the foci. You need the foci for some construction methods. Just draw radii of length a from the ends of the minor axis. Given the foci, however, you can't uniquely determine the axes. You need additional information such as the length of one axis. However, the major axis is always along the line through the foci and the minor axis always perpendicularly bisects the line between the foci.

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Created 28 December 1998, Last Update 14 September 2009
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