Ellipse

In working with ellipses it is useful to identify several special points, chords, measurements, and properties:

The major axis: The longest chord in an ellipse that passes through the foci. It is equal in length to the constant sum in Definition 1 above. In Definitions 3 and 4 the larger of the constants a or b is equal to the semimajor axis.

The center: The midpoint, C, of the major axis.

The vertices: The end points of the major axis.

The minor axis: The chord which is perpendicular to the major axis at the center. It is the shortest chord which passes through the center. In Definitions 3 and 4 the smaller of a or b is the semiminor axis.

The foci: The fixes points in Definitions 1 and 2. In any ellipse, these points lie on the major axis and are at a distance c on either side of the center. If a and b are the semimajor and semiminor axes respectively, then a² = b² + c^2.. In the examples in Definitions 3 and 4, the foci are 3 units from the center.

The eccentricity: A measure of the relative elongation of an ellipse. It is the ratio e in Definition 2, or the ratio FC/VC (center-to-focus divided by center-to-vertex). These two definitions are mathematically equivalent. When the eccentricity is close to zero, the ellipse is almost circular; when it is close to 1, the ellipse is almost a parabola. All ellipses having the same eccentricity are geometrically similar figures.

The angle measure of eccentricity: Another measure of eccentricity. It is the acute angle formed by the major axis and a line passing through one focus and an end point of the minor axis. This angle is the arc cosine of the eccentricity.

The area: The area of an ellipse is given by the simple formula PIab, where a and b are the semimajor and semiminor axes.

The perimeter: There is no simple formula for the perimeter of an ellipse. The formula is an elliptic integral which can be evaluated only by approximation.

The reflective property of an ellipse: If an ellipse is thought of as a mirror, any ray which passes through one focus and strikes the ellipse will be reflected through the other focus. This is the principle behind rooms designed so that a small sound made at one location can be easily heard at another, but not elsewhere in the room. The two locations are the foci of an ellipse.