Ellipse

Ellipse, in geometry, closed two-dimensional curve that looks like a flattened circle. An ellipse may be defined as all points, P, the sum of whose distances from two fixed points is a constant, C. The two fixed points that define an ellipse, labeled F₁ and F₂, are known as its foci:

The ellipse is one of the most important curves in physical science. In astronomy, the orbits of Earth and the other planets around the Sun are ellipses. The ellipse is used in engineering in the arches of some bridges and the design of gears for certain types of machinery such as Wankel engines and punch presses.

The major axis of an ellipse is a straight line that passes through the two foci lengthwise along the ellipse and extends to meet the curve at the farthest point of each end. Any ellipse is symmetrical with respect to its major axis—the portion on one side of the major axis is a mirror image of the portion on the other side. Ellipses are also symmetrical with respect to their minor axes, lines perpendicular to the major axis at the midpoint between the two foci.

An ellipse can be drawn by placing two tacks in a piece of paper at the two foci and loosely tying a length of string between them. A pencil holding the string tight will trace an ellipse as it moves.

An ellipse can also be drawn by graphing its equation on xy axes. The equation of an ellipse centered at (0, 0)—where the x and y axes meet—is

The more general equation of an ellipse centered at any point (h, k) is

A numerical value called eccentricity determines the shape of an ellipse. Eccentricity is the ratio of the distance between the foci to the length of the major axis. Ellipses range in shape from perfectly circular when eccentricity equals 0, to extremely long and narrow when eccentricity approaches 1. Eccentricity is always less than 1, and is equal to .

Ellipses are a type of conic section, a class of curves formed by a plane that cuts through a right circular cone. A circle, which is formed by a plane that is perpendicular to the axis of the cone, is a specialized form of ellipse. In an ellipse that is also a circle, the two foci coincide in a single point (the circle’s center), and the major and minor axes are of equal length.