Ellipse

The edge of an ellipse is a constant distance from its two foci.

An ellipse is a "conic section", a two-dimensional shape. The term conic sections refers to the two-dimensional shapes you get with a slice through a three-dimensional cone, The other three conic sections are the circle, the parabola and hyperbolas. Like the circle the slices that form ellipses slice completely through, but, unlike the circle, the cut is at an angle, so they are like stretched circles.

The measure of how stretched an ellipse is is its eccentricity. The eccentricity of ellipse ranges from just above zero to just less than one. This is because a circle has an eccentricy of zero. The parabola has an eccentricity of one. The very longest ellipses are hard to distinguish from parabolas, simply by measurement, which is why their eccentricities are close to one.

A circle has a single focus. Ellipses have two foci. The edge of a circle is a constant distance from its centre, its focus. The edge of an ellipse is a constant distance from its two foci. The line from a foci to the ellipses edge is called a "axis". For any point on the edge of an ellipse, the sum of the two axes is a constant.

All objects that orbit one another follow an elliptical path. Johann Kepler, working before the invention of calculus, determined from detailed study of astronomical observation, that as objects approached one another their relative velocity increased, and, as they receded from one another, their velocity decreased.

The diameter through the two foci is called the major axis, and its length equals the sum of its distances from the foci. Half the length of the major axis is called the semi-major axis. The diameter that is the perpendicular bisector of the major axis is called the minor axis. Half the length of the minor axis is called the semi-minor axis. These axes are the longet and shortest diameters.

1 minus the ratio of the minor to the major axis is called the ellipticity of the ellipse. It ranges from 0 for a circle to 1 for a straight mline.

The area of an ellipse equals the product of pi, the semi-major axis and the semi-minor axis. There is no simple formula for the circumference (except in the case of a circle or a straight line); its calculation requires elliptic functions.

If a mirror is made in the shape of an ellipse, a beam of light passing through one focus will, after reflection, pass through the other focus. Continuing on, it will after successive reflections pass through each focus alternately, while getting closer and closer to the major axis.

An ellipse is one type of conic section.

The study of ellipses has also proven useful to mechanical engineers, who make use of elliptical cams and elliptical gears. In the days before electronic computers battleships relied on analog computers to arrive at their firing solution -- ie where to aim their big cannons. These large analog computers relied on elliptically shaped cams. Technicians, relying on estimates of their enemy's distance and speed, and estimates of the speed of the wind, would tune settings on cams whose shape emulated the fall of shot, across a range of distances. If their estimates were close they could read out a firing solution from these analog computers that would be close enough to score a hit.