An ellipse keeps a constant sum to two foci
For every point on an ellipse, the distances to two fixed points called foci add up to the same total, and a circle is simply the case where the foci merge. Its area has a tidy formula, yet finding the exact perimeter requires integration.
Apollonius of Perga gave the curve its name in his Conics, from the Greek élleipsis, meaning omission. The ellipse is the closed member of the conic sections, the curves made by slicing a cone with a plane; parabolas and hyperbolas are the open, unbounded ones. Dandelin spheres prove that any plane cutting a cone at a gentler slope than its sides, without passing through the tip, gives an ellipse. Slicing a round cylinder at an angle does too.
The line through the foci is the major axis and the perpendicular line through the centre is the minor axis. Their full lengths, the widest and narrowest diameters, are usually written 2a and 2b. The major axis meets the curve at two vertices and the minor axis at two co-vertices. Centred at the origin with the major axis along x, the curve obeys x²/a² + y²/b² = 1 and can be traced as (a cos t, b sin t) as t runs from 0 to 2π.
Eccentricity measures elongation. It equals the distance from centre to focus divided by a; a circle scores zero, and stretching without limit ends not in an ellipse but a parabola. An equivalent definition uses one focus and a line outside the curve called the directrix, with the ratio of the two distances fixed at the eccentricity.
Each planet's orbit is roughly an ellipse with the Sun, or more exactly the Sun–planet centre of mass, at one focus, and the same holds for moons and other two-body systems. A circle seen at an angle projects to an ellipse, and matching horizontal and vertical oscillations of equal frequency trace one, the simplest Lissajous figure, an effect echoed in elliptically polarised light.
Source: Ellipse
