Ellipse

Not to be confused with Ellipsis.

An ellipse is a shape that looks like an oval or a flattened circle.

An ellipse obtained as the intersection of a cone with a plane.

In geometry, an ellipse is a plane curve which results from the intersection of a cone by a plane in a way that produces a closed curve.

Circles are special cases of ellipses, created when the cutting plane is perpendicular to the cone's axis.

An ellipse and its properties.

The foci (purple crosses) are at intersects of the major axis (red) and a circle (cyan) of radius equal to the semi-major axis (blue), centred on an end of the minor axis (grey)

A circle has one center, called a focus. An ellipse has two foci.

An ellipse is simply all points on a graph that the sum of the distances from 2 points are the same. For example, an ellipse can be made by putting two pins into cardboard and a circle of string around those two, then putting a pencil in the loop and pulling as far as possible without breaking the string in all directions. The orbits of the planets are ellipses, with the sun at one focus.

In analytic geometry the equation of an ellipse is

$${\frac {(x-h)^{2}}{a^{2}}}+{\frac {(y-k)^{2}}{b^{2}}}=1$$
where the center of the ellipse is (h,k). 2A is the length from each end of the longer skinnier side. 2b is the length of the 2 ends of the short side. A²-B²=C² for c is the length between the foci and the center.

An ellipse can also be expressed as

${\frac {(x-h)^{2}}{b^{2}}}+{\frac {(y-k)^{2}}{a^{2}}}=1$

In this case, it will be taller than it is wide.

Other websites

• Ellipse & Hyperbola Construction Archived 2008-02-11 at the Wayback Machine - An interactive sketch showing how to trace the curves of the ellipse and hyperbola. (Requires Java.)
• Ellipse Construction Archived 2007-12-02 at the Wayback Machine - Another interactive sketch, this time showing a different method of tracing the ellipse. (Requires Java.)
• Ellipse on MathWorld - More on Ellipse
• The Shape and History of The Ellipse in Washington, D.C. by Clark Kimberling
• Collection of animated ellipse demonstrations. Ellipse, axes, semi-axes, area, perimeter, tangent, foci.
• Woodworking videos showing how to work with ellipses in wood.