Ellipse

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

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

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.

The equation of an ellipse is

[math]\displaystyle{ \frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1 }[/math]
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

[math]\displaystyle{ \frac{(x-h)^{2}}{b^{2}} + \frac{(y-k)^{2}}{a^{2}} = 1 }[/math]

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

Ellipse Media

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  An ellipse (red) obtained as the intersection of a cone with an inclined plane.

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  Ellipse: Vertices, semi-axes, linear eccentricity

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  Ellipses: examples with increasing eccentricity

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  Ellipse: definition by sum of distances to foci

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  Ellipse: definition by focus and circular directrix

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  The construction of points based on the parametric equation and the interpretation of parameter t, which is due to de la Hire

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  Ellipse points calculated by the rational representation with equal spaced parameters (\Delta u = 0.2).

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  Ellipse as an affine image of the unit circle

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  Whirls: nested, scaled and rotated ellipses. The spiral is not drawn: we see it as the locus of points where the ellipses are especially close to each other.

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.