Ellipse

An ellipse is a conic section formed by cutting through a cone at an angle. Equivalently, it is defined as the locus, or set, of all points $P$ such that the sum of the distances from $P$ to two fixed foci (singular focus) is a constant. (The equivalence of these two definitions is a non-trivial fact.)

Ellipses tend to resemble circles which have been "flattened" or "stretched." They occur in nature as well as in mathematics: as was proven in Kepler's Laws, the planets all revolve about the sun in elliptical, not circular, orbits with the sun at one of the foci. Note that the circle is just a special case of the ellipse, just as a square is to a rectangle, and occurs when (in the first definition) the cut is perpendicular to the axis of the the cone, or (in the second definition) the two foci of the ellipse coincide.

For a given non-circular ellipse, there will be two points on the ellipse closest to the center and two points furthest away -- it will be "tall and skinny" or "short and fat." The segment connecting the center of the ellipse to one of the "farther away ends" is called the semimajor axis and the segment connecting the center to a closer end is called the semiminor axis. These two segments will be perpendicular. Drawing all four semi-axes divides the ellipse into 4 congruent quarters.

To draw an ellipse with two pushpins, a rubber band, pencil, and paper, stick the pushpins in the paper (these will be the "foci"), the rubber band around the pins, and trace out (please finish description, wording gets awkward after this for me).

The general equation of an ellipse with semi-minor and -major axes a and b and center C(h,k) is $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ , or, when centered at the origin, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ . (definition of eccentricity and polar equation needed)

The three-dimensional counterpart of the ellipse is the ellipsoid.

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Ellipse