Features
The three types of conics are the ellipse, parabola, and hyperbola. The circle can be considered as a fourth type (as it was by Apollonius) or as a kind of ellipse. The circle and the ellipse arise when the intersection of cone and plane is a closed curve. The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone – for a right cone as in the picture at the top of the page this means that the cutting plane is perpendicular to the symmetry axis of the cone. If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola. In this case, the plane will intersect both halves (nappes) of the cone, producing two separate unbounded curves.
Various parameters are associated with a conic section, as shown in the following table. (For the ellipse, the table gives the case of a>b, for which the major axis is horizontal; for the reverse case, interchange the symbols a and b. For the hyperbola the east-west opening case is given. In all cases, a and b are positive.)
| conic section | equation | eccentricity (e) | linear eccentricity (c) | semi-latus rectum (ℓ) | focal parameter (p) |
|---|---|---|---|---|---|
| circle | |||||
| ellipse | |||||
| parabola | |||||
| hyperbola |
Conic sections are exactly those curves that, for a point F, a line L not containing F and a non-negative number e, are the locus of points whose distance to F equals e times their distance to L. F is called the focus, L the directrix, and e the eccentricity.
The linear eccentricity (c) is the distance between the center and the focus (or one of the two foci).
The latus rectum (2ℓ) is the chord parallel to the directrix and passing through the focus (or one of the two foci).
The semi-latus rectum (ℓ) is half the latus rectum.
The focal parameter (p) is the distance from the focus (or one of the two foci) to the directrix.
The following relations hold:
Read more about this topic: Conic Section
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