Focus (geometry)

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In geometry, the foci (singular focus) are a pair of special points used in describing conic sections. The four types of conic sections are the circle, parabola, ellipse, and hyperbola.

The focus has two equivalent defining properties; and they always fall on the major axis of symmetry of the conic. The simpler depends on the type of conic:

In an ellipse, the sum of the distances from any point on the ellipse to the two foci is a constant (which is always the length of the major axis of the ellipse).
In a circle, there is only one focus, the center of the circle, and all the points of the circle are equidistant from it. (This can be viewed a special case of the above, with a circle being an ellipse with two foci at the same point; the sum of the distances is the diameter.)
In a hyperbola, the difference of the distances is always constant.
A parabola also only has one focus (although it is sometimes useful to speak of a focus at infinity); but there is a line called the directrix such that the distance from any point of the parabola to the focus is equal to the (perpendicular) distance from the point to the directrix.

The rule for the parabola can be generalized to other conics, and this is the other defining property: A conic section can be defined as the set of points such that the ratio of distance to its focus to the distance to the corresponding directrix is a constant, called the eccentricity. Even in the case of two foci, the described set, applied on a single focus-directrix combination, is the whole conic section.

The circle has eccentricity 0, and the directrix is a line at infinity. The focus-directrix property is thus true of the circle, but it is also true of every other point on the plane.

[edit] Conics in projective geometry

It is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix.

For the ellipse, both the focus and the center of the directrix circle have finite coordinates and the radius of the directrix circle is greater than the distance between the center of this circle and the focus; thus, the focus is inside the directrix circle. The ellipse thus generated has its second focus at the center of the directrix circle.

For the parabola, the center of the directrix moves to the point at infinity (see projective geometry. The directrix 'circle' becomes a curve with zero curvature, indistinguishable from a straight line. The two arms of the parabola become increasingly parallel as they extend, and 'at infinity' become parallel; using the principles of projective geometry, the two parallels intersect at the point at infinity and the parabola becomes a closed curve (elliptical projection).

To generate a hyperbola, the radius of the directrix circle is chosen to be less than the distance between the center of this circle and the focus; thus, the focus is outside the directrix circle. The arms of the hyperbola approach asymptotic lines and the 'right-hand' arm of one branch of a hyperbola meets the 'left-hand' arm of the other branch of a hyperbola at the point at infinity; this is based on the principle that, in projective geometry, a single line meets itself at a point at infinity. The two branches of a hyperbola are thus the two (twisted) halves of a curve closed over infinity.

In projective geometry, all conics are equivalent in the sense that every theorem that can be proved for one conic section applies to all the others.