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

A conic section can be defined as the set of points whose distance to its focus is equal to the eccentricity times the distance to the corresponding directrix. Even in the case of two foci, the described set, applied on a single focus-directrix combination, is the whole conic section.

Note that (non-circular) ellipses and hyperbolas each have a pair of foci. An ellipse can be described as the set of points for which the sum of the distances to the foci is constant, while a hyperbola is the set of points for which the absolute value of the difference of the distances to the foci is constant.

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.

Astronomical significance

In the gravitational two-body problem, the orbits of the two bodies are described by conic sections with foci at the center of mass.

Conic sections

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