foci

Foci, plural of focus, are fixed points used to define conic sections and to describe optical systems. In geometry, a conic is the locus of points whose distances to the foci satisfy certain conditions: ellipse, hyperbola, and parabola have two, two, and one foci, respectively. A circle is a degenerate case in which the two foci coincide at the center.

Ellipse: For an ellipse with major axis 2a and minor axis 2b (a ≥ b), the foci lie on the major axis at distance c from the center, where c^2 = a^2 − b^2. The eccentricity e = c/a < 1. For any point P on the ellipse, PF1 + PF2 = 2a.

Hyperbola: Hyperbolas have two fixed foci F1 and F2. The difference of distances from any point P on the hyperbola to the foci is constant: |PF1 − PF2| = 2a. In standard form x^2/a^2 − y^2/b^2 = 1, the foci are at (±c,0) with c^2 = a^2 + b^2 and e = c/a > 1.

Parabola: A parabola has a single focus F and a directrix. In y^2 = 4ax, the focus is at (a,0) and the directrix is x = −a. A point P on the parabola is equidistant from F and the directrix.

Optics: In lenses and mirrors, the focus is where light rays converge (real) or appear to diverge (virtual). The focal length f characterizes the system: for a thin lens, 1/f = (n − 1)(1/R1 − 1/R2); for a spherical mirror, f = R/2. These foci reflect the geometry of the surface and the refractive index.