FociEdit

In geometry, the foci (singular: focus) are fixed points that underlie the classical definitions of the conic sections. These points encode the way each curve is generated or constrained. The most familiar cases are the ellipse, the parabola, and the hyperbola, each of which can be described in terms of distances to one or more foci. The two foci of an Ellipse and the single focus of a Parabola play central roles in both the aesthetic geometry of these curves and their practical applications in optics, astronomy, and engineering. The foci also appear in the definition of a Conic section more generally through the relationship between distance to a focus and distance to a directrix, a line that helps organize the curve’s shape.

Historically, the idea of fixed focal points emerged from the ancient study of conic sections by Apollonius of Perga and later matured with the rise of analytic geometry in the early modern period. The ellipse, parabola, and hyperbola were understood not only as loci but as objects with precise, measurable relationships to fixed points and lines. The analytic approach—developed by figures such as René Descartes—made it possible to locate the foci explicitly in coordinate form and to derive their properties from equations. Over time, the concept of eccentricity, the ratio that measures how elongated a conic is, became a standard way to relate the foci to the overall shape.

Definition and basic properties

Ellipse

An ellipse is the set of points for which the sum of the distances to two fixed points, the foci, is a constant. The standard, axis-aligned form is given by the equation x^2/a^2 + y^2/b^2 = 1, where a > b. The foci lie along the major axis at positions (±c, 0), with c^2 = a^2 − b^2. The distance from the center to either focus is c, and the major axis length is 2a while the minor axis length is 2b. The eccentricity e = c/a satisfies 0 < e < 1 for an ellipse. In many descriptions, the foci are denoted as the two fixed points F1 and F2, so that PF1 + PF2 remains constant for every point P on the curve.

Hyperbola

A hyperbola is the locus of points for which the absolute difference of the distances to two fixed points, the foci, is a constant. The standard form x^2/a^2 − y^2/b^2 = 1 has foci at (±c, 0) with c^2 = a^2 + b^2. The difference PF1 − PF2 equals ±2a for points on the curve. The eccentricity e = c/a satisfies e > 1. The transverse axis length is 2a, and the asymptotes reflect the hyperbola’s open, two-branch structure.

Parabola

A parabola is the locus of points equidistant from a single focus and a fixed line called the directrix. In the standard orientation, the parabola y^2 = 4px has its focus at (p, 0) and its directrix at x = −p, with p > 0. All points P on the parabola satisfy PF = dist(P, directrix). The eccentricity is e = 1. The vertex is at the origin, and the focal length p determines the opening and spread of the curve.

General and rotated configurations

If a conic is rotated or translated, the same focal relationships persist, though the simple horizontal or vertical coordinate forms give way to more general equations. In all cases, each conic type is associated with one or more foci and, for the non-parabolic cases, a directrix or pair of directrices that encode the curve’s geometry. The broader concept can be captured by the ratio PF/L, where L is the distance to a corresponding directrix; this ratio defines the eccentricity e for the conic.

Historical notes

The idea that a conic section can be described by fixed points and a fixed line or distance condition goes back to the work of Apollonius of Perga in ancient Greece. His descriptions of ellipse, parabola, and hyperbola laid the groundwork for later, more algebraic treatments.
The advent of analytic geometry by René Descartes allowed mathematicians to translate geometric focal properties into coordinate equations. This enabled precise calculation of the foci for a given conic and made the constants a, b, and c tangible in the same framework.
In the broader history of science, the idea of a focus appears in optics and astronomical models. For example, elliptical mirrors concentrate light from one focus to the other, a principle that has practical applications in telescopes and sensors, and the orbital shapes of planets and comets are well described by conic sections with specific foci, as later refined in Kepler's laws.

Applications and significance

Optics and imaging: The collecting and focusing properties of mirrors and lenses rely on the focal structure of conic sections. Elliptical reflectors, for instance, take advantage of the fact that rays emanating from one focus reflect to the other focus, a principle used in devices ranging from old telescopes to modern optical instruments.
Astronomy and celestial mechanics: The orbits of planets and many comets are well approximated by ellipses with a central focus at a primary body like the sun. This focus-centric view is a cornerstone of orbital dynamics and is connected to Kepler's laws.
Engineering and design: The precise geometry of conics informs antenna design, satellite dishes, and other devices where controlled focusing and distribution of signals are important.
Mathematics and theory: The focal definitions illuminate properties such as the constant-sum or constant-difference conditions, which tie into the study of ellipse and hyperbola in Coordinate geometry and more general Conic section theory.