Conic Section

A conic section is a curve formed by the intersection of a plane with the surface of a circular cone. In geometry, its nondegenerate forms are the circle, ellipse, parabola, and hyperbola, with the circle regarded as a special ellipse. In analytic geometry, conics are represented by second-degree polynomial equations in two coordinates. This algebraic description also accommodates degenerate cases, such as points and pairs of straight lines. (openstax.org)

Geometric construction

The conventional construction uses a double cone: two infinitely extended conical surfaces, called nappes, sharing an apex and an axis. A plane perpendicular to the axis, but not passing through the apex, produces a circle. An oblique plane that cuts only one nappe and is not parallel to a generating line produces a noncircular ellipse. A generating line is a straight line on the cone passing through its apex. (openstax.org)

A plane parallel to a generating line produces a parabola, provided it avoids the apex. A plane intersecting both nappes produces a hyperbola, whose two disconnected branches lie on opposite nappes. If the cutting plane passes through the apex, the intersection degenerates into a point, a single line, or two intersecting lines. These cases distinguish genuine curved sections from limiting configurations of the same construction. (openstax.org)

Historical development

Conics were extensively studied in ancient Greek mathematics. Apollonius of Perga, active around the late third and early second centuries BCE, organized and substantially extended their theory in his treatise Conics. Its eight books examined diameters, tangents, asymptotes, normals, and other properties. The first four books survive in Greek, while Arabic transmission preserves material through Book VII; Book VIII is lost. (mathshistory.st-andrews.ac.uk)

Apollonius built on earlier work, including that of Euclid, while developing more general treatments and additional theorems. His terminology established the names ellipse, parabola, and hyperbola. His approach was geometric rather than based on modern symbolic coordinate equations. (mathshistory.st-andrews.ac.uk)

Distance definitions and eccentricity

Conics can also be characterized through Euclidean distances. A focus is a distinguished fixed point; a directrix is an associated fixed line. For a noncircular, nondegenerate conic, the ratio

is constant as the point $P$ moves along the curve. Here $F$ is a focus, $d$ its corresponding directrix, and $e$ is the eccentricity. Distance to the directrix means perpendicular distance. (openstax.org)

The ranges $0<e<1$, $e=1$, and $e>1$ identify ellipses, parabolas, and hyperbolas respectively. A circle has eccentricity zero, but it is not obtained from this ratio using a finite directrix. Noncircular ellipses and hyperbolas have two foci and two corresponding directrices; a parabola has one of each. Eccentricity determines the type of conic, not its size or position. (openstax.org)

An ellipse is alternatively the set of points whose distances to two fixed foci have a constant sum. For a hyperbola, the absolute difference of those distances is constant. A parabola consists of points equally distant from its focus and directrix. These definitions require no physical cone and lead directly to coordinate equations. (openstax.org)

Standard equations

With a suitable choice of origin and coordinate axes, the principal forms are:

For the ellipse, the foci are $(\pm c,0)$, where $c^2=a^2-b^2$, and $e=c/a$. For the hyperbola, $c^2=a^2+b^2$, again with $e=c/a$. Its asymptotes are $y=\pm(b/a)x$. The parabola’s directrix is $x=-p$. Translations replace $x,y$ by $x-h,y-k$, while rotations accommodate differently oriented axes. (openstax.org)

More generally, a conic has an equation of the form

where at least one of $A,B,C$ is nonzero. For a real nondegenerate conic, the discriminant $B^2-4AC$ is negative for an ellipse, zero for a parabola, and positive for a hyperbola. This expression is unchanged by rotation of axes. Degenerate or empty real loci require additional examination: the discriminant alone does not guarantee a nondegenerate curve. (openstax.org)

Polar form and applications

In polar coordinates, with a focus at the origin and an appropriately oriented axis, a conic can be written

where $\ell>0$ is the semilatus rectum. Changing the orientation introduces a minus sign or replaces cosine by sine. This form emphasizes eccentricity and the distance from a focus rather than from a center. (openstax.org)

In classical mechanics, ideal two-body motion under inverse-square gravity follows conic trajectories when angular momentum is nonzero. Bound trajectories are ellipses, including circles; unbound trajectories are parabolas or individual branches of hyperbolas. With gravitational potential set to zero at infinity, negative, zero, and positive orbital energy correspond to these three cases. Johannes Kepler’s first law identifies planetary orbits as ellipses with the Sun at one focus within this idealized description. (openstax.org)

In optics, conics have characteristic reflection properties. Rays parallel to a parabola’s axis reflect toward its focus, while rays originating at one focus of an ellipse reflect toward the other. These properties underlie parabolic reflectors and elliptical focusing arrangements. (openstax.org)