Conic section

Types and defining properties

Conics are commonly classified by eccentricity, a nonnegative number that compares distances to a focus and to a corresponding directrix. The classification is:

• Ellipse: eccentricity e with 0 < e < 1; the circle occurs when e = 0.
• Parabola: e = 1; the curve is equidistant from a single focus and a line (the directrix).
• Hyperbola: e > 1; formed by the intersection producing two separate branches.

Many geometric properties follow from these definitions: for example, parabolas reflect rays from their focus into parallel directions, and ellipses reflect rays from one focus to the other. Dandelin spheres give a direct geometric construction of foci for conics produced by plane-cone intersections.

Algebraic description and classification test

On the plane a conic is a nondegenerate solution of a general quadratic equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. The type can be read from the discriminant B^2 - 4AC:

• Ellipse (or circle) when B^2 - 4AC < 0,
• Parabola when B^2 - 4AC = 0,
• Hyperbola when B^2 - 4AC > 0.

Standard coordinate forms (after translation and rotation) are useful in calculation: for example, x^2/a^2 + y^2/b^2 = 1 for an ellipse, x^2/a^2 - y^2/b^2 = 1 for a hyperbola, and y^2 = 4px for a parabola in an appropriate orientation.

History and mathematical importance

The systematic investigation of conic sections is a classical subject in Greek mathematics. Apollonius of Perga (third century BC) wrote the foundational treatise that studied their properties and terminology. Conics later became central to mechanics and astronomy: the trajectories predicted by the Newtonian two-body model are conic sections, relating ellipses to planetary orbits and hyperbolas or parabolas to certain cometary paths and escape trajectories (two-body problem).

Applications and examples

Conic sections appear widely in science and engineering. Parabolic shapes are used for satellite dishes, headlights and telescopes because of their focusing property. Elliptical reflectors and whispering galleries exploit the two-focus reflection law. Hyperbolic mirrors and lenses can correct certain aberrations in optical systems. In navigation and orbital mechanics, the classification of possible trajectories (bound, parabolic escape, or hyperbolic escape) matches the conic types. For further foundational reading see links in geometry and classical treatments of the subject: geometry, cone, plane.

Notable distinctions and facts

Although often introduced via an intersection picture, conics are equally well characterized by purely planar definitions (focus-directrix and eccentricity) and by quadratic equations. The circle is the only conic with full rotational symmetry and is typically treated as a distinguished special case. Modern approaches connect conics to projective geometry, where they are seen as degree-two curves whose qualitative behavior is preserved under projection and affine transformations.

Related topics and resources: circle, parabola, hyperbola, ellipse.