Parabolic

Parabolic is an adjective that relates to a parabola, a plane curve that is a special type of conic section. A parabola can be defined as the locus of points equidistant from a fixed focus and a fixed directrix, or as the result of intersecting a right circular cone with a plane parallel to one of its generating lines. In analytic geometry, parabolas have simple standard forms and key features such as a vertex, axis of symmetry, and focal length.

In Cartesian form, a parabola with vertex at (h, k) and opening vertically is written as (x − h)^2 = 4p(y − k), where p is the focal length. Its focus is (h, k + p) and its directrix is the line y = k − p. If p > 0 the parabola opens upward; if p < 0 it opens downward. A horizontal orientation uses (y − k)^2 = 4p(x − h) with focus (h + p, k). Equivalently, the parabola can be expressed as y = a(x − h)^2 + k, with a = 1/(4p). All parabolas have eccentricity e = 1.

A three-dimensional analogue is the paraboloid, obtained by rotating a parabola about its axis. It can be described by z = (x^2 + y^2)/(4p), with focus at (0, 0, p). Parabolic reflectors exploit the property that incoming lines parallel to the axis reflect to the focus (and vice versa), a principle used in satellite dishes, telescope optics, and certain lighting devices. In physics, neglecting air resistance, projectile motion is described by parabolic trajectories.