parabool

Parabool is the Dutch term for parabola, a plane curve and a conic section. In English usage, the term parabola refers to the same shape.

It is defined as the locus of points equidistant from a fixed point called the focus and a fixed line called the directrix. The parabola is symmetric about its axis, the line through the vertex and focus. The vertex is the parabola’s closest point to the directrix.

In Cartesian coordinates, a vertical parabola with its vertex at the origin and opening upward has equation y = x^2/(4p), where p is the distance from the vertex to the focus; equivalently x^2 = 4py. A horizontal parabola has equation x = y^2/(4p) or y^2 = 4px, with focus at (p,0) and directrix x = -p. The parameter p (the focal length) determines the curvature and the direction in which the parabola opens.

Key properties include its symmetry about the axis, the focus and directrix defining the curve, and the latus rectum — a line segment through the focus perpendicular to the axis with length 4|p|. The eccentricity of a parabola is 1. Parabolas have the characteristic that light or sound rays parallel to the axis reflect to pass through the focus.

Applications are widespread: parabolic reflectors in satellite dishes and telescopes, car headlights and solar concentrators, and various optical and engineering contexts. In physics, projectile motion without air resistance is approximately parabolic under constant gravitational acceleration. In geometry, a parabola arises as a conic section obtained by intersecting a cone with a plane parallel to a generating line.