language-icon Old Web
English
Sign In

Parabola

In mathematics, a parabola is a plane curve that is mirror-symmetrical and is approximately U-shaped. It fits several superficially different other mathematical descriptions, which can all be proved to define exactly the same curves.The area enclosed between a parabola and a chord (see diagram) is two-thirds of the area of a parallelogram which surrounds it. One side of the parallelogram is the chord, and the opposite side is a tangent to the parabola. The slope of the other parallel sides is irrelevant to the area. Often, as here, they are drawn parallel with the parabola's axis of symmetry, but this is arbitrary.Image is inverted. AB is x-axis. C is origin. O is center. A is (x, y). OA = OC = R. PA = x. CP = y. OP = (R − y). Other points and lines are irrelevant for this purpose.The radius of curvature at the vertex is twice the focal length. The measurements shown on the above diagram are in units of the latus rectum, which is four times the focal length.Elliptic coneParabolic cylinderElliptic paraboloidHyperbolic paraboloidHyperboloid of one sheetHyperboloid of two sheetsA bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola.Parabolic trajectories of water in a fountain.The path (in red) of Comet Kohoutek as it passed through the inner solar system, showing its nearly parabolic shape. The blue orbit is the Earth'sThe supporting cables of suspension bridges follow a curve which is intermediate between a parabola and a catenary.The Rainbow Bridge across the Niagara River, connecting Canada (left) to the United States (right). The parabolic arch is in compression, and carries the weight of the road.Parabolic arches used in architectureParabolic shape formed by a liquid surface under rotation. Two liquids of different densities completely fill a narrow space between two sheets of transparent plastic. The gap between the sheets is closed at the bottom, sides and top. The whole assembly is rotating around a vertical axis passing through the centre. (See Rotating furnace)Solar cooker with parabolic reflectorParabolic antennaParabolic microphone with optically transparent plastic reflector, used to overhear referee conversations at an American college football game.Array of parabolic troughs to collect solar energyEdison's searchlight, mounted on a cart. The light had a parabolic reflector.Physicist Stephen Hawking in an aircraft flying a parabolic trajectory to simulate zero-gravity In mathematics, a parabola is a plane curve that is mirror-symmetrical and is approximately U-shaped. It fits several superficially different other mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane which is parallel to another plane that is tangential to the conical surface. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the 'axis of symmetry'. The point on the parabola that intersects the axis of symmetry is called the 'vertex', and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the 'focal length'. The 'latus rectum' is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar. Parabolas have the property that, if they are made of material that reflects light, then light which travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel ('collimated') beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy. This reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas. The earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolas. (The solution, however, does not meet the requirements of compass-and-straightedge construction.) The area enclosed by a parabola and a line segment, the so-called 'parabola segment', was computed by Archimedes via the method of exhaustion in the third century BC, in his The Quadrature of the Parabola. The name 'parabola' is due to Apollonius who discovered many properties of conic sections. It means 'application', referring to 'application of areas' concept, that has a connection with this curve, as Apollonius had proved. The focus–directrix property of the parabola and other conic sections is due to Pappus. Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a parabolic reflector could produce an image was already well known before the invention of the reflecting telescope. Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne, and James Gregory. When Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers. A parabola can be defined geometrically as a set of points (locus of points) in the Euclidean plane:

[ "Geometry", "Applied mathematics", "Mathematical optimization", "Mathematical analysis", "Algebra", "degenerate equation", "Parabolic partial differential equation", "Chernoff's distribution", "parabolic system", "parabolic wave equation" ]
Parent Topic
Child Topic
    No Parent Topic