 conic section

Geom.a curve formed by the intersection of a plane with a right circular cone; an ellipse, a circle, a parabola, or a hyperbola. Also called conic.[165565]
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Any twodimensional curve traced by the intersection of a right circular cone with a plane.If the plane is perpendicular to the cone's axis, the resulting curve is a circle. Intersections at other angles result in ellipses, parabolas, and hyperbolas. The conic sections are studied in Euclidean geometry to analyze their physical properties and in analytic geometry to derive their equations. In either context, they have useful applications to optics, antenna design, structural engineering, and architecture.* * *
▪ geometryIntroductionalso called conicin geometry, any curve produced by the intersection of a plane and a right circular cone. Depending on the angle of the plane relative to the cone, the intersection is a circle, an ellipse, a hyperbola, or a parabola. Special (degenerate) cases of intersection occur when the plane passes through only the apex (producing a single point) or through the apex and another point on the cone (producing one straight line or two intersecting straight lines). See the figure—>.The basic descriptions, but not the names, of the conic sections can be traced to Menaechmus (flourished c. 350 BC), a pupil of both Plato and Eudoxus of Cnidus. Apollonius of Perga (c. 262–190 BC), known as the “Great Geometer,” gave the conic sections their names and was the first to define the two branches of the hyperbola (which presuppose the double cone). Apollonius's eightvolume treatise on the conic sections, Conics, is one of the greatest scientific works from the ancient world.Analytic definitionConics may also be described as plane curves that are the paths (loci) of a point moving so that the ratio of its distance from a fixed point (the focus) to the distance from a fixed line (the directrix) is a constant, called the eccentricity of the curve. If the eccentricity is zero, the curve is a circle; if equal to one, a parabola; if less than one, an ellipse; and if greater than one, a hyperbola. See the figure—>.Every conic section corresponds to the graph of a second degree polynomial equation of the form Ax^{2} + By^{2} + 2Cxy + 2Dx + 2Ey + F = 0, where x and y are variables and A, B, C, D, E, and F are coefficients that depend upon the particular conic. By a suitable choice of coordinate axes, the equation for any conic can be reduced to one of three simple r forms:^{x2}/_{a2} + ^{y2}/_{b2} = 1, ^{x2}/_{a2} − ^{y2}/_{b2} = 1, or y^{2} = 2px,corresponding to an ellipse, a hyperbola, and a parabola, respectively. (An ellipse where a = b is in fact a circle.) The extensive use of coordinate systems for the algebraic analysis of geometric curves originated with René Descartes (Descartes, René) (1596–1650). See History of geometry: Cartesian geometry (geometry).Greek originsThe early history of conic sections is joined to the problem of “doubling the cube.” According to Eratosthenes of Cyrene (c. 276–190 BC), the people of Delos consulted the oracle of Apollo (oracle) for aid in ending a plague (c. 430 BC) and were instructed to build Apollo a new altar of twice the old altar's volume and with the same cubic shape. Perplexed, the Delians consulted Plato, who stated that “the oracle meant, not that the god wanted an altar of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt for geometry.” Hippocrates of Chios (c. 470–410 BC) first discovered that the “Delian problem” can be reduced to finding two mean proportionals between a and 2a (the volumes of the respective altars)—that is, determining x and y such that a:x = x:y = y:2a. This is equivalent to solving simultaneously any two of the equations x^{2} = ay, y^{2} = 2ax, and xy = 2a^{2}, which correspond to two parabolas and a hyperbola, respectively. Later, Archimedes (c. 290–211 BC) showed how to use conic sections to divide a sphere into two segments having a given ratio.Diocles (c. 200 BC) demonstrated geometrically that rays—for instance, from the Sun—that are parallel to the axis of a paraboloid of revolution (produced by rotating a parabola about its axis of symmetry) meet at the focus. Archimedes is said to have used this property to set enemy ships on fire. The focal properties of the ellipse were cited by Anthemius of Tralles, one of the architects for Hagia Sophia Cathedral (Hagia Sophia) in Constantinople (completed in AD 537), as a means of ensuring that an altar could be illuminated by sunlight all day.PostGreek applicationsConic sections found their first practical application outside of optics in 1609 when Johannes Kepler (Kepler, Johannes) derived his first law of planetary motion (Kepler's laws of planetary motion): A planet travels in an ellipse with the Sun at one focus. Galileo Galilei (Galileo) published the first correct description of the path of projectiles—a parabola—in his Dialogues of the Two New Sciences (1638). In 1639 the French engineer Girard Desargues (Desargues, Girard) initiated the study of those properties of conics that are invariant under projections (see projective geometry). Eighteenthcentury architects created a fad for whispering galleries—such as in the U.S. Capital and in St. Paul's Cathedral in London—in which a whisper at one focus of an ellipsoid (an ellipse rotated about one axis) can be heard at the other focus, but nowhere else. From the ubiquitous parabolic satellite dish (see the figure—>) to the use of ultrasound in lithotripsy (therapeutics), new applications for conic sections continue to be found.Christian Marinus TaisbakAdditional ReadingThe early history of conic sections can be traced in Wilbur Richard Knorr, The Ancient Tradition of Geometric Problems (1986, reissued 1993). Victor J. Katz, A History of Mathematics: An Introduction, 2nd ed. (1998), includes information on the later development of conic sections. The mathematics of conic sections is richly presented in W. Gellert et al. (eds.), The VNR Concise Encyclopedia of Mathematics, 2nd ed. (1989).* * *
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