Parabola (conic)
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| Parabola (conic) | |
|---|---|
| Name | Parabola (conic) |
| Caption | Standard parabola y = x^2 |
| Type | Conic section |
Parabola (conic) is a conic section formed by intersection of a plane with a right circular cone parallel to a generator of the cone. It appears across history in works by Apollonius of Perga, Archimedes, Euclid, Pappus of Alexandria and later developments by René Descartes, Isaac Newton, Johannes Kepler, and Blaise Pascal. The parabola has a single axis of symmetry and a unique focus and directrix, and its shape is determined by a constant eccentricity equal to one.
Definition and basic properties
A parabola is the locus of points equidistant from a fixed point (the focus) and a fixed line (the directrix), a concept used by Nicolaus Copernicus and refined by Galileo Galilei in studies of projectile motion. Classical treatments by Apollonius of Perga classified it among conic sections alongside figures studied by Ptolemy and later treated in analytic form by Descartes in coordinate geometry. Key properties include reflective symmetry used in designs by Leonardo da Vinci and Andrea Palladio, a single nondegenerate branch studied by Augustin-Louis Cauchy, and a quadratic relation in Cartesian coordinates exploited by Joseph-Louis Lagrange.
Geometric constructions and focus-directrix property
Geometric constructions of a parabola were described by Archimedes and illustrated in Renaissance texts by Albrecht Dürer and Giovanni Battista Piranesi. The focus-directrix definition underlies reflective properties used in engineering projects by Isambard Kingdom Brunel and Guglielmo Marconi and in optical designs by George Biddell Airy and James Clerk Maxwell. The tangent at any point bisects the angle between the line to the focus and the perpendicular to the directrix, a fact applied by Sadi Carnot in mechanics and by Friedrich Bessel in instrument design.
Algebraic equations and coordinates
In Cartesian coordinates, the standard parabola is given by y = ax^2 + bx + c, as formalized in analytic geometry by René Descartes and used in calculations by Pierre-Simon Laplace. Converting between vertex form and standard form employs techniques from Carl Friedrich Gauss and Adrien-Marie Legendre. Parametric equations x = t, y = at^2 + bt + c parallel parametric methods introduced by Leonhard Euler; polar and parametric forms were used in celestial mechanics by Johannes Kepler and in perturbation theory by Henri Poincaré.
Conic classification and eccentricity
Conic sections — ellipse, parabola, hyperbola — were compared by Apollonius of Perga and later by Kepler in planetary theory. The parabola is the boundary case with eccentricity e = 1, a convention appearing in classification schemes by John Couch Adams and Simon Newcomb. In projective geometry developed by Jean-Victor Poncelet and Felix Klein, a parabola is equivalent to other conics under appropriate transformations exploited by David Hilbert and Emmy Noether.
Applications and occurrences
Parabolas appear in the design of reflectors and antennas by Karl Jansky and Heinrich Hertz, in architecture by Santiago Calatrava and Buckminster Fuller, and in astronomy through orbital approximations used by Edmond Halley and William Herschel. In engineering, parabolic arches are used by Gustave Eiffel and in bridge design by John Roebling. Parabolic mirrors focus light in instruments by Isaac Newton and William Herschel, and parabolic trajectories model projectiles in studies by Galileo Galilei and in ballistics by Benjamin Robins.
Transformations and calculus properties
Under affine transformations studied by Augustin-Louis Cauchy and Hermann Grassmann, a parabola may map to other conics; projective treatments by Jean-Victor Poncelet relate to work of Felix Klein. Calculus properties including arc length, curvature, and envelope constructions were developed using methods of Gottfried Wilhelm Leibniz, Isaac Newton, and Joseph Fourier. The evolute and involute of a parabola were treated by Johann Bernoulli and applied in gear design by James Watt and Eli Whitney.