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parabola
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{{short description|Plane curve: conic section}}{{other uses}}In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U shaped. It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves.(File:Parts of Parabola.svg|thumb|right|upright=1.36|Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward.)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.{{efn|The tangential plane just touches the conical surface along a line which passes through the apex of the cone}}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 where the parabola intersects its 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.File:Conic Sections.svg|thumb|The parabola is a member of the family of conic sectionconic section

History

File:Leonardo parabolic compass.JPG|thumb|180px|Parabolic compass designed by Leonardo da VinciLeonardo da VinciThe earliest known work on conic sections was by Menaechmus in the 4th 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 3rd 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.WEB,weblink Can You Really Derive Conic Formulae from a Cone? - Deriving the Symptom of the Parabola - Mathematical Association of America, 30 September 2016, 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.BOOK, Reflecting Telescope Optics: Basic design theory and its historical development, 2, Ray N., Wilson, Springer, 2004, 3-540-40106-7, 3,weblink Extract of page 3
Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne,Stargazer, p. 115. and James Gregory.Stargazer, pp. 123 and 132 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.WEB,weblink Spherical Mirrors, Richard, Fitzpatrick, July 14, 2007, Electromagnetism and Optics, lectures, University of Texas at Austin, Paraxial Optics, October 5, 2011,
{{clear}}

Definition as a locus of points

A parabola can be defined geometrically as a set of points (locus of points) in the Euclidean plane:
  • A parabola is a set of points, such that for any point P of the set the distance |PF| to a fixed point F, the focus, is equal to the distance |Pl| to a fixed line l, the directrix:


{P : |PF| = |Pl|}
The midpoint V of the perpendicular from the focus F onto the directrix l is called vertex and the line FV the axis of symmetry of the parabola.

In a cartesian coordinate system

Axis of symmetry parallel to the y-axis

(File:Parabel-def-p-v.svg|thumb|Parabola: Definition, p: semi-latus rectum)(File:Parabel-py.svg|thumb|Parabola: axis parallel to y-axis)(File:Parabel-abc.svg|thumb|Parabola: general case)If one introduces cartesian coordinates, such that F=(0,f) ,f> 0, and the directrix has the equation y=-f one obtains for a point P=(x,y) from |PF|^2 = |Pl|^2 the equation x^2+(y-f)^2=(y+f)^2 . Solving for y yields
y=frac{1}{4f}x^2.
This parabola is U-shaped (opening to the top).The horizontal chord through the focus (see picture in opening section) is called the latus rectum; one half of it is the semi-latus rectum. The latus rectum is parallel to the directrix. The semi-latus rectum is designated by the letter p. From the picture one obtains
p=2f.
The latus rectum is defined similarly for the other two conics, namely the ellipse and the hyperbola, respectively. The latus rectum is the line drawn through a focus of a conic section parallel to the directrix and terminated both ways by the curve. For any case, p is the radius of the osculating circle at the vertex. For a parabola, the semi-latus rectum, p, is the distance of the focus from the directrix. Using the parameter p, the equation of the parabola can be rewritten as
x^2=2py.
More generally, if the vertex is V=(v_1,v_2), the focus F=(v_1,v_2+f) and the directrix y=v_2-f , one obtains the equation
y=frac{1}{4f}(x-v_1)^2+v_2=frac{1}{4f}x^2-frac{v_1}{2f}x+frac{v_1^2}{4f}+v_2.
Remark:
  1. In the case of f0 the parabolas are opening to the top and for a


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