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Encyclopedia > Parabola
A parabola
A parabola
A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue)
A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue)

In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: /pəˈrab(ə)lə/) is a conic section generated by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. A parabola can also be defined as locus of points in a plane which are equidistant from a given point (the focus) and a given line (the directrix). Image File history File links Parabola. ... Image File history File links Parabola. ... made with Graph 2. ... made with Graph 2. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... IPA may refer to: The International Phonetic Alphabet or India Pale Ale ... It has been suggested that Latus rectum be merged into this article or section. ... In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. ... Two intersecting planes in three-dimensional space In mathematics, a plane is a fundamental two-dimensional object. ... In mathematics, a locus (Latin for place, plural loci) is a collection of points which share a common property. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... Distance is a numerical description of how far apart things lie. ... In geometry, the focus (pl. ... In mathematics, a conic section (or just conic) is a curved locus of points, formed by intersecting a cone with a plane. ...


A particular case arises when the plane is tangent to the conical surface. In this case, the intersection is a degenerate parabola consisting of a straight line. In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. ... Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ...


The parabola is an important concept in abstract mathematics, but it is also seen with considerable frequency in the physical world, and there are many practical applications for the construct in engineering, physics, and other domains. Engineering is the design, analysis, and/or construction of works for practical purposes. ... Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the fundamental laws of the universe and their precise formulation in a mathematical framework. ...

Contents

Definitions and overview

Analytic geometry equations

In Cartesian coordinates, a parabola with an axis parallel to the y axis with vertex (h, k), focus (h, k + p), and directrix y = k - p, with p being the distance from the vertex to the focus, has the equation Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

(x - h)^2 = 4p(y + k) ,

or, alternatively

(y - k) = frac{1}{4p}(x-h)^2 ,

More generally, a parabola is a curve in the Cartesian plane defined by an irreducible equation of the form Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ...

A x^2 + B xy + C y^2 + D x + E y + F = 0 ,

such that B^2 = 4 AC ,, where all of the coefficients are real, where AC not= 0 ,, and where more than one solution, defining a pair of points (x, y) on the parabola, exists. That the equation is irreducible means it does not factor as a product of two not necessarily distinct linear factors.


Other geometric definitions

A parabolas are conic sections.
A parabolas are conic sections.

A parabola may also be characterised as a conic section with an eccentricity of 1. As a consequence of this, all parabolas are similar. A parabola can also be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction. In this sense, a parabola may be considered an ellipse that has one focus at infinity. The parabola is an inverse transform of a cardioid. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... It has been suggested that Latus rectum be merged into this article or section. ... (This page refers to eccentricity in mathematics. ... // Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ... For other uses, see Ellipse (disambiguation). ... This article or section is not written in the formal tone expected of an encyclopedia article. ... In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles. ... In geometry, the cardioid is an epicycloid which has one and only one cusp. ...


A parabola has a single axis of reflective symmetry, which passes through its focus and is perpendicular to its directrix. The point of intersection of this axis and the parabola is called the vertex. A parabola spun about this axis in three dimensions traces out a shape known as a paraboloid of revolution. Sphere symmetry group o. ... Paraboloid of revolution Hyperbolic paraboloid In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation: (elliptic paraboloid), or (hyperbolic paraboloid). ...


The parabola is found in numerous situations in the physical world (see below).


Equations

(with vertex (h, k) and distance p between vertex and focus - note that if the vertex is below the focus, or equivalently above the directrix, p is positive, otherwise p is negative; similarly with horizontal axis of symmetry p is positive if vertex is to the left of the focus, or equivalently to the right of the directrix)


Cartesian

Vertical axis of symmetry
(x - h)^2 = 4p(y - k) ,
y = a(xty-h)^2 + k ,
y = ax^2 + bx + c ,
mbox{where }a = frac{1}{4p}; b = frac{-h}{2p}; c = frac{h^2}{4p} + k;
h = frac{-b}{2a}; k = frac{4ac - b^2}{4a}.
x(t) = 2pt + h; y(t) = pt^2 + k ,

Horizontal axis of symmetry
(y - k)^2 = 4p(x - h) ,
x = a(y - k)^2 + h ,
x = ay^2 + by + c ,
mbox{where }a = frac{1}{4p}; b = frac{-k}{2p}; c = frac{k^2}{4p} + h;
h = frac{4ac - b^2}{4a}; k = frac{-b}{2a}.
x(t) = pt^2 + h; y(t) = 2pt + k ,

Semi-latus rectum and polar coordinates

In polar coordinates, a parabola with the focus at the origin and the top on the negative x-axis, is given by the equation This article describes some of the common coordinate systems that appear in elementary mathematics. ...

r (1 - cos theta) = l ,

where l is the semi-latus rectum: the distance from the focus to the parabola itself, measured along a line perpendicular to the axis. Note that this is twice the distance from the focus to the apex of the parabola or the perpendicular distance from the focus to the latus rectum. Semi- is a prefix from the Latin language meaning 1/2. ... In a conic section, the latus rectum is the chord parallel to the directrix through the focus. ...


Gauss-mapped form

A Gauss-mapped form: (tan2φ,2tanφ) has normal (cosφ,sinφ). In differential geometry, the Gauss map (named, like so many things, after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere . ...


Derivation of the focus

Parabolic curve showing directrix (L) and focus (F). The distance from a given point Pn to the focus is always the same as the distance from Pn to a point Qn directly below, on the directrix.
Parabolic curve showing directrix (L) and focus (F). The distance from a given point Pn to the focus is always the same as the distance from Pn to a point Qn directly below, on the directrix.

Given a parabola parallel to the y-axis with vertex (0,0) and with equation Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

y = a x^2, qquad qquad qquad (1)

then there is a point (0,f) — the focus — such that any point P on the parabola will be equidistant from both the focus and a line perpendicular to the axis of symmetry of the parabola (the linea directrix), in this case parallel to the x axis. Since the vertex is one of the possible points P, it follows that the linea directrix passes through the point (0,-f). So for any point P=(x,y), it will be equidistant from (0,f) and (x,-f). It is desired to find the value of f which has this property.


Let F denote the focus, and let Q denote the point at (x,-f). Line FP has the same length as line QP.

| FP | = sqrt{ x^2 + (y - f)^2 },
| QP | = y + f.
| FP | = | QP |
sqrt{x^2 + (a x^2 - f)^2 } = a x^2 + f qquad

Square both sides,

x^2 + (a x^2 - f)^2 = (a x^2 + f)^2 qquad
= a^2 x^4 + f^2 + 2 a x^2 f quad
x^2 + a^2 x^4 + f^2 - 2 a x^2 f = a^2 x^4 + f^2 + 2 a x^2 f quad

Cancel out terms from both sides,

x^2 - 2 a x^2 f = 2 a x^2 f, quad
x^2 = 4 a x^2 f. quad

Cancel out the x2 from both sides (x is generally not zero),

1 = 4 a f quad
f = {1 over 4 a }

Now let p=f and the equation for the parabola becomes

x^2 = 4 p y quad

Q.E.D. Q.E.D. is an abbreviation of the Latin phrase (literally, which was to be demonstrated). In simple terms, the use of this Latin phrase is to indicate that something has been definitively proven. ...


If the equation of the parabola is given in standard form Standard Forms in mathematics assist people in identifying types of equations. ...


ax2 + bx + c = y


then the focus is located at the point


left (frac{-b}{2a},frac{-b^2}{4a}+c+frac{1}{4a} right)


and the directrix is designated by the equation


y=frac{-b^2}{4a}+c+frac{1}{4a}


Reflective property of the tangent

The tangent of the parabola described by equation (1) has slope

{dy over dx} = 2 a x = {2 y over x}

This line intersects the y-axis at the point (0,-y) = (0, - a x2), and the x-axis at the point (x/2,0). Let this point be called G. Point G is also the midpoint of points F and Q:

F = (0,f), quad
Q = (x,-f), quad
{F + Q over 2} = {(0,f) + (x,-f) over 2} = {(x,0) over 2} = ({x over 2}, 0).

Since G is the midpoint of line FQ, this means that

| FG | cong | GQ |,

and it is already known that P is equidistant from both F and Q:

| PF | cong | PQ |,

and, thirdly, line GP is equal to itself, therefore:

Delta FGP cong Delta QGP

It follows that angle FPG cong angle GPQ.


Line QP can be extended beyond P to some point T, and line GP can be extended beyond P to some point R. Then angle RPT and angle GPQ are vertical, so they are equal (congruent). But angle GPQ is equal to angle FPG. Therefore angle RPT is equal to angle FPG. An object is in a vertical position when it is aligned in an up-down direction, perpendicular to the horizon. ...


The line RG is tangent to the parabola at P, so any light beam bouncing off point P will behave as if line RG were a mirror and it were bouncing off that mirror.


Let a light beam travel down the vertical line TP and bounce off from P. The beam's angle of inclination from the mirror is angle RPT, so when it bounces off, its angle of inclination must be equal to angle RPT. But angle FPG has been shown to be equal to angle RPT. Therefore the beam bounces off along the line FP: directly towards the focus.


Conclusion: Any light beam moving vertically downwards in the concavity of the parabola (parallel to the axis of symmetry) will bounce off the parabola moving directly towards the focus. (See parabolic reflector.) A parabolic reflector (also known as a parabolic dish or a parabolic mirror) is a reflective device formed in the shape of a paraboloid of revolution. ...


Analyzing the Parabola

Part I: Introduction

Analyzing the parabola is a way to solve quadratic equations that depends on the symmetry and other characteristics of the parabola. It is easiest to use this method when you are given a problem in which you must solve for the vertex first. This method can also be exploited to derive a second quadratic formula.


Part II: The “Height” of the Parabola

This is one of the most crucial concepts of using this method. The “height” of a parabola is really the absolute value of the distance along the axis of symmetry between the vertex of the function and the line connecting two solutions of the quadratic equation for any value of x. For example (y=2x²) the height between the origin (the vertex in this case) when x=2 is 8. In this paper, the formula for the height will be given as h = − ax2.


Part III: Examples of Applying the Method

Example 1: 2x²+3x-4

  • Step 1: Find the vertex
vtx = (-0.75, -5.125)
  • Step 2: Set the height of the parabola equal to the y-vertex value and solve for x
h = - ax²
-5.125 = -2x²
x = ±√(-5.125/-2) ≈ ±1.601
  • Step 3: Up until now the x value of the vertex has been presumed to be zero. We must now add the x value in the vertex to the solutions so that they are relative to the x-axis, not the line of symmetry.
x ≈ ±1.601 - 0.75 = 0.851, -2.351

Example 2: 9x²+2x+4

  • Step 1: Find the vertex
vtx = (-0.1111, 3.8888)
  • Step 2: Set the height of the parabola equal to the y-vertex value and solve for x
h = -ax²
3.8888 = -9x²
x = ±√(3.8888/-9) ≈ ±0.6573i
  • Step 3: Up until now the x value of the vertex has been presumed to be zero. We must now add the x value in the vertex to the solutions so that they are relative to the x-axis, not the line of symmetry.
x ≈ ±0.6573i-0.1111 = -0.1111-0.6573i, -0.1111+0.6573i

Part IV: The Second Quadratic Formula

Following these steps, it is possible to derive a second quadratic formula.

  • Step 1: Find the vertex
vtx = (j, k)
  • Step 2: Set the height of the parabola equal to the y-vertex value and solve for x
h = -ax²
k = -ax²
x = ±√(k/-a)
  • Step 3: Up until now the x value of the vertex has been presumed to be zero. We must now add the x value in the vertex to the solutions so that they are relative to the x-axis, not the parabola’s line of symmetry.
x = j±√(k/-a)

The a coefficient must be of the opposite sign in this formula or else the roots to an equation with two real solutions will be imaginary.


Part V: Sample Problems

  • Example 1
vtx = (-3, -9)
a = 2
x = j±√(k/-a)
x = -3 ± √(-9/-2) ≈ -0.8787, -5.1213
  • Example 2
vtx = (-2.25, 8.125)
a = -2
x = j±√(k/-a)
x = -2.25 ± √(8.125/2) ≈ 0.2344, 4.26556

Parabolas in the physical world

In nature, approximations of parabolas and paraboloids are found in many diverse situations. The most well-known instance of the parabola in the history of physics is the trajectory of a particle or body in motion under the influence of a uniform gravitational field without air resistance (for instance, a baseball flying through the air, neglecting air friction). The parabolic trajectory of projectiles was discovered experimentally by Galileo in the early 17th century, who performed experiments with balls rolling on inclined planes. The parabolic shape for projectiles was later proven mathematically by Isaac Newton. For objects extended in space, such as a diver jumping from a diving board, the object itself follows a complex motion as it rotates, but the center of mass of the object nevertheless forms a parabola. As in all cases in the physical world, the trajectory is always an approximation of a parabola. The presence of air resistance, for example, always distorts the shape, although at low speeds, the shape is a good approximation of a parabola. At higher speeds, such as in ballistics, the shape is highly distorted and does not resemble a parabola. Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the fundamental laws of the universe and their precise formulation in a mathematical framework. ... Mathematically the term trajectory refers to the ordered set of states which are assumed by a dynamical system over time (see e. ... The gravitational field is a field (physics), generated by massive objects, that determines the magnitude and direction of gravitation experienced by other massive objects. ... For a solid object moving through a fluid or gas, drag is the sum of all the aerodynamic or hydrodynamic forces in the direction of the external fluid flow. ... Friction is the force that opposes the relative motion or tendency toward such motion of two surfaces in contact. ... Galileo can refer to: Galileo Galilei, astronomer, philosopher, and physicist (1564 - 1642) the Galileo spacecraft, a NASA space probe that visited Jupiter and its moons the Galileo positioning system Life of Galileo, a play by Bertolt Brecht Galileo (1975) - screen adaptation of the play Life of Galileo by Bertolt Brecht... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... Sir Isaac Newton, (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist, regarded by many as the greatest figure in the history of science. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

Parabolic shape formed by the surface of a Newtonian liquid under rotation
Parabolic shape formed by the surface of a Newtonian liquid under rotation

Another situation in which parabola may arise in nature is in two-body orbits, for example, of a small planetoid or other object under the influence of the gravitation of the sun. Such parabolic orbits are a special case that are rarely found in nature. Orbits that form a hyperbola or an ellipse are much more common. In fact, the parabolic orbit is the borderline case between those two types of orbit. Image File history File linksMetadata Coriolis_effect11. ... Image File history File linksMetadata Coriolis_effect11. ... A graph of a hyperbola. ... For other uses, see Ellipse (disambiguation). ...


Approximations of parabolas are also found in the shape of cables of suspension bridges. Freely hanging cables do not describe parabolas, but rather catenary curves. Under the influence of a uniform load (for example, the deck of bridge), however, the cable is deformed towards a parabola. A suspension bridge is a type of bridge that has been created since ancient times as early as 100 AD. Simple suspension bridges, for use by pedestrians and livestock, are still constructed, based upon the ancient Inca rope bridge. ... In mathematics, the catenary is the shape of a hanging flexible chain or cable when supported at its ends and acted upon by a uniform gravitational force (its own weight). ...


Paraboloids arise in several physical situations as well. The most well-known instance is the parabolic reflector, which is a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point. The principle of the parabolic reflector may have been discovered in the 3rd century BC by the geometer Archimedes, who, according to a legend of debatable veracity,[1] constructed parabolic mirrors to defend Syracuse against the Roman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to telescopes in the 17th century. Today, paraboloid reflectors can be commonly observed throughout much of the world in microwave and satellite dish antennas. A parabolic reflector (also known as a parabolic dish or a parabolic mirror) is a reflective device formed in the shape of a paraboloid of revolution. ... Electromagnetic radiation can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... Child – 5:16 All I Need – 3:55 Drifting – 6:43 Hold On – 4:40 Open Me – 3:35 Beautiful – 5:44 Look In – 4:14 Without You – 4:55 Live It – 7:23 Dont Walk Away – 3:04 Lead Me On – 5:34 Rest – 5:06 Child [Piano... (2nd millennium BC - 1st millennium BC - 1st millennium) The 3rd century BC started on January 1, 300 BC and ended on December 31, 201 BC. // Events The Pyramid of the Moon, one of several monuments built in Teotihuacán Teotihuacán, Mexico begun The first two Punic Wars between Carthage... Archimedes (Greek: c. ... Syracuse (Italian, Siracusa, ancient Syracusa - see also List of traditional Greek place names) is a city on the eastern coast of Sicily and the capital of the province of Syracuse, Italy. ... Motto: Senatus Populusque Romanus (SPQR) The Roman Empire at its greatest extent, c. ... A telescope (from the Greek tele = far and skopein = to look or see; teleskopos = far-seeing) is an instrument designed for the observation of remote objects. ... Microwaves are electromagnetic waves with wavelengths longer than those of terahertz (THz) frequencies, but relatively short for radio waves. ...


Paraboloids are also observed in the surface of a liquid confined to a container and rotated around the central axis. In this case, the centrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind the liquid mirror telescope. Centrifugal force (from Latin centrum center and fugere to flee) is a term which may refer to two different forces which are related to rotation. ... A liquid mirror telescope is a reflecting telescope whose primary mirror is a rotating pool of a reflective liquid, usually mercury. ...


Aircraft used to create a weightless state for purposes of experimentation, such as NASA's “vomit comet,” follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall, which produces the same effect as zero gravity for most purposes. An Airbus A380, currently the worlds largest passenger airliner An aircraft is any vehicle or craft capable of atmospheric flight. ... Astronauts on the International Space Station display an example of weightlessness Weightlessness is the experience (by people and objects) during freefall, of having no weight. ... The National Aeronautics and Space Administration (NASA) is an agency of the United States Government, responsible for that nations public space program. ... Weightlessness inside the Vomit Comet The Vomit Comet was the nickname given to the aircraft used by NASAs Reduced Gravity Research Program. ... Free fall in its strictest sense is the condition of acceleration which is due only to gravity. ... Astronauts on the International Space Station display an example of weightlessness Weightlessness is the experience (by people and objects) during freefall, of having no weight. ...


See also

It has been suggested that Latus rectum be merged into this article or section. ... In mathematics, the catenary is the shape of a hanging flexible chain or cable when supported at its ends and acted upon by a uniform gravitational force (its own weight). ... A parabolic reflector (also known as a parabolic dish or a parabolic mirror) is a reflective device formed in the shape of a paraboloid of revolution. ... Paraboloid of revolution Hyperbolic paraboloid In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation: (elliptic paraboloid), or (hyperbolic paraboloid). ... A graph of a hyperbola. ... For other uses, see Ellipse (disambiguation). ...

External links

Wikisource has an original article from the 1911 Encyclopædia Britannica about:
Parabola
  • Eric W. Weisstein, Parabola at MathWorld.
  • Archimedes Triangle and Squaring of Parabola at cut-the-knot
  • Two Tangents to Parabola at cut-the-knot
  • Parabola As Envelope of Straight Lines at cut-the-knot
  • Parabolic Mirror at cut-the-knot
  • Three Parabola Tangents at cut-the-knot
  • Focal Properties of Parabola at cut-the-knot
  • Parabola As Envelope II at cut-the-knot
  • Parabola Construction - An interactive sketch showing how to trace a parabola. (Requires Java.)
  • Quadratic Bezier Construction - An interactive sketch showing how to trace the quadratic Bezier curve (a parabolic segment). (Requires Java.)
  • More Interactive Parabola Construction (Java-enabled)

  Results from FactBites:
 
Parabola as envelope of lines (500 words)
Parabola is a conic defined by its focal property: there is a point - focus - and a line - directrix - and parabola is the locus of points equidistant from the focus and the directrix.
It is known that the tangent to parabola at a point bisects the angle between the segments joining the point to the focus and the directrix.
This is the parabola with focus F and the apex on the x-axis.
  More results at FactBites »


 

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