In mathematics, the parabola (; plural parabolae or parabolas, from the Greek παραβολή) is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. Given a point (the focus) and a corresponding line (the directrix) on the plane, the locus of points in that plane that are equidistant from them is a parabola.

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 axis of symmetry that intersects the parabola is called the "vertex", and it is the point where the curvature is greatest. 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 similar.

The parabola has many important applications, from automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas.

History

.]] s.]] 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 parabolae. (The solution, however, does not meet the requirements imposed by 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. The focus–directrix property of the parabola and other conics 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.

Equation in Cartesian coordinates

Let the directrix be the line x = −p and let the focus be the point (p, 0). If (x, y) is a point on the parabola then, by Pappus' definition of a parabola, it is the same distance from the directrix as the focus; in other words: :$x+p=\backslash sqrt\{(x-p)^2+y^2\}$ Squaring both sides and simplifying produces :$y^2\; =\; 4px\backslash$ as the equation of the parabola.

By translation, the above is the general equation of a parabola with a horizontal axis; by interchanging the roles of x and y one obtains the corresponding equation of a parabola with a vertical axis as :$(x-h)^\{2\}=4p(y-k)\backslash ,$ The last equation can be rewritten :$y=ax^2+bx+c\backslash ,$ so the graph of any function which is a polynomial of degree 2 in x is a parabola with a vertical axis.

More generally, a parabola is a curve in the Cartesian plane defined by an irreducible equation — one that does not factor as a product of two not necessarily distinct linear equations — of the general conic form

:$A\; x^\{2\}\; +\; B\; xy\; +\; C\; y^\{2\}\; +\; D\; x\; +\; E\; y\; +\; F\; =\; 0\; \backslash ,$

with the parabola restriction that

:$B^\{2\}\; =\; 4\; AC,\backslash ,$

where all of the coefficients are real and where A and C are not both zero. The equation is irreducible if and only if the determinant of the 3×3 matrix : is non-zero: that is, if (AC - B²/4)F + BED/4 - CD²/4 - AE²/4 ≠ 0. The reducible case, also called the degenerate case, gives a pair of parallel lines, possibly real, possibly imaginary, and possibly coinciding with each other.

Other geometric definitions

A parabola may also be characterized as a conic section with an eccentricity of 1. As a consequence of this, all parabolae are similar, meaning that while they can be different sizes, they are all the same shape. 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.

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.

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

Equations

Cartesian

Vertical axis of symmetry

:$(x\; -\; h)^2\; =\; 4p(y\; -\; k)\; \backslash ,$ :$y\; =\backslash frac\{(x-h)^2\}\{4p\}+k\backslash ,$ :$y\; =\; ax^2\; +\; bx\; +\; c\; \backslash ,$ where :$a\; =\; \backslash frac\{1\}\{4p\};\; \backslash \; \backslash \; b\; =\; \backslash frac\{-h\}\{2p\};\; \backslash \; \backslash \; c\; =\; \backslash frac\{h^2\}\{4p\}\; +\; k;\; \backslash \; \backslash$ :$h\; =\; \backslash frac\{-b\}\{2a\};\; \backslash \; \backslash \; k\; =\; \backslash frac\{4ac\; -\; b^2\}\{4a\}$. Parametric form: :$x(t)\; =\; 2pt\; +\; h;\; \backslash \; \backslash \; y(t)\; =\; pt^2\; +\; k\; \backslash ,$

Horizontal axis of symmetry

:$(y\; -\; k)^2\; =\; 4p(x\; -\; h)\; \backslash ,$ :$x\; =\backslash frac\{(y\; -\; k)^2\}\{4p\}\; +\; h;\backslash \; \backslash ,$ :$x\; =\; ay^2\; +\; by\; +\; c\; \backslash ,$ where :$a\; =\; \backslash frac\{1\}\{4p\};\; \backslash \; \backslash \; b\; =\; \backslash frac\{-k\}\{2p\};\; \backslash \; \backslash \; c\; =\; \backslash frac\{k^2\}\{4p\}\; +\; h;\; \backslash \; \backslash$ :$h\; =\; \backslash frac\{4ac\; -\; b^2\}\{4a\};\; \backslash \; \backslash \; k\; =\; \backslash frac\{-b\}\{2a\}$. Parametric form: :$x(t)\; =\; pt^2\; +\; h;\; \backslash \; \backslash \; y(t)\; =\; 2pt\; +\; k\; \backslash ,$

General parabola

The general form for a parabola is :$(\backslash alpha\; x+\backslash beta\; y)^2\; +\; \backslash gamma\; x\; +\; \backslash delta\; y\; +\; \backslash epsilon\; =\; 0\; \backslash ,$ This result is derived from the general conic equation given below:

:$Ax^2\; +Bxy\; +\; Cy^2\; +\; Dx\; +\; Ey\; +\; F\; =\; 0\; \backslash ,$

and the fact that, for a parabola,

:$B^2=4AC\; \backslash ,$.

The equation for a general parabola with a focus point F(u, v), and a directrix in the form :$ax+by+c=0\; \backslash ,$ is :$\backslash frac\{\backslash left(ax+by+c\backslash right)^2\}=\backslash left(x-u\backslash right)^2+\backslash left(y-v\backslash right)^2\; \backslash ,$

Latus rectum, semi-latus rectum, and polar coordinates

In polar coordinates, a parabola with the focus at the origin and the directrix parallel to the y-axis, is given by the equation : $r\; (1\; +\; \backslash cos\; \backslash theta)\; =\; l\; \backslash ,$

where l is the semilatus 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 vertex of the parabola or the perpendicular distance from the focus to the latus rectum.

The latus rectum is the chord that passes through the focus and is perpendicular to the axis. It has a length of 2l.

Gauss-mapped form

A Gauss-mapped form: $(\backslash tan^2\backslash phi,2\backslash tan\backslash phi)$ has normal $(\backslash cos\backslash phi,\backslash sin\backslash phi)$.

Derivation of the focus

To derive the focus of a simple parabola, where the axis of symmetry is parallel to the y-axis with the vertex at (0,0), such as :$y\; =\; a\; x^2\backslash ,\backslash !$

then there is a point (0,f)—the focus, F—such that any point P on the parabola will be equidistant from both the focus and the linear directrix, L. The linear directrix is a line perpendicular to the axis of symmetry of the parabola (in this case parallel to the x axis) and passes through the point (0,-f). So any point P=(x,y) on the parabola will be equidistant both to (0,f) and (x,-f).

FP, a line from the focus to a point on the parabola, has the same length as QP, a line drawn from that point on the parabola perpendicular to the linea directrix, intersecting at point Q.

Imagine a right triangle with two legs, x and f-y (the vertical distance between F and P). The length of the hypotenuse, FP, is given by

:$\backslash |\; FP\; \backslash |\; =\; \backslash sqrt\{\; x^2\; +\; (f\; -\; y)^2\; \}\backslash ,\backslash !$

(Note that (f-y) and (y-f) produce the same result because it is squared.)

The line QP is given by adding y (the vertical distance between the point P and the x-axis) and f (the vertical distance between the x-axis and the linea directrix).

:$\backslash |\; QP\; \backslash |\; =\; f\; +\; y\backslash ,\backslash !$

These two line segments are equal, and, as indicated above, y=ax², thus

:$\backslash |\; FP\; \backslash |\; =\; \backslash |\; QP\; \backslash |\; \backslash ,\backslash !$ :$\backslash sqrt\{x^2\; +\; (f\; -\; a\; x^2\; )^2\; \}\; =\; f\; +\; a\; x^2\backslash ,\backslash !$

Square both sides, :$x^2\; +\; (f^2\; -\; 2\; a\; x^2\; f\; +\; a^2\; x^4)\; =\; (f^2\; +\; 2\; a\; x^2\; f\; +\; a^2\; x^4)\backslash ,\backslash !$ Cancel out terms from both sides, :$x^2\; -\; 2\; a\; x^2\; f\; =\; 2\; a\; x^2\; f\backslash ,\backslash !$ :$x^2\; =\; 4\; a\; x^2\; f\backslash ,\backslash !$ Divide out the x² from both sides (we assume that x is not zero), :$1\; =\; 4\; a\; f\backslash ,\backslash !$ :$f\; =\; \{1\; \backslash over\; 4\; a\; \}\backslash ,\backslash !$

So, for a parabola such as f(x)=x², the a coefficient is 1, so the focus F is (0,¼)

As stated above, this is the derivation of the focus for a simple parabola, one centered at the origin and with symmetry around the y-axis. For any generalized parabola, with its equation given in the standard form

:$y=ax^2+bx+c\backslash ,\backslash !$,

the focus is located at the point

:$\backslash left\; (\backslash frac\{-b\}\{2a\},\backslash frac\{-b^2\}\{4a\}+c+\backslash frac\{1\}\{4a\}\; \backslash right)\backslash ,\backslash !$

which may also be written as

:$\backslash left\; (\backslash frac\{-b\}\{2a\},c-\backslash frac\{b^2-1\}\{4a\}\; \backslash right)\backslash ,\backslash !$

and the directrix is designated by the equation

:$y=\backslash frac\{-b^2\}\{4a\}+c-\backslash frac\{1\}\{4a\}\backslash ,\backslash !$

which may also be written as

:$y=c-\backslash frac\{b^2+1\}\{4a\}\backslash ,\backslash !$

Reflective property of the tangent

The tangent of the parabola described by equation y=ax² has slope :$\{dy\; \backslash over\; dx\}\; =\; 2\; a\; x\; =\; \{2\; y\; \backslash over\; x\}$ 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 line segment FQ: :$F\; =\; (0,f),\; \backslash quad$ :$Q\; =\; (x,-f),\; \backslash quad$ :$\{F\; +\; Q\; \backslash over\; 2\}\; =\; \{(0,f)\; +\; (x,-f)\; \backslash over\; 2\}\; =\; \{(x,0)\; \backslash over\; 2\}\; =\; \backslash left(\{x\; \backslash over\; 2\},\; 0\backslash right).$ Since G is the midpoint of line FQ, this means that :$\backslash |\; FG\; \backslash |\; \backslash cong\; \backslash |\; GQ\; \backslash |,$ and it is already known that P is equidistant from both F and Q: :$\backslash |\; PF\; \backslash |\; \backslash cong\; \backslash |\; PQ\; \backslash |,$ and, thirdly, line GP is equal to itself, therefore: :$\backslash Delta\; FGP\; \backslash cong\; \backslash Delta\; QGP$

It follows that $\backslash angle\; FPG\; \backslash cong\; \backslash angle\; GPQ$.

Line QP can be extended beyond P to some point T, and line GP can be extended beyond P to some point R. Then $\backslash angle\; RPT$ and $\backslash angle\; GPQ$ are vertical, so they are equal (congruent). But $\backslash angle\; GPQ$ is equal to $\backslash angle\; FPG$. Therefore $\backslash angle\; RPT$ is equal to $\backslash angle\; FPG$.

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 $\backslash angle\; RPT$, so when it bounces off, its angle of inclination must be equal to $\backslash angle\; RPT$. But $\backslash angle\; FPG$ has been shown to be equal to $\backslash angle\; RPT$. 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.)

The same reasoning can be applied to a parabola whose axis is vertical, so that it can be specified by the equation :$y\; =\; ax^2\; +\; bx\; +\; c\; \backslash ,$.

The tangent has then a generic slope of :$m\_\{tan\}\; =\; 2ax\; +\; b\; \backslash ,$.

Reflection derivation, together with trigonometric angle addition rules, leads to the result that the reflected ray has a slope of :$m\_\{ref\}\; =\; \{m\_\{tan\}^2\; -\; 1\; \backslash over\; 2m\_\{tan\}\}$.

Another tangent property

Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then (1) the distance from F to T is 2f, and (2) a tangent to the parabola at point T intersects the line of symmetry at a 45° angle.

When b varies

The x-coordinate at the vertex is $x=-\backslash frac\{b\}\{2a\}$, which is found by deriving the original equation $y=ax^2+bx+c$, setting the resulting $dy/dx=2ax+b$ equal to zero (a critical point), and solving for $x$. Substitute this x-coordinate into the original equation to yield: :$y=a\backslash left\; (-\backslash frac\{b\}\{2a\}\backslash right\; )^2\; +\; b\; \backslash left\; (\; -\backslash frac\{b\}\{2a\}\; \backslash right\; )\; +\; c$ Simplifying: :$=\backslash frac\{ab^2\}\{4a^2\}\; -\backslash frac\{b^2\}\{2a\}\; +\; c$ :$=\backslash frac\{b^2\}\{4a\}\; -\backslash frac\{2\backslash cdot\; b^2\}\{2\backslash cdot\; 2a\}\; +\; c\backslash cdot\backslash frac\{4a\}\{4a\}$ :$=\backslash frac\{-b^2+4ac\}\{4a\}$ :$=-\backslash frac\{b^2-4ac\}\{4a\}=-\backslash frac\{D\}\{4a\}$

Thus, the vertex is at point :$\backslash left\; (-\backslash frac\{b\}\{2a\},-\backslash frac\{D\}\{4a\}\backslash right\; )$

Parabolae in the physical world

, causes the curve swept out to deviate slightly from the expected perfect parabola.]]

In nature, approximations of parabolae and paraboloids (such as catenary curves) are found in many diverse situations. The best-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. He also later proved this mathematically in his book Dialogue Concerning Two New Sciences. 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.

Another situation in which parabolae 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. An object following a parabolic orbit moves at the exact escape velocity of the object it is orbiting, while elliptical orbits are slower and hyperbolic orbits are faster.

, Florianópolis, Brazil. Suspension bridges follow a curve which is intermediate between a parabola and a catenary.]]

Approximations of parabolae are also found in the shape of the main cables on a simple suspension bridge. The curve of the chains of a suspension bridge is always an intermediate curve between an parabola and a catenary, but in practice the curve is generally nearer to a parabola, and in calculations the second degree parabola is used. Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise hyperbolic cable is deformed toward a parabola. Unlike an inelastic chain, a freely-hanging spring of zero unstressed length takes the shape of a parabola.

Paraboloids arise in several physical situations as well. The best-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, 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.

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.

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.

Vertical curves in roads are usually parabolic by design.

Generalizations

In algebraic geometry, the parabola is generalized by the rational normal curves, which have coordinates $(x,x^2,x^3,\backslash dots,x^n);$ the standard parabola is the case $n=2,$ and the case $n=3$ is known as the twisted cubic. A further generalization is given by the Veronese variety, when there are more than one input variable.

In the theory of quadratic forms, the parabola is the graph of the quadratic form $x^2$ (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form $x^2+y^2$ (or scalings) and the hyperbolic paraboloid is the graph of the indefinite quadratic form $x^2-y^2.$ Generalizations to more variables yield further such objects.

The curves $y=x^p$ for other values of p are traditionally referred to as the higher parabolas, and were originally treated implicitly, in the form $x^p=ky^q$ for p and q both positive integers, in which form they are seen to be algebraic curves. These correspond to the explicit formula $y=x^\{p/q\}$ for a positive fractional power of x. Negative fractional powers correspond to the implicit equation $x^py^q=k,$ and are traditionally referred to as higher hyperbolas. Analytically, x can also be raised to an irrational power (for positive values of x); the analytic properties are analogous to when x is raised to rational powers, but the resulting curve is no longer algebraic, and cannot be analyzed via algebraic geometry.

See also

Catenary

Ellipse

Hyperbola

Universal parabolic constant

Parabolic reflector

Parabolic partial differential equation

Paraboloid

Quadratic equation

Quadratic function

Notes

References

Lockwood, E. H. (1961): A Book of Curves, Cambridge University Press

External links

Apollonius' Derivation of the Parabola at Convergence

Interactive parabola-drag focus, see axis of symmetry, directrix, standard and vertex forms

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

Module for the Tangent Parabola

Focal Properties of Parabola at cut-the-knot

Parabola As Envelope II at cut-the-knot

The similarity of parabola at Dynamic Geometry Sketches

a method of drawing a parabola with string and tacks

Category:Conic sections Category:Curves

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