Parabola

Overview

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the

**parabola**(icon; plural

*parabolae*or

*parabolas*, from the Greek

Greek language

Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

*παραβολή*) is a conic section

Conic section

In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

, the intersection of a right circular conical surface

Conical surface

In geometry, a 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...

and a plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

parallel to a generating straight line of that surface. Given a point (the

**focus**

) and a corresponding line (the

Focus (geometry)

In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...

**directrix**) on the plane, the locus

Locus (mathematics)

In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....

of point

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

s in that plane that are equidistant

Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...

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".

Unanswered Questions

Encyclopedia

In mathematics

, the

, the intersection of a right circular conical surface

and a plane

parallel to a generating straight line of that surface. Given a point (the

of point

s 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.

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

, 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 mathematician

s 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.

Squaring both sides and simplifying produces

as the equation of the parabola. By interchanging the roles of

The equation can be generalized to allow the vertex to be at a point other than the origin by defining the vertex as the point (

The last equation can be rewritten

so the graph of any function which is a polynomial of degree 2 in

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

with the parabola restriction that

where all of the coefficients are real and where

is non-zero: that is, if (

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 ellipse

s 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).

where.

Parametric form:

where.

Parametric form:

This result is derived from the general conic equation given below:

and the fact that, for a parabola,

.

The equation for a general parabola with a focus point

is

where

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

form:

has normal

.

then there is a point (0,

Imagine a right triangle with two legs,

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

The line

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

Square both sides,

Cancel out terms from both sides,

Divide out the

So, for a parabola such as f(x)=x², the

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

,

the focus is located at the point

which may also be written as

and the directrix is designated by the equation

which may also be written as

This line intersects the

Since

and it is already known that

and, thirdly, line

It follows that

.

Line

, so they are equal (congruent). But is equal to . Therefore is equal to .

The line

Let a light beam travel down the vertical line

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.

The tangent has then a generic slope of.

Reflection derivation, together with trigonometric angle addition rules, leads to the result that the reflected ray has a slope of.

## When

The

the original equation , setting the resulting equal to zero (a critical point

), and solving for . Substitute this

Simplifying:

Thus, the vertex is at point

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

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.

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 a 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 telescope

s 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.

, the parabola is generalized by the rational normal curves, which have coordinates the standard parabola is the case and the case 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 form

s, the parabola is the graph of the quadratic form (or other scalings), while the elliptic paraboloid is the graph of the positive-definite

quadratic form (or scalings) and the hyperbolic paraboloid is the graph of the indefinite quadratic form Generalizations to more variables yield further such objects.

The curves for other values of

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the

**parabola**(icon; plural*parabolae*or*parabolas*, from the GreekGreek language

Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

*παραβολή*) is a conic sectionConic section

In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

, the intersection of a right circular conical surface

Conical surface

In geometry, a 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...

and a plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

parallel to a generating straight line of that surface. Given a point (the

**focus**

) and a corresponding line (theFocus (geometry)

In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...

**directrix**) on the plane, the locusLocus (mathematics)

In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....

of point

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

s in that plane that are equidistant

Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...

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

Vertex (curve)

In the geometry of curves, a vertex is a point of where the first derivative of curvature is zero. This is typically a local maximum or minimum of curvature. Other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant...

", and it is the point where the curvature

Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

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

Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...

.

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

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, engineering

Engineering

Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, and many other areas.

## History

The earliest known work on conic sections was by MenaechmusMenaechmus

Menaechmus was an ancient Greek mathematician and geometer born in Alopeconnesus in the Thracian Chersonese, who was known for his friendship with the renowned philosopher Plato and for his apparent discovery of conic sections and his solution to the then-long-standing problem of doubling the cube...

in the fourth century BC. He discovered a way to solve the problem of doubling the cube

Doubling the cube

Doubling the cube is one of the three most famous geometric problems unsolvable by compass and straightedge construction...

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

Archimedes

Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

via the method of exhaustion

Method of exhaustion

The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will...

in the third century BC, in his

*The Quadrature of the Parabola*

.The name "parabola" is due to ApolloniusThe Quadrature of the Parabola

The Quadrature of the Parabola is a treatise on geometry, written by Archimedes in the 3rd century BC. Written as a letter to his friend Dositheus, the work presents 24 propositions regarding parabolas, culminating in a proof that the area of a parabolic segment is 4/3 that of a certain inscribed...

.

Apollonius of Perga

Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...

, who discovered many properties of conic sections. The focus–directrix property of the parabola and other conics is due to Pappus

Pappus of Alexandria

Pappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...

.

Galileo

Galileo Galilei

Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity.

The idea that a parabolic reflector

Parabolic reflector

A parabolic reflector is a reflective device used to collect or project energy such as light, sound, or radio waves. Its shape is that of a circular paraboloid, that is, the surface generated by a parabola revolving around its axis...

could produce an image was already well known before the invention of the reflecting telescope

Reflecting telescope

A reflecting telescope is an optical telescope which uses a single or combination of curved mirrors that reflect light and form an image. The reflecting telescope was invented in the 17th century as an alternative to the refracting telescope which, at that time, was a design that suffered from...

. Designs were proposed in the early to mid seventeenth century by many mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

s including René Descartes

René Descartes

René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...

, Marin Mersenne

Marin Mersenne

Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics"...

, and James Gregory. When Isaac Newton

Isaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

built the first reflecting telescope

Newton's Reflector

The first reflecting telescope built by Sir Isaac Newton in 1668 is a landmark in the history of telescopes, being the first known successful reflecting telescope. It was the prototype for a design that later came to be called a newtonian 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:Squaring both sides and simplifying produces

as the equation of the parabola. By interchanging the roles of

*x*and*y*one obtains the corresponding equation of a parabola with a vertical axis asThe equation can be generalized to allow the vertex to be at a point other than the origin by defining the vertex as the point (

*h*,*k*). The equation of a parabola with a vertical axis then becomesThe last equation can be rewritten

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

Irreducible polynomial

In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....

equation — one that does not factor as a product of two not necessarily distinct linear equations — of the general conic form

with the parabola restriction that

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 matrixis non-zero: that is, if (

*AC*-*B*^{2}/4)*F*+*BED*/4 -*CD*^{2}/4 -*AE*^{2}/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 eccentricityEccentricity (mathematics)

In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...

of 1. As a consequence of this, all parabolae are similar

Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...

, meaning that while they can be different sizes, they are all the same shape. A parabola can also be obtained as the limit

Limit of a function

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

of a sequence of ellipse

Ellipse

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

s 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

Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...

. The parabola is an inverse transform of a cardioid

Cardioid

A cardioid is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It is therefore a type of limaçon and can also be defined as an epicycloid having a single cusp...

.

A parabola has a single axis of reflective symmetry

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

, 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

Paraboloid

In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....

of revolution.

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

#### Vertical axis of symmetry

where.

Parametric form:

#### Horizontal axis of symmetry

where.

Parametric form:

#### General parabola

The general form for a parabola isThis result is derived from the general conic equation given below:

and the fact that, for a parabola,

.

The equation for a general parabola with a focus point

*F*(*u*,*v*), and a directrix in the formis

### 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 equationwhere

*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-mappedGauss map

In differential geometry, the Gauss map maps a surface in Euclidean space R3 to the unit sphere S2. Namely, given a surface X lying in R3, the Gauss map is a continuous map N: X → S2 such that N is a unit vector orthogonal to X at p, namely the normal vector to X at p.The Gauss map can be defined...

form:

has normal

.

## 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 asthen 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 linear 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(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 linear directrix).These two line segments are equal, and, as indicated above, y=ax², thus

Square both sides,

Cancel out terms from both sides,

Divide out the

*x²*from both sides (we assume that*x*is not zero),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

Standard form

Standard form may refer to:*The more common name for scientific notation in British English*Standard form – a common form of a linear equation*Canonical form...

,

the focus is located at the point

which may also be written as

and the directrix is designated by the equation

which may also be written as

## Reflective property of the tangent

The tangent of the parabola described by equation y=ax^{2}has slopeThis 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*:Since

*G*is the midpoint of line*FQ*, this means thatand it is already known that

*P*is equidistant from both*F*and*Q*:and, thirdly, line

*GP*is equal to itself, therefore:It follows that

.

Line

*QP*can be extended beyond*P*to some point*T*, and line*GP*can be extended beyond*P*to some point*R*. Then and are verticalVertical (angles)

In geometry, a pair of angles is said to be vertical if the angles are formed from two intersecting lines and the angles are not adjacent. The two angles share a vertex...

, so they are equal (congruent). But is equal to . Therefore is equal to .

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 , so when it bounces off, its angle of inclination must be equal to . But has been shown to be equal to . 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

Parabolic reflector

A parabolic reflector is a reflective device used to collect or project energy such as light, sound, or radio waves. Its shape is that of a circular paraboloid, that is, the surface generated by a parabola revolving around its axis...

.)

The same reasoning can be applied to a parabola whose axis is vertical, so that it can be specified by the equation.

The tangent has then a generic slope of.

Reflection derivation, together with trigonometric angle addition rules, leads to the result that the reflected ray has a slope of.

## 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 2*f*, 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 , which is found by derivingDerivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

the original equation , setting the resulting equal to zero (a critical point

Critical point (mathematics)

In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...

), and solving for . Substitute this

*x*-coordinate into the original equation to yield:Simplifying:

Thus, the vertex is at point

## Parabolae in the physical world

In nature, approximations of parabolae and paraboloids (such as catenary curves

Catenary

In physics and geometry, the catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight. The curve is the graph of the hyperbolic cosine function, and has a U-like shape, superficially similar in appearance to a parabola...

) are found in many diverse situations. The best-known instance of the parabola in the history of physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

is the trajectory

Trajectory

A trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...

of a particle or body in motion under the influence of a uniform gravitational field

Gravitational field

The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

without air resistance (for instance, a baseball flying through the air, neglecting air friction

Friction

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of 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 massCenter of mass

In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

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

Hyperbola

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

or an ellipse

Ellipse

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

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

Escape velocity

In physics, escape velocity is the speed at which the kinetic energy plus the gravitational potential energy of an object is zero gravitational potential energy is negative since gravity is an attractive force and the potential is defined to be zero at infinity...

of the object it is orbiting, while elliptical orbits are slower and hyperbolic orbits are faster.

Approximations of parabolae are also found in the shape of the main cables on a simple suspension bridge

Suspension bridge

A suspension bridge is a type of bridge in which the deck is hung below suspension cables on vertical suspenders. Outside Tibet and Bhutan, where the first examples of this type of bridge were built in the 15th century, this type of bridge dates from the early 19th century...

. The curve of the chains of a suspension bridge is always an intermediate curve between a parabola and a catenary

Catenary

In physics and geometry, the catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight. The curve is the graph of the hyperbolic cosine function, and has a U-like shape, superficially similar in appearance to a parabola...

, 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

Parabolic reflector

A parabolic reflector is a reflective device used to collect or project energy such as light, sound, or radio waves. Its shape is that of a circular paraboloid, that is, the surface generated by a parabola revolving around its axis...

, which is a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation

Electromagnetic radiation

Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

to a common focal point

Focus (optics)

In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...

. The principle of the parabolic reflector may have been discovered in the 3rd century BC by the geometer Archimedes

Archimedes

Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

, who, according to a legend of debatable veracity, constructed parabolic mirrors to defend Syracuse

Syracuse, Italy

Syracuse is a historic city in Sicily, the capital of the province of Syracuse. The city is notable for its rich Greek history, culture, amphitheatres, architecture, and as the birthplace of the preeminent mathematician and engineer Archimedes. This 2,700-year-old city played a key role in...

against the Roman

Roman Empire

The Roman Empire was the post-Republican period of the ancient Roman civilization, characterised by an autocratic form of government and large territorial holdings in Europe and around the Mediterranean....

fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to telescope

Telescope

A telescope is an instrument that aids in the observation of remote objects by collecting electromagnetic radiation . The first known practical telescopes were invented in the Netherlands at the beginning of the 1600s , using glass lenses...

s in the 17th century. Today, paraboloid reflectors can be commonly observed throughout much of the world in microwave

Microwave

Microwaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz and 300 GHz. This broad definition includes both UHF and EHF , and various sources use different boundaries...

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

Centrifugal force

Centrifugal force can generally be any force directed outward relative to some origin. More particularly, in classical mechanics, the centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame...

causes the liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind the liquid mirror telescope.

Aircraft

Aircraft

An aircraft is a vehicle that is able to fly by gaining support from the air, or, in general, the atmosphere of a planet. An aircraft counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines.Although...

used to create a weightless state for purposes of experimentation, such as NASA

NASA

The National Aeronautics and Space Administration is the agency of the United States government that is responsible for the nation's civilian space program and for aeronautics and aerospace research...

's “Vomit Comet

Vomit Comet

A Reduced Gravity Aircraft is a type of fixed-wing aircraft that briefly provides a nearly weightless environment in which to train astronauts, conduct research and film motion pictures....

,” follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall

Free fall

Free fall is any motion of a body where gravity is the only force acting upon it, at least initially. These conditions produce an inertial trajectory so long as gravity remains the only force. Since this definition does not specify velocity, it also applies to objects initially moving upward...

, which produces the same effect as zero gravity

Weightlessness

Weightlessness is the condition that exists for an object or person when they experience little or no acceleration except the acceleration that defines their inertial trajectory, or the trajectory of pure free-fall...

for most purposes.

Vertical curves in roads are usually parabolic by design.

## Generalizations

In algebraic geometryAlgebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, the parabola is generalized by the rational normal curves, which have coordinates the standard parabola is the case and the case is known as the twisted cubic

Twisted cubic

In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation...

. A further generalization is given by the Veronese variety, when there are more than one input variable.

In the theory of quadratic form

Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

s, the parabola is the graph of the quadratic form (or other scalings), while the elliptic paraboloid is the graph of the positive-definite

Definite bilinear form

In mathematics, a definite bilinear form is a bilinear form B over some vector space V such that the associated quadratic formQ=B \,...

quadratic form (or scalings) and the hyperbolic paraboloid is the graph of the indefinite quadratic form Generalizations to more variables yield further such objects.

The curves for other values of

*p*are traditionally referred to as the**higher parabolas**, and were originally treated implicitly, in the form for*p*and*q*both positive integers, in which form they are seen to be algebraic curves. These correspond to the explicit formula for a positive fractional power of*x.*Negative fractional powers correspond to the implicit equation 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

- CatenaryCatenaryIn physics and geometry, the catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight. The curve is the graph of the hyperbolic cosine function, and has a U-like shape, superficially similar in appearance to a parabola...
- EllipseEllipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
- HyperbolaHyperbolaIn mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
- Universal parabolic constantUniversal parabolic constantThe universal parabolic constant is a mathematical constant.It is defined as the ratio, for any parabola, of the arc length of the parabolic segment formed by the latus rectum to the focal parameter...
- Parabolic reflectorParabolic reflector
- Parabolic partial differential equationParabolic partial differential equationA parabolic partial differential equation is a type of second-order partial differential equation , describing a wide family of problems in science including heat diffusion, ocean acoustic propagation, in physical or mathematical systems with a time variable, and which behave essentially like heat...
- ParaboloidParaboloidIn mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....
- Quadratic equationQuadratic equationIn mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
- Quadratic functionQuadratic functionA quadratic function, in mathematics, is a polynomial function of the formf=ax^2+bx+c,\quad a \ne 0.The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis....

## 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-knotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
- Two Tangents to Parabola at cut-the-knotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
- Parabola As Envelope of Straight Lines at cut-the-knotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
- Parabolic Mirror at cut-the-knotCut-the-knot
- Three Parabola Tangents at cut-the-knotCut-the-knot
- Module for the Tangent Parabola
- Focal Properties of Parabola at cut-the-knotCut-the-knot
- Parabola As Envelope II at cut-the-knotCut-the-knot
- The similarity of parabola at Dynamic Geometry Sketches
- a method of drawing a parabola with string and tacks