# Parabola

File:Parabola with focus and directrix.svg
Parabolic curve showing directrix (L) and focus (F). The distance from any point on the parabola to the focus (PnF) equals the perpendicular distance from the same point on the parabola to the directrix (PnQn).
File:Parabola with focus and arbitrary line.svg
Parabolic curve showing chord (L), focus (F), and vertex (V). L is an arbitrary chord of the parabola perpendicular to its axis of symmetry, which passes through V and F. (The ends of the chord are not shown here.) The lengths of all paths Qn - Pn - F are the same, equalling the distance between the chord L and the directrix. (See previous diagram above.) This is similar to saying that a parabola is an ellipse, but with one focal point at infinity. It also directly implies, by the wave nature of light, that parallel light arriving along the lines Qn - Pn will be reflected to converge at F. A linear wavefront along L is concentrated, after reflection, to the one point where all parts of it have travelled equal distances and are in phase, namely F. No consideration of angles is required.
File:Parabel som keglesnit.jpg
A parabola obtained as the intersection of a cone with a plane parallel to a straight line on its surface.

In mathematics, a parabola (Template:IPAc-en; plural parabolae or parabolas, from the Greek παραβολή) is a conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. Another way to generate a parabola is to examine a point (the focus) and a line (the directrix). The locus of points in that plane that are equidistant from both the line and point is a parabola. In algebra, parabolas are frequently encountered as graphs of quadratic functions, such as $y=x^2.$

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. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". 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.

Parabolas have the property that, if they are made of material that reflects light, then light which enters a parabola travelling parallel to its axis of symmetry is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected ("collimated") into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy. This reflective property is the basis of many practical uses of parabolas.

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

## History

File:Parabola construction - parallelogram method.gif
Parabola construction using parallelogram method. Click magnify icon for explanation.

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. It means "application", referring to "application of areas" concept, that has a connection with this curve, as Apollonius had proved.<ref>Apollonius' Derivation of the Parabola at Convergence</ref> 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.<ref>Template:Cite book, Extract of page 3 </ref> Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne,<ref>Stargazer, p. 115.</ref> and James Gregory.<ref>Stargazer, pp. 123 and 132</ref> 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.<ref>Template:Cite web</ref> Template:Clear left

## Equation in Cartesian coordinates

Let the directrix be the line x = −p and let the focus be the point (p, 0). If (xy) 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:

$x+p=\sqrt{(x-p)^2+y^2}.$

Squaring both sides and simplifying produces

$y^2 = 4px\$

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 as

$x^2 = 4py.\$

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 (hk). The equation of a parabola with a vertical axis then becomes

$(x-h)^{2}=4p(y-k).\,$

The last equation can be rewritten

$y=ax^2+bx+c\,$

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

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

with the parabola restriction that

$B^{2} = 4 AC,\,$

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

$\begin{bmatrix} A & B/2 & D/2 \\ B/2 & C & E/2 \\ D/2 & E/2 & F \end{bmatrix}.$ 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.<ref>Lawrence, J. Dennis, A Catalog of Special Plane Curves, Dover Publ., 1972.</ref> Template:Clear

## Conic section and quadratic form

File:Las parábolas son cuadráticas.svg
Stock diagram. The upper circle, HQK, is irrelevant in this context.

With reference to the diagram, suppose that there is a horizontal cross-section of the cone passing through P, and that its radius is $r.$ The angle of inclination of the side of the cone, and also of the plane of the parabola, from the vertical is $\theta.$ The chord BC is a diameter of the circle, passing through the point M, which is the midpoint of the chord ED. Let us call the lengths of the line segments EM and DM $x,$ and the length of PM is $y.$

Clearly:

$BM=2y\sin{\theta}.$   (The triangle BPM is isosceles.)
$CM=2r.$   (PM is parallel to AC.)

Using the intersecting chords theorem on the chords BC and DE, we get:

$EM \cdot DM=BM \cdot CM$
$\therefore x^2=4ry\sin{\theta}$

Rearranging:

$y=\frac{x^2}{4r\sin{\theta}}$

For any given cone and parabola, $r$ and $\theta$ are constants, so this last equation is a simple quadratic one relating $x$ and $y.$ This shows that the definition of a parabola as a conic section is equivalent to its definition as the graph of a quadratic function. Both definitions produce curves of exactly the same shape. The focal length of the parabola (see below) is $r\sin{\theta}.$ It should be noted that many different combinations of $r$ and $\theta$ produce the same focal length. In this sense, this situation is different from the one involving the focus and directrix. The focal length of the parabola uniquely specifies the distance between the focus and directrix.

### Position of the focus

If a line is perpendicular to the plane of the parabola and passes through the centre of the horizontal cross-section of the cone passing through P, then the point where this line intersects the plane of the parabola is the focus of the parabola. This is a straightforward consequence of the fact that the focal length is $r\sin{\theta}.$

## 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

In the following equations $h$ and $k$ are the coordinates of the vertex, $(h,k)$, of the parabola and $p$ is the distance from the vertex to the focus and the vertex to the directrix.

#### Vertical axis of symmetry

$(x - h)^2 = 4p(y - k) \,$
$y =\frac{(x-h)^2}{4p}+k\,$
$y = ax^2 + bx + c \,$

where

$a = \frac{1}{4p}; \ \ b = \frac{-h}{2p}; \ \ c = \frac{h^2}{4p} + k; \ \$
$h = \frac{-b}{2a}; \ \ k = \frac{4ac - b^2}{4a}$.

Parametric form:

$x(t) = 2pt + h; \ \ y(t) = pt^2 + k \,$

#### Horizontal axis of symmetry

$(y - k)^2 = 4p(x - h) \,$
$x =\frac{(y - k)^2}{4p} + h;\ \,$
$x = ay^2 + by + c \,$

where

$a = \frac{1}{4p}; \ \ b = \frac{-k}{2p}; \ \ c = \frac{k^2}{4p} + h; \ \$
$h = \frac{4ac - b^2}{4a}; \ \ k = \frac{-b}{2a}$.

Parametric form:

$x(t) = pt^2 + h; \ \ y(t) = 2pt + k \,$

#### General parabola

The general form for a parabola is

$(\alpha x+\beta y)^2 + \gamma x + \delta y + \epsilon = 0 \,$

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

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

and the fact that, for a parabola,

$B^2=4AC \,$.

The equation for a general parabola with a focus point F(u, v), and a directrix in the form

$ax+by+c=0 \,$

is

$\frac{\left(ax+by+c\right)^2}{{a}^{2}+{b}^{2}}=\left(x-u\right)^2+\left(y-v\right)^2 \,$

### Latus rectum, semilatus 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

$r (1 + \cos \theta) = l \,$

where l is the semilatus rectum: the distance from the focus to the parabola itself, measured along a line perpendicular to the axis of symmetry. Note that this equals the perpendicular distance from the focus to the directrix, and is twice the focal length, which is the distance from the focus to the vertex of the parabola.

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

### Gauss-mapped form

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

## Proof of the reflective property

File:Parabel 2.svg
Reflective property of a parabola

The reflective property states that, if a parabola can reflect light, then light which enters it travelling parallel to the axis of symmetry is reflected to the focus. This is derived from the wave nature of light in the caption to a diagram near the top of this article. This derivation is valid, but may not be satisfying to readers who would prefer a mathematical approach. In the following proof, the fact that every point on the parabola is equidistant from the focus and from the directrix is taken as axiomatic.

Consider the parabola $y=x^2.$ Since all parabolas are similar, this simple case represents all others. The right-hand side of the diagram shows part of this parabola.

Construction and definitions

The point E is an arbitrary point on the parabola, with coordinates $(x,x^2).$ The focus is F, the vertex is A (the origin), and the line FA (the y-axis) is the axis of symmetry. The line EC is parallel to the axis of symmetry, and intersects the x-axis at D. The point C is located on the directrix (which is not shown, to minimize clutter). The point B is the midpoint of the line segment FC.

Deductions

Measured along the axis of symmetry, the vertex, A, is equidistant from the focus, F, and from the directrix. Correspondingly, since C is on the directrix, the y-coordinates of F and C are equal in absolute value and opposite in sign. B is the midpoint of FC, so its y-coordinate is zero, so it lies on the x-axis. Its x-coordinate is half that of E, D, and C, i.e. $\fracTemplate:XTemplate:2.$ The slope of the line BE is the quotient of the lengths of ED and BD, which is $\frac{x^2}{\left(\frac{x}{2}\right)},$ which comes to $2x.$

But $2x$ is also the slope (first derivative) of the parabola at E. Therefore the line BE is the tangent to the parabola at E.

The distances EF and EC are equal because E is on the parabola, F is the focus and C is on the directrix. Therefore, since B is the midpoint of FC, triangles FEB and CEB are congruent (three sides), which implies that the angles marked $\alpha$ are equal. (The angle above E is vertically opposite angle BEC.) This means that a ray of light which enters the parabola and arrives at E travelling parallel to the axis of symmetry will be reflected by the line BE so it travels along the line EF, as shown in red in the diagram (assuming that the lines can somehow reflect light). Since BE is the tangent to the parabola at E, the same reflection will be done by an infinitessimal arc of the parabola at E. Therefore, light that enters the parabola and arrives at E travelling parallel to the axis of symmetry of the parabola is reflected by the parabola toward its focus.

The point E has no special characteristics. This conclusion about reflected light applies to all points on the parabola, as is shown on the left side of the diagram. This is the reflective property.

## Two tangent properties

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.<ref>Downs, J. W., Practical Conic Sections, Dover Publ., 2003.</ref>Template:Rp

## Orthoptic property

File:Isoptic.png
Perpendicular tangents intersect on the directrix

Template:Main If two tangents to a parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect on the directrix are perpendicular.

Proof

Without loss of generality, consider the parabola $y=x^2.$ Suppose that two tangents contact this parabola at the points $(p,p^2)$ and $(q,q^2).$ Their slopes are $2p$ and $2q,$ respectively. Thus the equation of the first tangent is of the form $y=2px+C,$ where $C$ is a constant. In order to make the line pass through $(p,p^2),$ the value of $C$ must be $-p^2,$ so the equation of this tangent is $y=2px-p^2.$ Likewise, the equation of the other tangent is $y=2qx-q^2.$ At the intersection point of the two tangents, $2px-p^2=2qx-q^2.$ Thus $2x(p-q)=p^2-q^2.$ Factoring the difference of squares, cancelling, and dividing by 2 gives $x=\frac{p+q}{2}.$ Substituting this into one of the equations of the tangents gives an expression for the y-coordinate of the intersection point: $y=2p\left(\frac{p+q}{2}\right)-p^2.$ Simplifying this gives $y=pq.$

We now use the fact that these tangents are perpendicular. The product of the slopes of perpendicular lines is -1, assuming that both of the slopes are finite. The slopes of our tangents are $2p$ and $2q,$, so $(2p)(2q)=-1,$ so $pq=-\frac{1}{4}.$ Thus the y-coordinate of the intersection point of the tangents is given by $y=-\frac{1}{4}.$ This is also the equation of the directrix of this parabola, so the two perpendicular tangents intersect on the directrix.

## Dimensions of parabolas with axes of symmetry parallel to the y-axis

These parabolas have equations of the form $y=ax^2+bx+c.$ By interchanging $x$ and $y,$ the parabolas' axes of symmetry become parallel to the x-axis.

### Coordinates of the vertex

The x-coordinate at the vertex is $x=-\frac{b}{2a}$, which is found by differentiating 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:

$y=a\left (-\frac{b}{2a}\right )^2 + b \left ( -\frac{b}{2a} \right ) + c.$

Simplifying:

$=\frac{ab^2}{4a^2} -\frac{b^2}{2a} + c$
$=\frac{b^2}{4a} -\frac{2\cdot b^2}{2\cdot 2a} + c\cdot\frac{4a}{4a}$
$=\frac{-b^2+4ac}{4a}$
$=-\frac{b^2-4ac}{4a}=-\frac{D}{4a}$

where $D$ is the discriminant, $(b^2-4ac).$

Thus, the vertex is at point

$\left (-\frac{b}{2a},-\frac{D}{4a}\right ).$

### Coordinates of the focus

Since the axis of symmetry of this parabola is parallel with the y-axis, the x-coordinates of the focus and the vertex are equal. The coordinates of the vertex are calculated in the preceding section. The x-coordinate of the focus is therefore also $-\frac{b}{2a}.$

To find the y-coordinate of the focus, consider the point, P, located on the parabola where the slope is 1, so the tangent to the parabola at P is inclined at 45 degrees to the axis of symmetry. Using the reflective property of a parabola, we know that light which is initially travelling parallel to the axis of symmetry is reflected at P toward the focus. The 45-degree inclination causes the light to be turned 90 degrees by the reflection, so it travels from P to the focus along a line that is perpendicular to the axis of symmetry and to the y-axis. This means that the y-coordinate of P must equal that of the focus.

By differentiating the equation of the parabola and setting the slope to 1, we find the x-coordinate of P:

$y=ax^2+bx+c,$
$\frac{dy}{dx}=2ax+b=1$
$\therefore x=\frac{1-b}{2a}$

Substituting this value of $x$ in the equation of the parabola, we find the y-coordinate of P, and also of the focus:

$y=a\left(\frac{1-b}{2a}\right)^2+b\left(\frac{1-b}{2a}\right)+c$
$=a\left(\frac{1-2b+b^2}{4a^2}\right)+\left(\frac{b-b^2}{2a}\right)+c$
$=\left(\frac{1-2b+b^2}{4a}\right)+\left(\frac{2b-2b^2}{4a}\right)+c$
$=\frac{1-b^2}{4a}+c=\frac{1-(b^2-4ac)}{4a}=\frac{1-D}{4a}$

where $D$ is the discriminant, $(b^2-4ac),$ as is used in the "Coordinates of the vertex" section.

The focus is therefore the point:

$\left(-\frac{b}{2a},\frac{1-D}{4a}\right)$

### Axis of symmetry, focal length, and directrix

The above coordinates of the focus of a parabola of the form:

$y=ax^2+bx+c$

can be compared with the coordinates of its vertex, which are derived in the section "Coordinates of the vertex", above, and are:

$\left(\frac{-b}{2a},\frac{-D}{4a}\right)$

where $D=b^2-4ac.$

The axis of symmetry is the line which passes through both the focus and the vertex. In this case, it is vertical, with equation:

$x=-\frac{b}{2a}$.

The focal length of the parabola is the difference between the y-coordinates of the focus and the vertex:

$f=\left(\frac{1-D}{4a}\right)-\left(\frac{-D}{4a}\right)$
$=\frac{1}{4a}$

Measured along the axis of symmetry, the vertex is the midpoint between the focus and the directrix. Therefore, the equation of the directrix is:

$y=-\frac{D}{4a}-\frac{1}{4a}=-\frac{1+D}{4a}$

## Length of an arc of a parabola

If a point X is located on a parabola which has focal length $f,$ and if $p$ is the perpendicular distance from X to the axis of symmetry of the parabola, then the lengths of arcs of the parabola which terminate at X can be calculated from $f$ and $p$ as follows, assuming they are all expressed in the same units.

$h=\frac{p}{2}$
$q=\sqrt{f^2+h^2}$
$s=\frac{hq}{f}+f\ln\left(\frac{h+q}{f}\right)$

This quantity, $s$, is the length of the arc between X and the vertex of the parabola.

The length of the arc between X and the symmetrically opposite point on the other side of the parabola is $2s.$

The perpendicular distance, $p$, can be given a positive or negative sign to indicate on which side of the axis of symmetry X is situated. Reversing the sign of $p$ reverses the signs of $h$ and $s$ without changing their absolute values. If these quantities are signed, the length of the arc between any two points on the parabola is always shown by the difference between their values of $s.$

This can be useful, for example, in calculating the size of the material needed to make a parabolic reflector or parabolic trough.

This calculation can be used for a parabola in any orientation. It is not restricted to the situation where the axis of symmetry is parallel to the y-axis.

(Note: In the above calculation, the square-root, $q$, must be positive. The quantity ln(a), sometimes written as loge(a), is the natural logarithm of a, i.e. its logarithm to base "e".)

## Parabolae in the physical world

File:Bouncing ball strobe edit.jpg
A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola.
File:ParabolicWaterTrajectory.jpg
Parabolic trajectories of water in a fountain.
File:Celler de Sant Cugat lateral.JPG
Parabolic arches used in architecture

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.<ref>Dialogue Concerning Two New Sciences (1638) (The Motion of Projectiles: Theorem 1); see [1]</ref><ref>However, this parabolic shape, as Newton recognized, is only an approximation of the actual elliptical shape of the trajectory, and is obtained by assuming that the gravitational force is constant (not pointing toward the center of the earth) in the area of interest. Often, this difference is negligible, and leads to a simpler formula for tracking motion.</ref> 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.

File:Ponte Hercilio Luz - Dezembro 1996 - by Sérgio Schmiegelow.jpg
Hercilio Luz Bridge, Florianópolis, Brazil. Suspension bridges follow a curve which is intermediate between a parabola and a catenary.<ref name="Troyano" />

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.<ref name="Troyano">Template:Cite book, Chapter 8 page 536 </ref><ref>Template:Cite book, Extract of page 159 </ref> Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise catenary-shaped cable is deformed toward a parabola. Unlike an inelastic chain, a freely hanging spring of zero unstressed length takes the shape of a parabola. Suspension-bridge cables are, ideally, purely in tension, without having to carry other forces. Similarly, the structures of parabolic arches are purely in compression.

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, or conversely, collimates light from a point source at the focus into a parallel beam. 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,<ref>Template:Cite journal</ref> 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 receiving and transmitting antennas.

It can be seen in architecture as well. For instance the Oval Office of the The White House is essentially two Parabolas facing one another and as such an interesting effect happens. Two people, each standing at one of the focal points 21 feet apart, can whisper secretly to one another despite entertaining the company of others.

File:Coriolis effect11.jpg
Parabolic shape formed by a liquid surface under rotation. Two liquids of different densities completely fill a narrow space between two sheets of plexiglass. The gap between the sheets is closed at the bottom, sides and top. The whole assembly is rotating around a vertical axis passing through the centre. (See Rotating furnace)

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 $(x,x^2,x^3,\dots,x^n);$ 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 is 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.