Editing Parabola

{{short description|Plane curve: conic section}}
{{other uses}}
[[File:Parts of Parabola.svg|thumb|right|upright=1.36|Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward.]]
[[File:Conic Sections.svg|thumb|The parabola is a member of the family of [[conic section]]s.]]

In [[mathematics]], a '''parabola''' is a [[plane curve]] which is [[Reflection symmetry|mirror-symmetrical]] and is approximately U-shaped. It fits several superficially different [[Mathematics|mathematical]] descriptions, which can all be proved to define exactly the same curves.

One description of a parabola involves a [[Point (geometry)|point]] (the [[Focus (geometry)|focus]]) and a [[Line (geometry)|line]] (the [[Directrix (conic section)|directrix]]). The focus does not lie on the directrix. The parabola is the [[locus (mathematics)|locus of points]] in that plane that are [[equidistant]] from the directrix and the focus. Another description of a parabola is as a [[conic section]], created from the intersection of a right circular [[conical surface]] and a [[plane (geometry)|plane]] [[Parallel (geometry)|parallel]] to another plane that is [[tangent]]ial to the conical surface.{{efn|The tangential plane just touches the conical surface along a line, which passes through the apex of the cone.}}

The [[graph of a function|graph]] of a [[quadratic function]] <math>y=ax^2+bx+ c</math> (with <math>a\neq 0 </math>) is a parabola with its axis parallel to the {{mvar|y}}-axis. Conversely, every such  parabola is the graph of a quadratic function.

The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "[[vertex (curve)|vertex]]" and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "[[Conic section#Conic parameters|latus rectum]]" is the [[Chord (geometry)|chord]] of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically [[Similarity (geometry)|similar]].

Parabolas have the property that, if they are made of material that [[Reflection (physics)|reflects]] [[light]], then light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel ("[[collimated]]") beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with [[sound]] and other [[wave]]s. This reflective property is the basis of many practical uses of parabolas.

The parabola has many important applications, from a [[parabolic antenna]] or [[parabolic microphone]] to automobile [[Headlamp|headlight]] reflectors and the design of [[ballistic missiles]]. It is frequently used in [[physics]], [[engineering]], and many other areas.

== History ==
[[File:Leonardo parabolic compass.JPG|thumb|180px|Parabolic compass designed by [[Leonardo da Vinci]]]]
The earliest known work on conic sections was by [[Menaechmus]] in the 4th century BC. He discovered a way to solve the problem of [[doubling the cube]] using parabolas. (The solution, however, does not meet the requirements of [[compass-and-straightedge construction]].) The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by [[Archimedes]] by the [[method of exhaustion]] in the 3rd century BC, in his ''[[The Quadrature of the Parabola]]''. The name "parabola" is due to [[Apollonius of Perga|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>{{cite web |url=http://www.maa.org/press/periodicals/convergence/can-you-really-derive-conic-formulae-from-a-cone-deriving-the-symptom-of-the-parabola |title=Can You Really Derive Conic Formulae from a Cone? – Deriving the Symptom of the Parabola – Mathematical Association of America |access-date=30 September 2016}}</ref> The focus–directrix property of the parabola and other conic sections was mentioned in the works of [[Pappus of Alexandria|Pappus]].

[[Galileo Galilei|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>{{cite book
|title=Reflecting Telescope Optics: Basic design theory and its historical development
|edition=2
|first1=Ray N.
|last1=Wilson
|publisher=Springer
|year=2004
|isbn=3-540-40106-7
|page=3
|url=https://books.google.com/books?id=PuN7l2A2uzQC}} [https://books.google.com/books?id=PuN7l2A2uzQC&pg=PA3 Extract of page 3].
</ref> Designs were proposed in the early to mid-17th century by many [[mathematician]]s, including [[René Descartes]], [[Marin Mersenne]],<ref>''Stargazer'', [https://books.google.com/books?id=2LZZginzib4C&pg=PA115&dq=mersenne+zucchi+parallel#PPA115,M1 p.&nbsp;115].</ref> and [[James Gregory (mathematician)|James Gregory]].<ref>''Stargazer'', [https://books.google.com/books?id=2LZZginzib4C&pg=PA132&dq=Gregory++telescope+French+convex pp.&nbsp;123, 132].</ref> When [[Isaac Newton]] built the [[Newton's reflector|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 dish]]es and [[radar]] receivers.<ref>{{cite web
|url        = http://farside.ph.utexas.edu/teaching/316/lectures/node136.html
|title      = Spherical Mirrors
|first      = Richard
|last       = Fitzpatrick
|date       = July 14, 2007
|work       = Electromagnetism and Optics, lectures
|publisher  = [[University of Texas at Austin]]
|at         = Paraxial Optics
|access-date = October 5, 2011}}</ref>
{{clear}}

== Definition as a locus of points ==
A parabola can be defined geometrically as a set of points ([[locus (mathematics)|locus]]) in the Euclidean plane, as follows.

A parabola is the set of the points whose distance to a fixed point, the ''focus'', equals the distance to a fixed [[line (mathematics)|line]], the ''directrix''. That is, if {{tmath|F}} is the focus and {{tmath|l}} is the directrix, the parabola is the set of all points {{tmath|P}} such that
<math display="block">d(P,F) = d(P,l),</math> where {{tmath|d}} denotes [[Euclidean distance]].

The point where this distance is minimal is the midpoint <math>V</math> of the perpendicular from the focus <math>F</math> to the directrix <math>l.</math> It is called the ''vertex'', and its distance to both the focus and the directrix is the ''focal length'' of the parabola. 

The line <math>FV</math> is the uniqus  [[axis of symmetry]] of the parabola and called the ''axis'' of the parabola.

== In a Cartesian coordinate system ==
=== Axis of symmetry parallel to the ''y'' axis ===
[[File:Parabel-def-p-v.svg|thumb|Parabola with axis parallel to {{math|''y''}}-axis; {{math|''p''}} is the ''semi-latus rectum'']]
If one introduces [[Cartesian coordinates]], such that the vertex {{tmath|V}} is the origin and the focus {{tmath|F}} is on the positive {{tmath|y}}-axis, one has <math>F = (0, f),</math> where {{tmath|f}} is the focal length. The directrix has the equation <math>y = -f.</math> 

The above geometric characterization implies that a point  <math>P = (x, y)</math> is on the parabola if and only if
<math display=block>x^2 + (y - f)^2 = (y + f)^2</math>. Solving for <math>y</math> yields
<math display="block">y = \frac{1}{4f} x^2.</math>

This parabola is U-shaped (''opening to the top'').

The horizontal [[chord (geometry)|chord]] through the focus is on the line of equation {{tmath|1=t=f}} (see picture in opening section); it is called the ''latus rectum''; one half of it is the ''[[Conic section#Conic parameters|semi-latus rectum]]''. The latus rectum is parallel to the directrix. The semi-latus rectum is denoted by <math>p</math>. From the equation satisfied by the endpoints of the latus rectum, one gets
<math display="block">p = 2f.</math>
Thus, the semi-lactus rectum is the distance from the focus to the directrix. Using the parameter <math>p</math>, the equation of the parabola can be rewritten as
<math display="block">x^2 = 2py.</math>

More generally, if the vertex is <math>V = (v_1, v_2)</math>, the focus <math>F = (v_1, v_2 + f)</math>, and the directrix <math>y = v_2 - f </math>, one obtains the equation
<math display="block">y = \frac{1}{4f} (x - v_1)^2 + v_2 = \frac{1}{4f} x^2 - \frac{v_1}{2f} x + \frac{v_1^2}{4f} + v_2.</math>

'''Remarks''':
* If <math>f < 0</math> in the above equations one gets parabola with a downward opening.
* The hypothesis that the axis is parallel to the {{tmath|y}}-axis implies that the parabola is the graph of a [[quadratic function]]. Conversely, the graph of an arbitrary quadratic function is a parabola (see next section).
* If one exchanges <math>x</math> and <math>y</math>, one obtains equations of the form <math>y^2 = 2px</math>. These parabolas open to the left (if <math>p < 0</math>) or to the right (if <math>p > 0</math>).

=== General position ===
[[File:Parabel-abc.svg|thumb|Parabola: general position]]
If the focus is <math>F = (f_1, f_2)</math>, and the directrix <math>ax + by + c = 0</math>, then one obtains the equation
<math display="block">\frac{(ax + by + c)^2}{a^2 + b^2} = (x - f_1)^2 + (y - f_2)^2</math>

(the left side of the equation uses the [[Hesse normal form]] of a line to calculate the distance <math>|Pl|</math>).

For a [[parametric equation]] of a parabola in general position see {{slink||2=As the affine image of the unit parabola}}.

The [[implicit equation]] of a parabola is defined by an [[irreducible polynomial]] of degree two:
<math display="block">ax^2 + bxy + cy^2 + dx + ey + f = 0,</math>
such that <math>b^2 - 4ac = 0,</math> or, equivalently, such that <math>ax^2 + bxy + cy^2</math> is the square of a [[linear polynomial]].

== As a graph of a function ==
[[File:Parabeln-var-s.svg|thumb|Parabolas <math>y = ax^2</math>]]
The previous section shows that any parabola with the origin as vertex and the ''y'' axis as axis of symmetry can be considered as the graph of a function
<math display="block">f(x) = a x^2 \text{ with } a \ne 0.</math>

For <math>a > 0</math> the parabolas are opening to the top, and for <math>a < 0</math> are opening to the bottom (see picture). From the section above one obtains:
* The ''focus '' is <math>\left(0, \frac{1}{4a}\right)</math>,
* the ''focal length'' <math>\frac{1}{4a}</math>, the ''semi-latus rectum'' is <math>p = \frac{1}{2a}</math>,
* the ''vertex'' is <math>(0, 0)</math>,
* the ''directrix'' has the equation <math>y = -\frac{1}{4a}</math>,
* the ''[[tangent]]'' at point <math>(x_0, ax^2_0)</math> has the equation <math>y = 2a x_0 x - a x^2_0</math>.

For <math>a = 1</math> the parabola is the '''unit parabola''' with equation <math>y = x^2</math>.
Its focus is <math>\left(0, \tfrac{1}{4}\right)</math>, the semi-latus rectum <math>p = \tfrac{1}{2}</math>, and the directrix has the equation <math>y = -\tfrac{1}{4}</math>.

The general function of degree 2 is
<math display="block">f(x) = ax^2 + bx + c ~~\text{ with }~~ a, b, c \in \R,\ a \ne 0.</math>
[[Completing the square]] yields
<math display="block">f(x) = a \left(x + \frac{b}{2a}\right)^2 + \frac{4ac - b^2}{4a},</math>
which is the equation of a parabola with
* the axis <math>x = -\frac{b}{2a} </math> (parallel to the ''y'' axis),
* the ''focal length'' <math>\frac{1}{4a}</math>, the ''semi-latus rectum'' <math>p = \frac{1}{2a}</math>,
* the ''vertex'' <math>V = \left(-\frac{b}{2a}, \frac{4ac - b^2}{4a}\right)</math>,
* the ''focus'' <math>F = \left(-\frac{b}{2a}, \frac{4ac - b^2 + 1}{4a}\right)</math>,
* the ''directrix'' <math>y = \frac{4ac - b^2 - 1}{4a}</math>,
* the point of the parabola intersecting the ''y'' axis has coordinates <math>(0, c)</math>,
* the ''tangent'' at a point on the ''y'' axis has the equation <math>y = bx + c</math>.

== Similarity to the unit parabola ==
[[File:Parabel-scal2.svg|thumb|When the parabola <math>\color{blue}{y = 2x^2}</math> is uniformly scaled by factor 2, the result is the parabola <math>\color{red}{y = x^2}</math>]]
Two objects in the Euclidean plane are ''[[Similarity (geometry)|similar]]'' if one can be transformed to the other by a ''similarity'', that is, an arbitrary [[Composition of functions|composition]] of rigid motions ([[Translation of axes|translations]] and [[Rotation of axes|rotations]]) and [[uniform scaling]]s.

A parabola <math>\mathcal P</math> with vertex <math>V = (v_1, v_2)</math> can be transformed by the translation <math>(x, y) \to (x - v_1, y - v_2)</math> to one with the origin as vertex. A suitable rotation around the origin can then transform the parabola to one that has the {{mvar|y}} axis as axis of symmetry. Hence the parabola <math>\mathcal P</math> can be transformed by a rigid motion to a parabola with an equation <math>y = ax^2,\ a \ne 0</math>. Such a parabola can then be transformed by the [[uniform scaling]] <math>(x, y) \to (ax, ay)</math> into the unit parabola with equation <math>y = x^2</math>. Thus, any parabola can be mapped to the unit parabola by a similarity.<ref name="Kumpel">{{citation |first=P. G. |last=Kumpel |title=Do similar figures always have the same shape? |journal=The Mathematics Teacher |year=1975 |volume=68 |issue=8 |pages=626–628 |doi=10.5951/MT.68.8.0626 |issn=0025-5769}}.</ref>

A [[Synthetic geometry|synthetic]] approach, using similar triangles, can also be used to establish this result.<ref>{{citation |first1=Atara |last1=Shriki |first2=Hamatal |last2=David |title=Similarity of Parabolas – A Geometrical Perspective |journal=Learning and Teaching Mathematics |year=2011 |volume=11 |pages=29–34}}.</ref>

The general result is that two conic sections (necessarily of the same type) are similar if and only if they have the same eccentricity.<ref name="Kumpel" /> Therefore, only circles (all having eccentricity 0) share this property with parabolas (all having eccentricity 1), while general ellipses and hyperbolas do not.

There are other simple affine transformations that map the parabola <math>y = ax^2</math> onto the unit parabola, such as <math>(x, y) \to \left(x, \tfrac{y}{a}\right)</math>. But this mapping is not a similarity, and only shows that all parabolas are affinely equivalent (see {{slink||2=As the affine image of the unit parabola}}).

== As a special conic section ==
[[File:Kegelschnitt-schar-ev.svg|thumb|Pencil of conics with a common vertex]]
The [[Pencil (mathematics)|pencil]] of [[conic section]]s with the ''x'' axis as axis of symmetry, one vertex at the origin (0,&nbsp;0) and the same semi-latus rectum <math>p</math> can be represented by the equation 
<math display="block"> y^2 = 2px +(e^2 - 1) x^2, \quad e \ge 0,</math>
with <math>e</math> the [[Eccentricity (mathematics)|eccentricity]].
* For <math>e = 0</math> the conic is a ''circle'' (osculating circle of the pencil),
* for <math>0 < e < 1</math> an ''ellipse'',
* for <math>e = 1</math> the '''parabola''' with equation <math>y^2 = 2px,</math>
* for <math>e > 1</math> a hyperbola (see picture).

== In polar coordinates ==
[[File:Kegelschnittschar-polar-e.svg|thumb|Pencil of conics with a common focus]]
If {{math|''p'' > 0}}, the parabola with equation <math>y^2 = 2px</math> (opening to the right) has the [[Polar coordinate system|polar]] representation
<math display="block">r = 2p \frac{\cos\varphi}{\sin^2\varphi}, \quad \varphi \in \left[ -\tfrac{\pi}{2} , \tfrac{\pi}{2} \right] \setminus \{0\}</math>
where <math>r^2 = x^2 + y^2,\ x = r\cos\varphi</math>.

Its vertex is <math>V = (0, 0)</math>, and its focus is <math>F = \left(\tfrac{p}{2}, 0\right)</math>.

If one shifts the origin into the focus, that is, <math>F = (0, 0)</math>, one obtains the equation
<math display="block">r = \frac{p}{1 - \cos\varphi}, \quad \varphi \ne 2\pi k.</math>

''Remark 1:'' Inverting this polar form shows that a parabola is the [[Inversive geometry|inverse]] of a [[cardioid]].

''Remark 2:'' The second polar form is a special case of a pencil of conics with focus <math>F = (0, 0)</math> (see picture):
<math display="block">r = \frac{p}{1 - e\cos\varphi}</math> (<math>e</math> is the eccentricity).

== Conic section and quadratic form ==
=== Diagram, description, and definitions ===
[[image:Parabolic conic section.svg|thumb|Cone with cross-sections]]

The diagram represents a [[cone]] with its axis {{overline|AV}}. The point A is its [[apex (geometry)|apex]]. An inclined [[Cross section (geometry)|cross-section]] of the cone, shown in pink, is inclined from the axis by the same angle {{mvar|θ}}, as the side of the cone. According to the definition of a parabola as a conic section, the boundary of this pink cross-section EPD is a parabola.

A cross-section perpendicular to the axis of the cone passes through the vertex P of the parabola. This cross-section is circular, but appears [[Ellipse|elliptical]] when viewed obliquely, as is shown in the diagram. Its centre is V, and {{overline|PK}} is a diameter. We will call its radius&nbsp;{{mvar|r}}.

Another perpendicular to the axis, circular cross-section of the cone is farther from the apex A than the one just described. It has a [[Chord (geometry)|chord]] {{overline|DE}}, which joins the points where the parabola [[Intersection (Euclidean geometry)|intersects]] the circle. Another chord {{overline|BC}} is the [[perpendicular bisector]] of {{overline|DE}} and is consequently a diameter of the circle. These two chords and the parabola's axis of symmetry {{overline|PM}} all intersect at the point&nbsp;M.

All the labelled points, except D and E, are [[coplanar]]. They are in the plane of symmetry of the whole figure. This includes the point F, which is not mentioned above. It is defined and discussed below, in {{slink||Position of the focus}}.

Let us call the length of {{overline|DM}} and of {{overline|EM}} {{mvar|x}}, and the length of {{overline|PM}}&nbsp;{{mvar|y}}.

=== Derivation of quadratic equation ===
The lengths of {{overline|BM}} and {{overline|CM}} are:
{{unbulleted list | style = padding-left:1.6em;
| <math>\overline\mathrm{BM} = 2y\cos\theta</math>{{pad|1em}}(triangle BPM is [[isosceles]], because <math>\overline{PM} \parallel \overline{AC} \implies \angle PMB = \angle ACB = \angle ABC</math>
| <math>\overline\mathrm{CM} = 2r</math>{{pad|1em}}(PMCK is a [[parallelogram]]).
}}
Using the [[Chord theorem|intersecting chords theorem]] on the chords {{overline|BC}} and {{overline|DE}}, we get
<math display="block">\overline\mathrm{BM} \cdot \overline\mathrm{CM} = \overline\mathrm{DM} \cdot \overline\mathrm{EM}.</math>

Substituting:
<math display="block">4ry\cos\theta = x^2.</math>

Rearranging:
<math display="block">y = \frac{x^2}{4r\cos\theta}.</math>

For any given cone and parabola, {{mvar|r}} and {{mvar|θ}} are constants, but {{mvar|x}} and {{mvar|y}} are variables that depend on the arbitrary height at which the horizontal cross-section BECD is made. This last equation shows the relationship between these variables. They can be interpreted as [[Cartesian coordinate system|Cartesian coordinates]] of the points D and E, in a system in the pink plane with P as its origin. Since {{mvar|x}} is squared in the equation, the fact that D and E are on opposite sides of the {{mvar|y}} axis is unimportant. If the horizontal cross-section moves up or down, toward or away from the apex of the cone, D and E move along the parabola, always maintaining the relationship between {{mvar|x}} and {{mvar|y}} shown in the equation. The parabolic curve is therefore the [[Locus (mathematics)|locus]] of points where the equation is satisfied, which makes it a [[Graph of a function|Cartesian graph]] of the quadratic function in the equation.

=== Focal length ===
It is proved in a [[#In a cartesian coordinate system|preceding section]] that if a parabola has its vertex at the origin, and if it opens in the positive {{mvar|y}} direction, then its equation is {{math|1=''y'' = {{sfrac|''x''<sup>2</sup>|4''f''}}}}, where {{mvar|f}} is its focal length.{{efn|As stated above in the lead, the focal length of a parabola is the distance between its vertex and focus.}} Comparing this with the last equation above shows that the focal length of the parabola in the cone is {{math|''r'' cos ''θ''}}.

=== Position of the focus ===
In the diagram above, the point&nbsp;V is the [[Perpendicular#Foot of a perpendicular|foot of the perpendicular]] from the vertex of the parabola to the axis of the cone. ''The point&nbsp;F is the foot of the perpendicular from the point&nbsp;V to the plane of the parabola.''{{efn|The point&nbsp;V is the centre of the smaller circular cross-section of the cone. The point&nbsp;F is in the (pink) plane of the parabola, and the line {{overline|VF}} is perpendicular to the plane of the parabola.}} By symmetry, F is on the axis of symmetry of the parabola. Angle VPF is [[Complementary angles|complementary]] to {{mvar|θ}}, and angle PVF is complementary to angle VPF, therefore angle PVF is {{mvar|θ}}. Since the length of {{overline|PV}} is {{mvar|r}}, the distance of F from the vertex of the parabola is {{math|''r'' sin ''θ''}}. It is shown above that this distance equals the focal length of the parabola, which is the distance from the vertex to the focus. The focus and the point&nbsp;F are therefore equally distant from the vertex, along the same line, which implies that they are the same point. Therefore, ''the point&nbsp;F, defined above, is the focus of the parabola''.

This discussion started from the definition of a parabola as a conic section, but it has now led to a description as a graph of a quadratic function. This shows that these two descriptions are equivalent. They both define curves of exactly the same shape.

=== Alternative proof with Dandelin spheres ===
[[image:Dandelin-parabel.svg|thumb|Parabola (red): side projection view and top projection view of a cone with a Dandelin sphere]]
An alternative proof can be done using [[Dandelin spheres]]. It works without calculation and uses elementary geometric considerations only (see the derivation below).

The intersection of an upright cone by a plane <math>\pi</math>, whose inclination from vertical is the same as a [[generatrix]] (a.k.a. generator line, a line containing the apex and a point on the cone surface) <math>m_0</math> of the cone, is a parabola (red curve in the diagram).

This generatrix <math>m_0</math> is the only generatrix of the cone that is parallel to plane <math>\pi</math>. Otherwise, if there are two generatrices parallel to the intersecting plane, the intersection curve will be a [[hyperbola]] (or [[Degenerate conic|degenerate hyperbola]], if the two generatrices are in the intersecting plane). If there is no generatrix parallel to the intersecting plane, the intersection curve will be an [[ellipse]] or a [[circle]] (or [[Degenerate conic|a point]]).

Let plane <math>\sigma</math> be the plane that contains the vertical axis of the cone and line <math>m_0</math>. The inclination of plane <math>\pi</math> from vertical is the same as line <math>m_0</math> means that, viewing from the side (that is, the plane <math>\pi</math> is perpendicular to plane <math>\sigma</math>), <math>m_0 \parallel \pi</math>.

In order to prove the directrix property of a parabola (see {{slink||Definition as a locus of points}} above), one uses a [[Dandelin spheres|Dandelin sphere]] <math>d</math>, which is a sphere that touches the cone along a circle <math>c</math> and plane <math>\pi</math> at point <math>F</math>. The plane containing the circle <math>c</math> intersects with plane <math>\pi</math> at line <math>l</math>. There is a [[Reflection symmetry|mirror symmetry]] in the system consisting of plane <math>\pi</math>, Dandelin sphere <math>d</math> and the cone (the [[plane of symmetry]] is <math>\sigma</math>).

Since the plane containing the circle <math>c</math> is perpendicular to plane <math>\sigma</math>, and <math>\pi \perp \sigma</math>, their intersection line <math>l</math> must also be perpendicular to plane <math>\sigma</math>. Since line <math>m_0</math> is in plane <math>\sigma</math>, <math>l \perp m_0</math>.

It turns out that <math>F</math> is the ''focus'' of the parabola, and <math>l</math> is the ''directrix'' of the parabola.

# Let <math>P</math> be an arbitrary point of the intersection curve.
# The [[generatrix]] of the cone containing <math>P</math> intersects circle <math>c</math> at point <math>A</math>.
# The line segments <math>\overline{PF}</math> and <math>\overline{PA}</math> are tangential to the sphere <math>d</math>, and hence are of equal length.
# Generatrix <math>m_0</math> intersects the circle <math>c</math> at point <math>D</math>. The line segments <math>\overline{ZD}</math> and <math>\overline{ZA}</math> are tangential to the sphere <math>d</math>, and hence are of equal length.
# Let line <math>q</math> be the line parallel to <math>m_0</math> and passing through point <math>P</math>. Since <math>m_0 \parallel \pi</math>, and point <math>P</math> is in plane <math>\pi</math>, line <math>q</math> must be in plane <math>\pi</math>. Since <math>m_0 \perp l</math>, we know that <math>q \perp l</math> as well.
# Let point <math>B</math> be ''the foot of the perpendicular'' from point <math>P</math> to line <math>l</math>, that is, <math>\overline{PB}</math> is a segment of line <math>q</math>, and hence <math>\overline{PB} \parallel \overline{ZD}</math>.
# From [[intercept theorem]] and <math>\overline{ZD} = \overline {ZA}</math> we know that <math>\overline{PA} = \overline {PB}</math>. Since <math>\overline{PA} = \overline {PF}</math>, we know that <math>\overline{PF} = \overline {PB}</math>, which means that the distance from <math>P</math> to the focus <math>F</math> is equal to the distance from <math>P</math> to the directrix <math>l</math>.

== Proof of the reflective property ==
[[File:Parabel 2.svg|thumb|right|Reflective property of a parabola]]
The reflective property states that if a parabola can reflect light, then light that enters it travelling parallel to the axis of symmetry is reflected toward the focus. This is derived from [[geometrical optics]], based on the assumption that light travels in rays.

Consider the parabola {{math|1=''y'' = ''x''<sup>2</sup>}}. Since all parabolas are similar, this simple case represents all others.

=== Construction and definitions ===
The point E is an arbitrary point on the parabola. The focus is F, the vertex is A (the origin), and the line {{overline|FA}}  is the axis of symmetry. The line {{overline|EC}} is parallel to the axis of symmetry, intersects the {{mvar|x}} axis at D and intersects the directrix at C.  The point B is the midpoint of the line segment {{overline|FC}}.

=== Deductions ===
The vertex A is equidistant from the focus F and from the directrix. Since C is on the directrix, the {{mvar|y}} coordinates of F and C are equal in absolute value and opposite in sign. B is the midpoint of {{overline|FC}}. Its {{mvar|x}} coordinate is half that of D,  that is, {{math|''x''/2}}. The slope of the line {{overline|BE}} is the quotient of the lengths of {{overline|ED}} and {{overline|BD}}, which is {{math|1={{sfrac|''x''<sup>2</sup>|''x''/2}} = 2''x''}}. But {{math|2''x''}} is also the slope (first derivative) of the parabola at E. Therefore, the line {{overline|BE}} is the tangent to the parabola at E.

The distances {{overline|EF}} and {{overline|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 {{overline|FC}}, triangles △FEB and △CEB are congruent (three sides), which implies that the angles marked {{mvar|α}} are congruent. (The angle above E is vertically opposite angle ∠BEC.) This means that a ray of light that enters the parabola and arrives at E travelling parallel to the axis of symmetry will be reflected by the line {{overline|BE}} so it travels along the line {{overline|EF}}, as shown in red in the diagram (assuming that the lines can somehow reflect light). Since {{overline|BE}} is the tangent to the parabola at E, the same reflection will be done by an infinitesimal 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.

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.

=== Other consequences ===
There are other theorems that can be deduced simply from the above argument.

==== Tangent bisection property ====
The above proof and the accompanying diagram show that the tangent {{overline|BE}} bisects the angle ∠FEC. In other words, the tangent to the parabola at any point bisects the angle between the lines joining the point to the focus and perpendicularly to the directrix.

==== Intersection of a tangent and perpendicular from focus ====
[[File:Parabola-antipodera.gif|thumb|right|200px|Perpendicular from focus to tangent]]
Since triangles △FBE and △CBE are congruent, {{overline|FB}} is perpendicular to the tangent {{overline|BE}}. Since B is on the {{mvar|x}} axis, which is the tangent to the parabola at its vertex, it follows that the point of intersection between any tangent to a parabola and the perpendicular from the focus to that tangent lies on the line that is tangential to the parabola at its vertex. See animated diagram<ref name=ET>{{cite journal |last=Tsukerman |first=Emmanuel |title=On Polygons Admitting a Simson Line as Discrete Analogs of Parabolas |journal=Forum Geometricorum |volume=13 |date=2013 |pages=197–208 | url=http://forumgeom.fau.edu/FG2013volume13/FG201321.pdf}}</ref> and [[pedal curve]].

==== Reflection of light striking the convex side ====
If light travels along the line {{overline|CE}}, it moves parallel to the axis of symmetry and strikes the convex side of the parabola at E. It is clear from the above diagram that this light will be reflected directly away from the focus, along an extension of the segment {{overline|FE}}.

=== Alternative proofs ===
[[File:Parábola y tangente-prueba.svg|thumb|right|200px|Parabola and tangent]]
The above proofs of the reflective and tangent bisection properties use a line of calculus. Here a geometric proof is presented.

In this diagram, F is the focus of the parabola, and T and U lie on its directrix. P is an arbitrary point on the parabola. {{overline|PT}} is perpendicular to the directrix, and the line {{overline|MP}} bisects angle ∠FPT. Q is another point on the parabola, with {{overline|QU}} perpendicular to the directrix. We know that {{overline|FP}}&nbsp;=&nbsp;{{overline|PT}} and {{overline|FQ}}&nbsp;=&nbsp;{{overline|QU}}. Clearly, {{overline|QT}}&nbsp;>&nbsp;{{overline|QU}}, so {{overline|QT}}&nbsp;>&nbsp;{{overline|FQ}}. All points on the bisector {{overline|MP}} are equidistant from F and T, but Q is closer to F than to T. This means that Q is to the left of {{overline|MP}}, that is, on the same side of it as the focus. The same would be true if Q were located anywhere else on the parabola (except at the point P), so the entire parabola, except the point P, is on the focus side of {{overline|MP}}. Therefore, {{overline|MP}} is the tangent to the parabola at P. Since it bisects the angle ∠FPT, this proves the tangent bisection property.

The logic of the last paragraph can be applied to modify the above proof of the reflective property. It effectively proves the line {{overline|BE}} to be the tangent to the parabola at E if the angles {{mvar|α}} are equal. The reflective property follows as shown previously.

== Pin and string construction ==
[[File:Parabola-pin-string.svg|thumb|Parabola: pin string construction]]
The definition of a parabola by its focus and directrix can be used for drawing it with help of pins and strings:<ref>Frans van Schooten: ''Mathematische Oeffeningen'', Leyden, 1659, p. 334.</ref>
# Choose the ''focus'' <math>F</math> and the ''directrix'' <math>l</math> of the parabola.
# Take a triangle of a ''set square'' and prepare a ''string'' with length <math>|AB|</math> (see diagram).
# Pin one end of the string at point <math>A</math> of the triangle and the other one to the focus <math>F</math>.
# Position the triangle such that the second edge of the right angle is free to ''slide'' along the directrix.
# Take a ''pen'' and hold the string tight to the triangle.
# While moving the triangle along the directrix, the pen ''draws'' an arc of a parabola, because of <math>|PF| = |PB|</math> (see definition of a parabola).

== Properties related to Pascal's theorem ==
A parabola can be considered as the affine part of a non-degenerated projective conic with a point <math>Y_\infty</math> on the line of infinity <math>g_\infty</math>, which is the tangent at <math>Y_\infty</math>. The 5-, 4- and 3- point degenerations of [[Pascal's theorem]] are properties of a conic dealing with at least one tangent. If one considers this tangent as the line at infinity and its point of contact as the point at infinity of the ''y'' axis, one obtains three statements for a parabola.

The following properties of a parabola deal only with terms ''connect'', ''intersect'', ''parallel'', which are invariants of [[Similarity (geometry)|similarities]]. So, it is sufficient to prove any property for the ''unit parabola'' with equation <math>y = x^2</math>.

=== 4-points property ===
[[File:Parabel-pk-s.svg|thumb|4-points property of a parabola]]
Any parabola can be described in a suitable coordinate system by an equation <math>y = ax^2</math>.

{{block indent | em = 1.5 | text = Let <math>P_1 = (x_1, y_1),\ P_2 = (x_2, y_2),\ P_3 = (x_3, y_3),\ P_4 = (x_4, y_4)</math> be four points of the parabola <math>y = ax^2</math>, and <math>Q_2</math> the intersection of the secant line <math>P_1 P_4</math> with the line <math>x = x_2,</math> and let <math>Q_1</math> be the intersection of the secant line <math>P_2 P_3</math> with the line <math>x = x_1</math> (see picture). Then the secant line <math>P_3 P_4</math> is parallel to line <math>Q_1 Q_2</math>.

(The lines <math>x = x_1</math> and <math>x = x_2</math> are parallel to the axis of the parabola.)}}

''Proof:'' straightforward calculation for the unit parabola <math>y = x^2</math>.

''Application:'' The 4-points property of a parabola can be used for the construction of point <math>P_4</math>, while <math>P_1, P_2, P_3</math> and <math>Q_2</math> are given.

''Remark:'' the 4-points property of a parabola is an affine version of the 5-point degeneration of Pascal's theorem.

=== 3-points–1-tangent property ===
[[File:Parabel-tk-s.svg|thumb|3-points–1-tangent property]]

Let <math>P_0=(x_0,y_0),P_1=(x_1,y_1),P_2=(x_2,y_2)</math> be three points of the parabola with equation <math>y = ax^2</math> and <math>Q_2</math> the intersection of the secant line <math>P_0P_1</math> with the line <math>x = x_2</math> and <math>Q_1</math> the intersection of the secant line <math>P_0P_2</math> with the line <math>x = x_1</math> (see picture). Then the tangent at point <math>P_0</math> is parallel to the line <math>Q_1 Q_2</math>.
(The lines <math>x=x_1</math> and <math>x = x_2</math> are parallel to the axis of the parabola.)

''Proof:'' can be performed for the unit parabola <math>y=x^2</math>. A short calculation shows: line <math>Q_1Q_2</math> has slope <math>2x_0</math> which is the slope of the tangent at point <math>P_0</math>.

''Application:'' The 3-points-1-tangent-property of a parabola can be used for the construction of the tangent at point <math>P_0</math>, while <math>P_1,P_2,P_0</math> are given.

''Remark:'' The 3-points-1-tangent-property of a parabola is an affine version of the 4-point-degeneration of Pascal's theorem.

=== 2-points–2-tangents property ===
[[File:Parabel-tk-2-s.svg|thumb|2-points–2-tangents property]]

Let <math>P_1 = (x_1, y_1),\ P_2 = (x_2, y_2)</math> be two points of the parabola with equation <math>y = ax^2</math>, and <math>Q_2</math> the intersection of the tangent at point <math>P_1</math> with the line <math>x = x_2</math>, and <math>Q_1</math> the intersection of the tangent at point <math>P_2</math> with the line <math>x = x_1</math> (see picture). Then the secant <math>P_1 P_2</math> is parallel to the line <math>Q_1 Q_2</math>.
(The lines <math>x = x_1</math> and <math>x = x_2</math> are parallel to the axis of the parabola.)

''Proof:'' straight forward calculation for the unit parabola <math>y = x^2</math>.

''Application:'' The 2-points–2-tangents property can be used for the construction of the tangent of a parabola at point <math>P_2</math>, if <math>P_1, P_2</math> and the tangent at <math>P_1</math> are given.

''Remark 1:'' The 2-points–2-tangents property of a parabola is an affine version of the 3-point degeneration of Pascal's theorem.

''Remark 2:'' The 2-points–2-tangents property should not be confused with the following property of a parabola, which also deals with 2 points and 2 tangents, but is ''not'' related to Pascal's theorem.

=== Axis direction ===
[[File:Parabel-ak-s.svg|thumb|Construction of the axis direction]]

The statements above presume the knowledge of the axis direction of the parabola, in order to construct the points <math>Q_1, Q_2</math>. The following property determines the points <math>Q_1, Q_2</math> by two given points and their tangents only, and the result is that the line <math>Q_1 Q_2</math> is parallel to the axis of the parabola.

Let
# <math>P_1 = (x_1, y_1),\ P_2 = (x_2, y_2)</math> be two points of the parabola <math>y = ax^2</math>, and <math>t_1, t_2</math> be their tangents;
# <math>Q_1</math> be the intersection of the tangents <math>t_1, t_2</math>,
# <math>Q_2</math> be the intersection of the parallel line to <math>t_1</math> through <math>P_2</math> with the parallel line to <math>t_2</math> through <math>P_1</math> (see picture).
Then the line <math>Q_1 Q_2</math> is parallel to the axis of the parabola and has the equation <math>x = (x_1 + x_2) / 2.</math>

''Proof:'' can be done (like the properties above) for the unit parabola <math>y = x^2</math>.

''Application:'' This property can be used to determine the direction of the axis of a parabola, if two points and their tangents are given. An alternative way is to determine the midpoints of two parallel chords, see [[Parabola#Corollary concerning midpoints and endpoints of chords|section on parallel chords]].

''Remark:'' This property is an affine version of the theorem of two ''perspective triangles'' of a non-degenerate conic.<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski-planes], p. 36.</ref>

Related: Chord <math>P_1 P_2</math> has two additional properties:
# Its slope is the harmonic average of the slopes of tangents <math>t_1</math> and <math>t_2</math>.
# It is parallel to the tangent at the intersection of <math> Q_1 Q_2 </math> with the parabola.

== Steiner generation ==
=== Parabola ===
[[File:Parabel-steiner-s.svg|thumb|Steiner generation of a parabola]]

[[Jakob Steiner|Steiner]] established the following procedure for the construction of a non-degenerate conic (see [[Steiner conic]]):
{{block indent | em = 1.5 | text = Given two [[Pencil (mathematics)|pencils]] <math>B(U), B(V)</math> of lines at two points <math>U, V</math> (all lines containing <math>U</math> and <math>V</math> respectively) and a projective but not perspective mapping <math>\pi</math> of <math>B(U)</math> onto <math>B(V)</math>, the intersection points of corresponding lines form a non-degenerate projective conic section.}}

This procedure can be used for a simple construction of points on the parabola <math>y = ax^2</math>:
* Consider the pencil at the vertex <math>S(0, 0)</math> and the set of lines <math>\Pi_y</math> that are parallel to the ''y'' axis.
*# Let <math>P = (x_0, y_0)</math> be a point on the parabola, and <math>A = (0, y_0)</math>, <math>B = (x_0, 0)</math>.
*# The line segment <math>\overline{BP}</math> is divided into ''n'' equally spaced segments, and this division is projected (in the direction <math>BA</math>) onto the line segment <math>\overline{AP}</math> (see figure). This projection gives rise to a projective mapping <math>\pi</math> from pencil <math>S</math> onto the pencil <math>\Pi_y</math>.
*# The intersection of the line <math>SB_i</math> and the ''i''-th parallel to the ''y'' axis is a point on the parabola.

''Proof:'' straightforward calculation.

''Remark:'' Steiner's generation is also available for [[ellipse]]s and [[hyperbola]]s.

=== Dual parabola ===
[[File:Parabel-bezier.svg|400px|thumb|Dual parabola and Bézier curve of degree 2 (right: curve point and division points <math>Q_0, Q_1</math> for parameter <math>t = 0.4</math>)]]

A ''dual parabola'' consists of the set of tangents of an ordinary parabola.

The Steiner generation of a conic can be applied to the generation of a dual conic by changing the meanings of points and lines:
{{block indent | em = 1.5 | text = Let be given two point sets on two lines <math>u, v</math>, and a projective but not perspective mapping <math>\pi</math> between these point sets, then the connecting lines of corresponding points form a non degenerate dual conic.}}

In order to generate elements of a dual parabola, one starts with
# three points <math>P_0, P_1, P_2</math> not on a line,
# divides the line sections <math>\overline{P_0 P_1}</math> and <math>\overline{P_1 P_2}</math> each into <math>n</math> equally spaced line segments and adds numbers as shown in the picture.
# Then the lines <math>P_0 P_1, P_1 P_2, (1,1), (2,2), \dotsc</math> are tangents of a parabola, hence elements of a dual parabola.
# The parabola is a [[Bézier curve]] of degree 2 with the control points <math>P_0, P_1, P_2</math>.

The ''proof'' is a consequence of the ''[[de Casteljau algorithm]]'' for a Bézier curve of degree 2.

== Inscribed angles and the 3-point form ==
[[File:Parabel-pws-s.svg|thumb|Inscribed angles of a parabola]]
A parabola with equation <math>y = ax^2 + bx + c,\ a \ne 0</math> is uniquely determined by three points <math>(x_1, y_1), (x_2, y_2), (x_3, y_3)</math> with different ''x'' coordinates. The usual procedure to determine the coefficients <math>a, b, c</math> is to insert the point coordinates into the equation. The result is a linear system of three equations, which can be solved by [[Gaussian elimination]] or [[Cramer's rule]], for example. An alternative way uses the ''inscribed angle theorem'' for parabolas.

In the following, the angle of two lines will be measured by the difference of the slopes of the line with respect to the directrix of the parabola. That is, for a parabola of equation <math>y = ax^2 + bx + c,</math> the angle between two lines of equations <math>y = m_1 x + d_1,\ y = m_2x + d_2</math> is measured by <math>m_1 - m_2.</math>

Analogous to the [[inscribed angle theorem]] for circles, one has the ''inscribed angle theorem for parabolas'':<ref>E. Hartmann, [http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Lecture Note ''Planar Circle Geometries'', an Introduction to Möbius-, Laguerre- and Minkowski Planes], p. 72.</ref><ref>W. Benz, ''Vorlesungen über Geomerie der Algebren'', [[Springer Science+Business Media|Springer]] (1973).</ref>
{{block indent | em = 1.5 | text = Four points <math>P_i = (x_i, y_i),\ i = 1, \ldots, 4,</math> with different {{mvar|x}} coordinates (see picture) are on a parabola with equation <math>y = ax^2 + bx + c</math> if and only if the angles at <math>P_3</math> and <math>P_4</math> have the same measure, as defined above. That is,
<math display="block">\frac{y_4 - y_1}{x_4 - x_1} - \frac{y_4 - y_2}{x_4 - x_2} = \frac{y_3 - y_1}{x_3 - x_1} - \frac{y_3 - y_2}{x_3 - x_2}.</math>}}

(Proof: straightforward calculation: If the points are on a parabola, one may translate the coordinates for having the equation <math>y = ax^2</math>, then one has <math>\frac{y_i - y_j}{x_i - x_j} = x_i + x_j</math> if the points are on the parabola.)

A consequence is that the equation (in <math>{\color{green}x}, {\color{red}y}</math>) of the parabola determined by 3 points <math>P_i = (x_i, y_i),\ i = 1, 2, 3,</math> with different {{mvar|x}} coordinates is (if two {{mvar|x}} coordinates are equal, there is no parabola with directrix parallel to the {{mvar|x}} axis, which passes through the points)
<math display="block">\frac{{\color{red}y} - y_1}{{\color{green}x} - x_1} - \frac{{\color{red}y} - y_2}{{\color{green}x} - x_2} = \frac{y_3 - y_1}{x_3 - x_1} - \frac{y_3 - y_2}{x_3 - x_2}.</math>
Multiplying by the denominators that depend on <math>{\color{green}x},</math> one obtains the more standard form
<math display="block">(x_1 - x_2){\color{red}y} = ({\color{green}x} - x_1)({\color{green}x} - x_2) \left(\frac{y_3 - y_1}{x_3 - x_1} - \frac{y_3 - y_2}{x_3 - x_2}\right) + (y_1 - y_2){\color{green}x} + x_1 y_2 - x_2 y_1.</math>

== Pole–polar relation ==
[[File:Parabel-pol-s.svg|thumb|Parabola: pole–polar relation]]
In a suitable coordinate system any parabola can be described by an equation <math>y = ax^2</math>. The equation of the tangent at a point <math>P_0 = (x_0, y_0),\ y_0 = ax^2_0</math> is
<math display="block">y = 2ax_0(x - x_0) + y_0 = 2ax_0x - ax^2_0 = 2ax_0x - y_0.</math>
One obtains the function
<math display="block">(x_0, y_0) \to y = 2ax_0x - y_0</math>
on the set of points of the parabola onto the set of tangents.

Obviously, this function can be extended onto the set of all points of <math>\R^2</math> to a bijection between the points of <math>\R^2</math> and the lines with equations <math>y = mx + d, \ m, d \in \R</math>. The inverse mapping is 
<math display="block">\text{line } y = mx + d ~~ \rightarrow ~~ \text{point } (\tfrac{m}{2a}, -d).</math>
This relation is called the ''[[Pole and polar|pole–polar relation]] of the parabola'', where the point is the ''pole'', and the corresponding line its ''polar''.

By calculation, one checks the following properties of the pole–polar relation of the parabola:
* For a point (pole) ''on'' the parabola, the polar is the tangent at this point (see picture: <math>P_1,\ p_1</math>).
* For a pole <math>P</math> ''outside'' the parabola the intersection points of its polar with the parabola are the touching points of the two tangents passing <math>P</math> (see picture: <math>P_2,\ p_2</math>).
* For a point ''within'' the parabola the polar has no point with the parabola in common (see picture: <math>P_3,\ p_3</math> and <math>P_4,\ p_4</math>).
* The intersection point of two polar lines (see picture: <math>p_3, p_4</math>) is the pole of the connecting line of their poles (see picture: <math>P_3, P_4</math>).
* Focus and directrix of the parabola are a pole–polar pair.

''Remark:'' Pole–polar relations also exist for ellipses and hyperbolas.

== Tangent properties ==
=== Two tangent properties related to the latus rectum ===
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 {{mvar|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 {{math|2''f''}}, and (2) a tangent to the parabola at point T intersects the line of symmetry at a 45° angle.<ref>{{cite book|last=Downs |first=J. W. |title=Practical Conic Sections |publisher=Dover Publishing |date=2003}}{{ISBN missing}}</ref>{{rp|p=26}}
[[File:Parabel-orthop.svg|thumb|right|Perpendicular tangents intersect on the directrix]]

=== Orthoptic property ===
{{main|Orthoptic (geometry)}}
If two tangents to a parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents that intersect on the directrix are perpendicular. In other words, at any point on the directrix the whole parabola subtends a right angle.

=== Lambert's theorem ===
Let three tangents to a parabola form a triangle. Then '''[[Johann Heinrich Lambert|Lambert's]] theorem''' states that the focus of the parabola lies on the [[circumcircle]] of the triangle.<ref>{{cite journal|last=Sondow |first=Jonathan |arxiv=1210.2279 |title=The parbelos, a parabolic analog of the arbelos |journal=[[American Mathematical Monthly]] |volume=120 |date=2013 |issue=10 |pages=929–935 |doi=10.4169/amer.math.monthly.120.10.929|s2cid=33402874 }}</ref><ref name=ET/>{{rp|at=Corollary 20}}

Tsukerman's converse to Lambert's theorem states that, given three lines that bound a triangle, if two of the lines are tangent to a parabola whose focus lies on the circumcircle of the triangle, then the third line is also tangent to the parabola.<ref name=ET2>{{cite journal|last=Tsukerman |first=Emmanuel |title=Solution of Sondow's problem: a synthetic proof of the tangency property of the parbelos |journal=[[American Mathematical Monthly]] |volume=121 |date=2014 |issue=5 |pages=438–443 |arxiv=1210.5580 |doi=10.4169/amer.math.monthly.121.05.438|s2cid=21141837 }}</ref>

== Facts related to chords and arcs {{anchor|Chords|Arcs}} ==

=== Focal length calculated from parameters of a chord ===
Suppose a [[Chord (geometry)|chord]] crosses a parabola perpendicular to its axis of symmetry. Let the length of the chord between the points where it intersects the parabola be {{mvar|c}} and the distance from the vertex of the parabola to the chord, measured along the axis of symmetry, be {{mvar|d}}. The focal length, {{mvar|f}}, of the parabola is given by
<math display="block">f = \frac{c^2}{16d}.</math>

{{math proof | proof = Suppose a system of Cartesian coordinates is used such that the vertex of the parabola is at the origin, and the axis of symmetry is the {{mvar|y}} axis. The parabola opens upward. It is shown elsewhere in this article that the equation of the parabola is {{math|1=4''fy'' = ''x''<sup>2</sup>}}, where {{mvar|f}} is the focal length. At the positive {{mvar|x}} end of the chord, {{math|1=''x'' = {{sfrac|''c''|2}}}} and {{math|1=''y'' = ''d''}}. Since this point is on the parabola, these coordinates must satisfy the equation above. Therefore, by substitution, <math>4fd = \left(\tfrac{c}{2}\right)^2</math>. From this, <math>f = \tfrac{c^2}{16d}</math>.}}

=== Area enclosed between a parabola and a chord ===
[[File:Area between a parabola and a chord.svg|thumb|Parabola (magenta) and line (lower light blue) including a chord (blue). The area enclosed between them is in pink. The chord itself ends at the points where the line intersects the parabola.]]

The area enclosed between a parabola and a chord (see diagram) is two-thirds of the area of a parallelogram that surrounds it. One side of the parallelogram is the chord, and the opposite side is a tangent to the parabola.<ref>{{cite web |url=http://www.mathwarehouse.com/geometry/parabola/area-of-parabola.php |title=Sovrn Container |publisher=Mathwarehouse.com |access-date=2016-09-30}}</ref><ref>{{cite web |url=http://mysite.du.edu/~jcalvert/math/parabola.htm |title=Parabola |publisher=Mysite.du.edu |access-date=2016-09-30}}</ref> The slope of the other parallel sides is irrelevant to the area. Often, as here, they are drawn parallel with the parabola's axis of symmetry, but this is arbitrary.

A theorem equivalent to this one, but different in details, was derived by [[Archimedes]] in the 3rd century BCE. He used the areas of triangles, rather than that of the parallelogram.{{efn|Archimedes proved that the area of the enclosed parabolic segment was 4/3 as large as that of a triangle that he inscribed within the enclosed segment. It can easily be shown that the parallelogram has twice the area of the triangle, so Archimedes' proof also proves the theorem with the parallelogram.}} See [[The Quadrature of the Parabola]].

If the chord has length {{mvar|b}} and is perpendicular to the parabola's axis of symmetry, and if the perpendicular distance from the parabola's vertex to the chord is {{mvar|h}}, the parallelogram is a rectangle, with sides of {{mvar|b}} and {{mvar|h}}. The area {{mvar|A}} of the parabolic segment enclosed by the parabola and the chord is therefore
<math display="block">A = \frac{2}{3} bh.</math>

This formula can be compared with the area of a triangle: {{math|{{sfrac|1|2}}''bh''}}.

In general, the enclosed area can be calculated as follows. First, locate the point on the parabola where its slope equals that of the chord. This can be done with calculus, or by using a line that is parallel to the axis of symmetry of the parabola and passes through the midpoint of the chord. The required point is where this line intersects the parabola.{{efn|This method can be easily proved correct by calculus. It was also known and used by Archimedes, although he lived nearly 2000 years before calculus was invented.}} Then, using the formula given in [[Distance from a point to a line]], calculate the perpendicular distance from this point to the chord. Multiply this by the length of the chord to get the area of the parallelogram, then by 2/3 to get the required enclosed area.

=== Corollary concerning midpoints and endpoints of chords ===
[[File:Parabel-psehnen-s.svg|thumb|Midpoints of parallel chords]]

A corollary of the above discussion is that if a parabola has several parallel chords, their midpoints all lie on a line parallel to the axis of symmetry. If tangents to the parabola are drawn through the endpoints of any of these chords, the two tangents intersect on this same line parallel to the axis of symmetry (see [[parabola#Axis-direction of a parabola|Axis-direction of a parabola]]).{{efn|A proof of this sentence can be inferred from the proof of the [[#Orthoptic property|orthoptic property]], above. It is shown there that the tangents to the parabola {{math|1=''y'' = ''x''<sup>2</sup>}} at {{math|(''p'', ''p''<sup>2</sup>)}} and {{math|(''q'', ''q''<sup>2</sup>)}} intersect at a point whose {{mvar|x}} coordinate is the mean of {{mvar|p}} and {{mvar|q}}. Thus if there is a chord between these two points, the intersection point of the tangents has the same {{mvar|x}} coordinate as the midpoint of the chord.}}

=== Arc length ===
If a point X is located on a parabola with focal length {{mvar|f}}, and if {{mvar|p}} is the [[Distance from a point to a line|perpendicular distance]] from X to the axis of symmetry of the parabola, then the lengths of [[Arc (geometry)|arcs]] of the parabola that terminate at X can be calculated from {{mvar|f}} and {{mvar|p}} as follows, assuming they are all expressed in the same units.{{efn|In this calculation, the [[square root]] {{mvar|q}} must be positive. The quantity {{math|ln ''a''}} is the [[natural logarithm]] of&nbsp;{{mvar|a}}.}}
<math display="block">\begin{align}
 h &= \frac{p}{2}, \\
 q &= \sqrt{f^2 + h^2}, \\
 s &= \frac{hq}{f} + f \ln\frac{h + q}{f}.
\end{align}</math>

This quantity {{mvar|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 {{math|2''s''}}.

The perpendicular distance {{mvar|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 {{mvar|p}} reverses the signs of {{mvar|h}} and {{mvar|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 {{mvar|s}}''. The calculation can be simplified by using the properties of logarithms:
<math display="block">s_1 - s_2 = \frac{h_1 q_1 - h_2 q_2}{f} + f \ln\frac{h_1 + q_1}{h_2 + q_2}.</math>

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.

== A geometrical construction to find a sector area ==
[[File: Sector Area Prop 30.png|400px|Sector area proposition 30]]

S is the focus, and V is the principal vertex of the parabola VG. Draw VX perpendicular to SV.

Take any point B on VG and drop a perpendicular BQ from B to VX. Draw perpendicular ST intersecting BQ, extended if necessary, at T. At B draw the perpendicular BJ, intersecting VX at J.

For the parabola, the segment VBV, the area enclosed by the chord VB and the arc VB, is equal to ∆VBQ / 3, also <math>BQ = \frac{VQ^2}{4SV}</math>.

The area of the parabolic sector <math>SVB = \triangle SVB + \frac{\triangle VBQ}{3} = \frac{SV \cdot VQ}{2} + \frac{VQ \cdot BQ}{6}</math>.

Since triangles TSB and QBJ are similar,
<math display="block">VJ = VQ - JQ = VQ - \frac{BQ \cdot TB}{ST} = VQ - \frac{BQ \cdot (SV - BQ)}{VQ} = \frac{3VQ}{4} + \frac{VQ \cdot BQ}{4SV}.</math>

Therefore, the area of the parabolic sector <math>SVB = \frac{2SV \cdot VJ}{3}</math> and can be found from the length of VJ, as found above.

A circle through S, V and B also passes through J.

Conversely, if a point, B on the parabola VG is to be found so that the area of the sector SVB is equal to a specified value, determine the point J on VX and construct a circle through S, V and J. Since SJ is the diameter, the center of the circle is at its midpoint, and it lies on the perpendicular bisector of SV, a distance of one half VJ from SV. The required point B is where this circle intersects the parabola.

If a body traces the path of the parabola due to an inverse square force directed towards S, the area SVB increases at a constant rate as point B moves forward. It follows that J moves at constant speed along VX as B moves along the parabola.

If the speed of the body at the vertex where it is moving perpendicularly to SV is ''v'', then the speed of J is equal to {{math|3''v''/4}}.

The construction can be extended simply to include the case where neither radius coincides with the axis SV as follows. Let A be a fixed point on VG between V and B, and point H be the intersection on VX with the perpendicular to SA at A. From the above, the area of the parabolic sector <math>SAB = \frac{2SV \cdot (VJ - VH)}{3} = \frac{2SV \cdot HJ}{3}</math>.

Conversely, if it is required to find the point B for a particular area SAB, find point J from HJ and point B as before. By Book 1, Proposition 16, Corollary 6 of Newton's [[Philosophiæ Naturalis Principia Mathematica|''Principia'']], the speed of a body moving along a parabola with a force directed towards the focus is inversely proportional to the square root of the radius. If the speed at A is ''v'', then at the vertex V it is <math>\sqrt{\frac{SA}{SV}} v</math>, and point J moves at a constant speed of <math>\frac{3v}{4} \sqrt{\frac{SA}{SV}}</math>.

The above construction was devised by Isaac Newton and can be found in Book 1 of [[Philosophiæ Naturalis Principia Mathematica]] as Proposition 30.

== Focal length and radius of curvature at the vertex ==
The focal length of a parabola is half of its [[Radius of curvature (mathematics)|radius of curvature]] at its vertex.

;Proof:
<gallery widths="300px" heights="200px">
File:Huygens + Snell + van Ceulen - regular polygon doubling.svg|Image is inverted. AB is {{mvar|x}} axis. C is origin. O is center. A is {{math|(''x'', ''y'')}}. OA = OC = {{mvar|R}}. PA = {{mvar|x}}. CP = {{mvar|y}}. OP = {{math|(''R'' − ''y'')}}. Other points and lines are irrelevant for this purpose.
File:Parabola circle.svg|The radius of curvature at the vertex is twice the focal length. The measurements shown on the above diagram are in units of the latus rectum, which is four times the focal length.
File:Concave mirror.svg
</gallery>
Consider a point {{math|(''x'', ''y'')}} on a circle of radius {{mvar|R}} and with center at the point {{math|(0, ''R'')}}. The circle passes through the origin. If the point is near the origin, the [[Pythagorean theorem]] shows that
<math display="block">\begin{align}
 x^2 + (R - y)^2 &= R^2, \\[1ex]
 x^2 + R^2 - 2Ry + y^2 &= R^2, \\[1ex]
 x^2 + y^2 &= 2Ry.
\end{align}</math>

But if {{math|(''x'', ''y'')}} is extremely close to the origin, since the {{mvar|x}} axis is a tangent to the circle, {{mvar|y}} is very small compared with {{mvar|x}}, so {{math|''y''<sup>2</sup>}} is negligible compared with the other terms. Therefore, extremely close to the origin
{{NumBlk||<math display="block">x^2 = 2Ry.</math>|{{EquationRef|1}}}}

Compare this with the parabola
{{NumBlk||<math display="block">x^2 = 4fy,</math>|{{EquationRef|2}}}}
which has its vertex at the origin, opens upward, and has focal length {{mvar|f}} (see preceding sections of this article).

Equations {{EquationNote|(1)}} and {{EquationNote|2|(2)}} are equivalent if {{math|1=''R'' = 2''f''}}. Therefore, this is the condition for the circle and parabola to coincide at and extremely close to the origin. The radius of curvature at the origin, which is the vertex of the parabola, is twice the focal length.

; Corollary:
A concave mirror that is a small segment of a sphere behaves approximately like a parabolic mirror, focusing parallel light to a point midway between the centre and the surface of the sphere.

== As the affine image of the unit parabola ==
[[File:Parabel-aff-s.svg|thumb|Parabola as an affine image of the unit parabola]]
Another definition of a parabola uses [[affine transformation]]s:
{{block indent | em = 1.5 | text = Any ''parabola'' is the affine image of the unit parabola with equation <math>y = x^2</math>.}}

=== Parametric representation ===
An affine transformation of the Euclidean plane has the form <math>\vec x \to \vec f_0 + A \vec x</math>, where <math>A</math> is a regular matrix ([[determinant]] is not 0), and <math>\vec f_0</math> is an arbitrary vector. If <math>\vec f_1, \vec f_2</math> are the column vectors of the matrix <math>A</math>, the unit parabola <math>(t, t^2),\ t \in \R</math> is mapped onto the parabola
<math display="block">\vec x = \vec p(t) = \vec f_0 +\vec f_1 t +\vec f_2 t^2,</math>
where
* <math>\vec f_0</math> is a ''point'' of the parabola,
* <math>\vec f_1</math> is a ''tangent vector'' at point <math>\vec f_0</math>,
* <math>\vec f_2</math> is ''parallel to the axis'' of the parabola (axis of symmetry through the vertex).

=== Vertex ===
In general, the two vectors <math>\vec f_1, \vec f_2</math> are not perpendicular, and <math>\vec f_0</math> is ''not'' the vertex, unless the affine transformation is a [[Similarity (geometry)|similarity]].

The tangent vector at the point <math>\vec p(t)</math> is <math>\vec p'(t) = \vec f_1 + 2t \vec f_2</math>. At the vertex the tangent vector is orthogonal to <math>\vec f_2</math>. Hence the parameter <math>t_0</math> of the vertex is the solution of the equation
<math display="block">\vec p'(t) \cdot \vec f_2 = \vec f_1 \cdot \vec f_2 + 2t f_2^2 = 0,</math>
which is
<math display="block">t_0 = -\frac{\vec f_1 \cdot \vec f_2}{2 f_2^2},</math>
and the ''vertex'' is
<math display="block">\vec p(t_0) = \vec f_0 - \frac{\vec f_1 \cdot \vec f_2}{2 f_2^2} \vec f_1 + \frac{(\vec f_1 \cdot \vec f_2)^2}{4(f_2^2)^2} \vec f_2.</math>

=== Focal length and focus ===
The ''focal length'' can be determined by a suitable parameter transformation (which does not change the geometric shape of the parabola). The focal length is
<math display="block">f = \frac{f_1^2 \, f_2^2 - (\vec f_1 \cdot \vec f_2)^2}{4|f_2|^3}.</math>
Hence the ''focus'' of the parabola is
<math display="block">F:\ \vec f_0 - \frac{\vec f_1 \cdot \vec f_2}{2 f_2^2} \vec f_1 + \frac{f_1^2 \, f_2^2}{4(f_2^2)^2} \vec f_2.</math>

=== Implicit representation ===
Solving the parametric representation for <math>\; t, t^2\;</math> by [[Cramer's rule]] and using <math>\;t\cdot t-t^2 =0\; </math>, one gets the implicit representation
<math display="block">\det(\vec x\!-\!\vec f\!_0,\vec f\!_2)^2-\det(\vec f\!_1,\vec x\!-\!\vec f\!_0)\det(\vec f\!_1,\vec f\!_2) = 0.</math>

=== Parabola in space ===
The definition of a parabola in this section gives a parametric representation of an arbitrary parabola, even in space, if one allows <math>\vec f\!_0, \vec f\!_1, \vec f\!_2</math> to be vectors in space.

== As quadratic Bézier curve ==
[[File: Bezier quadratic anim.gif|thumb|right|Quadratic Bézier curve and its control points]]

A [[Bézier curve#Quadratic Bézier curves|quadratic Bézier curve]] is a curve <math>\vec c(t)</math> defined by three points <math>P_0: \vec p_0</math>, <math>P_1: \vec p_1</math> and <math>P_2: \vec p_2</math>, called its ''control points'':
<math display="block">\begin{align}
 \vec c(t) &= \sum_{i=0}^2 \binom{2}{i} t^i (1 - t)^{2-i} \vec p_i \\[1ex]
           &= (1 - t)^2 \vec p_0 + 2t(1 - t) \vec p_1 + t^2 \vec p_2 \\[2ex]
           &= \left(\vec p_0 - 2\vec p_1 + \vec p_2\right) t^2 + \left(-2\vec p_0 + 2\vec p_1\right) t + \vec p_0, \quad t \in [0, 1].
\end{align}</math>

This curve is an arc of a parabola (see {{slink||As the affine image of the unit parabola}}).

== Numerical integration ==
[[File:Simpsons method illustration.svg|thumb|Simpson's rule: the graph of a function is replaced by an arc of a parabola]]

In one method of [[numerical integration]] one replaces the graph of a function by arcs of parabolas and integrates the parabola arcs. A parabola is determined by three points. The formula for one arc is
<math display="block">\int_a^b f(x)\,dx \approx \frac{b - a}{6} \cdot \left( f(a) + 4f\left( \frac{a + b}{2} \right) + f(b) \right).</math>

The method is called [[Simpson's rule]].

== As plane section of quadric ==
The following [[quadric]]s contain parabolas as plane sections:
* elliptical [[cone]],
* parabolic [[cylinder]],
* elliptical [[paraboloid]],
* hyperbolic paraboloid,
* [[hyperboloid]] of one sheet,
* hyperboloid of two sheets.

<gallery>
File:Quadric Cone.jpg|Elliptic cone
File:Parabolic Cylinder Quadric.png|Parabolic cylinder
File:Paraboloid.png|Elliptic paraboloid
File:Hyperbol Paraboloid.pov.png|Hyperbolic paraboloid
File:Hyperboloid1.png|Hyperboloid of one sheet
File:Hyperboloid2.png|Hyperboloid of two sheets
</gallery>

== As trisectrix ==
[[File:Angle trisection with parabola2.svg|thumb|upright=1.25|Angle trisection with a parabola]]
A parabola can be used as a [[trisectrix]], that is it allows the [[Angle trisection|exact trisection of an arbitrary angle]] with straightedge and compass. This is not in contradiction to the impossibility of an angle trisection with [[compass-and-straightedge construction]]s alone, as the use of parabolas is not allowed in the classic rules for compass-and-straightedge constructions.

To trisect <math>\angle AOB</math>, place its leg <math>OB</math> on the ''x'' axis such that the vertex <math>O</math> is in the coordinate system's origin. The coordinate system also contains the parabola <math>y = 2x^2</math>. The unit circle with radius&nbsp;1 around the origin intersects the angle's other leg <math>OA</math>, and from this point of intersection draw the perpendicular onto the ''y'' axis. The parallel to ''y'' axis through the midpoint of that perpendicular and the tangent on the unit circle in <math>(0, 1)</math> intersect in <math>C</math>. The circle around <math>C</math> with radius <math>OC</math> intersects the parabola at <math>P_1</math>. The perpendicular from <math>P_1</math> onto the ''x'' axis intersects the unit circle at <math>P_2</math>, and <math>\angle P_2OB</math> is exactly one third of <math>\angle AOB</math>.

The correctness of this construction can be seen by showing that the ''x'' coordinate of <math>P_1</math> is <math>\cos(\alpha)</math>. Solving the equation system given by the circle around <math>C</math> and the parabola leads to the cubic equation <math>4x^3 - 3x - \cos(3\alpha) = 0</math>. The [[List of trigonometric identities#Triple-angle formulae|triple-angle formula]] <math>\cos(3\alpha) = 4 \cos(\alpha)^3 - 3 \cos(\alpha)</math> then shows that <math>\cos(\alpha)</math> is indeed a solution of that cubic equation.

This trisection goes back to [[René Descartes]], who described it in his book {{lang|fr|La Géométrie}} (1637).<ref>{{cite journal |first=Robert C. |last=Yates |title=The Trisection Problem |journal=National Mathematics Magazine |volume=15 |issue=4 |year=1941 |pages=191–202 |doi=10.2307/3028133 |jstor=3028133 }}</ref>

== Generalizations ==
If one replaces the real numbers by an arbitrary [[Field (mathematics)|field]], many geometric properties of the parabola <math> y = x^2</math> are still valid:
# A line intersects in at most two points. 
# At any point <math>(x_0, x_0^2)</math> the line <math>y = 2 x_0 x - x_0^2</math> is the tangent.
Essentially new phenomena arise, if the field has characteristic 2 (that is, <math>1 + 1 = 0</math>): the tangents are all parallel.

In [[algebraic geometry]], the parabola is generalized by the [[rational normal curve]]s, which have coordinates {{math|(''x'', ''x''<sup>2</sup>, ''x''<sup>3</sup>, ..., ''x<sup>n</sup>'')}}; the standard parabola is the case {{math|1=''n'' = 2}}, and the case {{math|1=''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 form]]s, the parabola is the graph of the quadratic form {{math|''x''<sup>2</sup>}} (or other scalings), while the [[elliptic paraboloid]] is the graph of the [[Definite bilinear form|positive-definite]] quadratic form {{math|''x''<sup>2</sup> + ''y''<sup>2</sup>}} (or scalings), and the [[hyperbolic paraboloid]] is the graph of the [[indefinite quadratic form]] {{math|''x''<sup>2</sup> − ''y''<sup>2</sup>}}. Generalizations to more variables yield further such objects.

The curves {{math|1=''y'' = ''x''{{isup|''p''}}}} for other values of {{mvar|p}} are traditionally referred to as the '''higher parabolas''' and were originally treated implicitly, in the form {{math|1=''x''{{isup|''p''}} = ''ky''{{isup|''q''}}}} for {{mvar|p}} and {{mvar|q}} both positive integers, in which form they are seen to be algebraic curves. These correspond to the explicit formula {{math|1=''y'' = ''x''{{isup|''p''/''q''}}}} for a positive fractional power of {{mvar|x}}. Negative fractional powers correspond to the implicit equation {{math|1=''x''{{isup|''p''}} ''y''{{isup|''q''}} = ''k''}} and are traditionally referred to as '''higher hyperbolas'''. Analytically, {{mvar|x}} can also be raised to an irrational power (for positive values of {{mvar|x}}); the analytic properties are analogous to when {{mvar|x}} is raised to rational powers, but the resulting curve is no longer algebraic and cannot be analyzed by algebraic geometry.

== In the physical world ==
In nature, approximations of parabolas and paraboloids 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 ball flying through the air, neglecting air [[friction]]).

The [[Projectile motion|parabolic trajectory of projectiles]] was discovered experimentally in the early 17th century by [[Galileo]], who performed experiments with balls rolling on inclined planes. He also later proved this [[mathematical]]ly in his book ''Dialogue Concerning Two New Sciences''.<ref>Dialogue Concerning Two New Sciences (1638) (The Motion of Projectiles: Theorem 1).</ref>{{efn|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.}} 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 moves along 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 [[hypothetical]] situation in which parabolas might arise, according to the theories of physics described in the 17th and 18th centuries by [[Sir Isaac Newton]], is in [[two-body orbit]]s, for example, the path of a small planetoid or other object under the influence of the gravitation of the [[Sun]]. [[Parabolic orbit]]s do not occur in nature; simple orbits most commonly resemble [[hyperbola]]s or [[ellipse]]s. The parabolic orbit is the [[degeneracy (math)|degenerate]] intermediate case between those two types of ideal orbit. An object following a parabolic orbit would travel at the exact [[escape velocity]] of the object it orbits; objects in [[elliptical orbit|elliptical]] or [[hyperbolic orbit|hyperbolic]] orbits travel at less or greater than escape velocity, respectively. Long-period [[comet]]s travel close to the Sun's escape velocity while they are moving through the inner Solar system, so their paths are nearly parabolic.

Approximations of parabolas 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 due to the weight of the load (i.e. the road) being much larger than the cables themselves, and in calculations the second-degree polynomial formula of a parabola is used.<ref name="Troyano">{{cite book
|title=Bridge engineering: a global perspective
|first1=Leonardo Fernández
|last1=Troyano
|publisher=Thomas Telford
|year=2003
|isbn=0-7277-3215-3
|page=536
|url=https://books.google.com/books?id=0u5G8E3uPUAC&pg=PA536}}
</ref><ref>{{cite book
|title=A memoir of suspension bridges
|first1=Charles Stewart
|last1=Drewry
|publisher=Oxford University
|year=1832
|page=[https://archive.org/details/amemoironsuspen00drewgoog/page/n183 159]
|url=https://archive.org/details/amemoironsuspen00drewgoog}}
</ref> Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise catenary-shaped cable is deformed toward a parabola (see {{slink|Catenary#Suspension bridge curve}}). 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, for example, bending. 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 [[Focus (optics)|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 dubious legend,<ref>{{cite journal
|last      = Middleton
|first     = W. E. Knowles
|date      = December 1961
|title     = Archimedes, Kircher, Buffon, and the Burning-Mirrors
|journal   = Isis
|volume    = 52
|issue     = 4
|publisher = Published by: The University of Chicago Press on behalf of The History of Science Society
|pages     = 533–543
|doi = 10.1086/349498
|jstor = 228646|s2cid = 145385010
}}</ref> constructed parabolic mirrors to defend [[Syracuse, Italy|Syracuse]] against the [[Roman Empire|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 receiving and transmitting antennas.

In [[parabolic microphone]]s, a parabolic reflector is used to focus sound onto a microphone, giving it highly directional performance.

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 [[Weightlessness|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.

=== Gallery ===

<gallery mode="packed" heights="200px" style="text-align:left">
File:Bouncing ball strobe edit.jpg|A [[bouncing ball]] captured with a stroboscopic flash at 25 images per second. 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:Comet Kohoutek orbit p391.svg|The path (in red) of [[Comet Kohoutek]] as it passed through the inner Solar system, showing its nearly parabolic shape. The blue orbit is the Earth's.
File:Laxmanjhula.jpg|The supporting cables of [[suspension bridge]]s follow a curve that is intermediate between a parabola and a [[catenary]].
File:Rainbow Bridge(2).jpg|The [[Rainbow Bridge (Niagara Falls)|Rainbow Bridge]] across the [[Niagara River]], connecting [[Canada]] (left) to the [[United States]] (right). The parabolic arch is in compression and carries the weight of the road.
File:Celler de Sant Cugat lateral.JPG|Parabolic arches used in architecture
File:Parabola shape in rotating layers of fluid.jpg|Parabolic shape formed by a liquid surface under rotation. Two liquids of different densities completely fill a narrow space between two sheets of transparent plastic. 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]])
File:ALSOL.jpg|[[Solar cooker]] with [[parabolic reflector]]
File:Antenna 03.JPG|[[Parabolic antenna]]
File:ParabolicMicrophone.jpg|[[Parabolic microphone]] with optically transparent plastic reflector used at an American college football game.
File:Solar Array.jpg|Array of [[parabolic trough]]s to collect [[solar energy]]
File:Ed d21m.jpg|[[Thomas Edison|Edison]]'s searchlight, mounted on a cart. The light had a parabolic reflector.
File:Physicist Stephen Hawking in Zero Gravity NASA.jpg|Physicist [[Stephen Hawking]] in an aircraft flying a parabolic trajectory to simulate zero gravity
</gallery>

{{clear}}

== See also ==
* {{slink|Confocal conic sections#Confocal parabolas}}
* [[Degenerate conic]]
* {{slink|Dome#Paraboloid dome}}
* [[Parabolic partial differential equation]]
* [[Quadratic equation]]
* [[Quadratic function]]
* [[Universal parabolic constant]]

== Footnotes ==
{{notelist}}

== References ==
{{reflist}}

== Further reading ==
* {{cite book|last=Lockwood |first=E. H. |date=1961 |title=A Book of Curves |publisher=Cambridge University Press}}

== External links ==
{{Wiktionary|parabola}}
{{Commons category|Parabolas}}
{{EB1911 poster|Parabola}}
* {{springer|title=Parabola|id=p/p071150}}
* {{MathWorld|title=Parabola|urlname=Parabola}}
* [http://www.mathwarehouse.com/quadratic/parabola/interactive-parabola.php Interactive parabola-drag focus, see axis of symmetry, directrix, standard and vertex forms]
* [http://www.cut-the-knot.org/Curriculum/Geometry/ArchimedesTriangle.shtml Archimedes Triangle and Squaring of Parabola] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaLambert.shtml Two Tangents to Parabola] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaEnvelope.shtml Parabola As Envelope of Straight Lines] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaMirror.shtml Parabolic Mirror] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/Curriculum/Geometry/ThreeParabolaTangents.shtml Three Parabola Tangents] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaFocal.shtml Focal Properties of Parabola] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaMesh.shtml Parabola As Envelope II] at [[cut-the-knot]]
* [http://dynamicmathematicslearning.com/similarparabola.html The similarity of parabola] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches], interactive dynamic geometry sketch.
* [http://objects.library.uu.nl/reader/index.php?obj=1874-20606&lan=en#page//11/65/13/116513161330830239976915216825059484204.jpg/mode/2up Frans van Schooten: ''Mathematische Oeffeningen'', 1659]

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[[Category:Parabolas| ]]
[[Category:Conic sections]]
[[Category:Algebraic curves]]