Parabola

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.

thumb|The parabola is a member of the family of [[conic sections.]]

In mathematics, a parabola ( ) is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the 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 parallel to another plane that is tangential to the conical surface.

The graph of a quadratic function <math>y=ax^2+bx+ c</math> (with <math>a\neq 0 </math>) is a parabola with its axis of symmetry coincident with the -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" 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 "latus rectum" is the 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 similar.

Parabolas have the property that, if they are made of material that 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 waves. 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 headlight reflectors and the design of ballistic missiles. It is frequently used in physics, engineering, and many other areas.

History

thumb|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, 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. The focus–directrix property of the parabola and other conic sections was mentioned in the works of Pappus.

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

The idea that a parabolic reflector could produce an image was already well known before the invention of the reflecting telescope. Designs were proposed in the early to mid-17th century by many mathematicians, including René Descartes, Marin Mersenne, and James Gregory. When Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers.

Definition as a locus of points

A parabola can be defined geometrically as a set of points (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, the directrix. That is, if is the focus and is the directrix, the parabola is the set of all points such that

<math display="block">d(P,F) = d(P,l),</math> where 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 unique 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

thumb|Parabola with axis parallel to -axis; is the semi-latus rectum

In Cartesian coordinates, if the vertex is the origin and the directrix has the equation <math>y = -f</math>, then, by examining the case <math>x = 0</math>, the focus is on the positive -axis, with <math>F = (0, f)</math>, where is the focal length.

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

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

The horizontal chord through the focus is on the line of equation (see picture in opening section); it is called the latus rectum; one half of it is the 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

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

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

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 -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

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

(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 .

The implicit equation of a parabola is defined by an irreducible polynomial of degree two:

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

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

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

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

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 similar if one can be transformed to the other by a similarity, that is, an arbitrary composition of rigid motions (translations and rotations) and uniform scalings.

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 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.

A synthetic approach, using similar triangles, can also be used to establish this result.

The general result is that two conic sections (necessarily of the same type) are similar if and only if they have the same eccentricity. and pedal curve.

Reflection of light striking the convex side

If light travels along the line , 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 .

Alternative proofs

thumb|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. is perpendicular to the directrix, and the line bisects angle ∠FPT. Q is another point on the parabola, with perpendicular to the directrix. We know that  =  and  = . Clearly,  > , so  > . All points on the bisector are equidistant from F and T, but Q is closer to F than to T. This means that Q is to the left of , 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 . Therefore, 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 to be the tangent to the parabola at E if the angles are equal. The reflective property follows as shown previously.

Pin and string construction

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:

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).

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 similarities. So, it is sufficient to prove any property for the unit parabola with equation <math>y = x^2</math>.

4-points property

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>.

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

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

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

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 section on parallel chords.

Remark: This property is an affine version of the theorem of two perspective triangles of a non-degenerate conic.

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

thumb|Steiner generation of a parabola

Steiner established the following procedure for the construction of a non-degenerate conic (see Steiner conic):

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 ellipses and hyperbolas.

Dual parabola

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>)|frame

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:

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

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:

(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 coordinates is (if two coordinates are equal, there is no parabola with directrix parallel to the axis, which passes through the points)

<math display="block">\frac