Parabola

In mathematics, a parabola (/pəˈræbələ/ pə-RA-bə-lə) 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 \(y=ax^2+bx+ c\) (with \(a\neq 0\)) is a parabola with its axis of symmetry coincident with the 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" 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.

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.

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 ⁠\(F\)⁠ is the focus and ⁠\(l\)⁠ is the directrix, the parabola is the set of all points ⁠\(P\)⁠ such that \[d(P,F) = d(P,l),\] where ⁠\(d\)⁠ denotes Euclidean distance.

The point where this distance is minimal is the midpoint \(V\) of the perpendicular from the focus \(F\) to the directrix \(l.\) It is called the vertex, and its distance to both the focus and the directrix is the focal length of the parabola.

The line \(FV\) is the unique axis of symmetry of the parabola and called the axis of the parabola.

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

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

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

The horizontal chord through the focus is on the line of equation ⁠\(y=f\)⁠ (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 \(p\). From the equation satisfied by the endpoints of the latus rectum, one gets \[p = 2f.\] Thus, the semi-lactus rectum is the distance from the focus to the directrix. Using the parameter \(p\), the equation of the parabola can be rewritten as \[x^2 = 2py.\]

More generally, if the vertex is \(V = (v_1, v_2)\), the focus \(F = (v_1, v_2 + f)\), and the directrix \(y = v_2 - f\), one obtains the equation \[y = \frac{1}{4f} (x - v_1)^2 + v_2 = \frac{1}{4f} x^2 - \frac{v_1}{2f} x + \frac{v_1^2}{4f} + v_2.\]

Remarks:

• If \(f < 0\) in the above equations one gets parabola with a downward opening.
• The hypothesis that the axis is parallel to the ⁠\(y\)⁠-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 \(x\) and \(y\), one obtains equations of the form \(y^2 = 2px\). These parabolas open to the left (if \(p < 0\)) or to the right (if \(p > 0\)).

If the focus is \(F = (f_1, f_2)\), and the directrix \(ax + by + c = 0\), then one obtains the equation \[\frac{(ax + by + c)^2}{a^2 + b^2} = (x - f_1)^2 + (y - f_2)^2\]

(the left side of the equation uses the Hesse normal form of a line to calculate the distance \(|Pl|\)).

For a parametric equation of a parabola in general position see § As the affine image of the unit parabola.

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

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 \[f(x) = a x^2 \text{ with } a \ne 0.\]

For \(a > 0\) the parabolas are opening to the top, and for \(a < 0\) are opening to the bottom (see picture). From the section above one obtains:

• The focus is \(\left(0, \frac{1}{4a}\right)\),
• the focal length \(\frac{1}{4a}\), the semi-latus rectum is \(p = \frac{1}{2a}\),
• the vertex is \((0, 0)\),
• the directrix has the equation \(y = -\frac{1}{4a}\),
• the tangent at point \((x_0, ax^2_0)\) has the equation \(y = 2a x_0 x - a x^2_0\).

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

The general function of degree 2 is \[f(x) = ax^2 + bx + c ~~\text{ with }~~ a, b, c \in \R,\ a \ne 0.\] Completing the square yields \[f(x) = a \left(x + \frac{b}{2a}\right)^2 + \frac{4ac - b^2}{4a},\] which is the equation of a parabola with

• the axis \(x = -\frac{b}{2a}\) (parallel to the y axis),
• the focal length \(\frac{1}{4a}\), the semi-latus rectum \(p = \frac{1}{2a}\),
• the vertex \(V = \left(-\frac{b}{2a}, \frac{4ac - b^2}{4a}\right)\),
• the focus \(F = \left(-\frac{b}{2a}, \frac{4ac - b^2 + 1}{4a}\right)\),
• the directrix \(y = \frac{4ac - b^2 - 1}{4a}\),
• the point of the parabola intersecting the y axis has coordinates \((0, c)\),
• the tangent at a point on the y axis has the equation \(y = bx + c\).

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 \(\mathcal P\) with vertex \(V = (v_1, v_2)\) can be transformed by the translation \((x, y) \to (x - v_1, y - v_2)\) to one with the origin as vertex. A suitable rotation around the origin can then transform the parabola to one that has the y axis as axis of symmetry. Hence the parabola \(\mathcal P\) can be transformed by a rigid motion to a parabola with an equation \(y = ax^2,\ a \ne 0\). Such a parabola can then be transformed by the uniform scaling \((x, y) \to (ax, ay)\) into the unit parabola with equation \(y = x^2\). 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. 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 \(y = ax^2\) onto the unit parabola, such as \((x, y) \to \left(x, \tfrac{y}{a}\right)\). But this mapping is not a similarity, and only shows that all parabolas are affinely equivalent (see § As the affine image of the unit parabola).

The pencil of conic sections with the x axis as axis of symmetry, one vertex at the origin (0, 0) and the same semi-latus rectum \(p\) can be represented by the equation \[y^2 = 2px +(e^2 - 1) x^2, \quad e \ge 0,\] with \(e\) the eccentricity.

• For \(e = 0\) the conic is a circle (osculating circle of the pencil),
• for \(0 < e < 1\) an ellipse,
• for \(e = 1\) the parabola with equation \(y^2 = 2px,\)
• for \(e > 1\) a hyperbola (see picture).

If p > 0, the parabola with equation \(y^2 = 2px\) (opening to the right) has the polar representation \[r = 2p \frac{\cos\varphi}{\sin^2\varphi}, \quad \varphi \in \left[ -\tfrac{\pi}{2} , \tfrac{\pi}{2} \right] \setminus \{0\}\] where \(r^2 = x^2 + y^2,\ x = r\cos\varphi\).

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

If one shifts the origin into the focus, that is, \(F = (0, 0)\), one obtains the equation \[r = \frac{p}{1 - \cos\varphi}, \quad \varphi \ne 2\pi k.\]

Remark 1: Inverting this polar form shows that a parabola is the inverse of a cardioid.

Remark 2: The second polar form is a special case of a pencil of conics with focus \(F = (0, 0)\) (see picture): \[r = \frac{p}{1 - e\cos\varphi}\] (\(e\) is the eccentricity).

The diagram represents a cone with its axis AV. The point A is its apex. An inclined cross-section of the cone, shown in pink, is inclined from the axis by the same angle 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 elliptical when viewed obliquely, as is shown in the diagram. Its centre is V, and PK is a diameter. We will call its radius 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 DE, which joins the points where the parabola intersects the circle. Another chord BC is the perpendicular bisector of DE and is consequently a diameter of the circle. These two chords and the parabola's axis of symmetry PM all intersect at the point 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 § Position of the focus.

Let us call the length of DM and of EM x, and the length of PM y.

The lengths of BM and CM are:

• \(\overline\mathrm{BM} = 2y\cos\theta\) (triangle BPM is isosceles, because \(\overline{PM} \parallel \overline{AC} \implies \angle PMB = \angle ACB = \angle ABC\)
• \(\overline\mathrm{CM} = 2r\) (PMCK is a parallelogram).

Using the intersecting chords theorem on the chords BC and DE, we get \[\overline\mathrm{BM} \cdot \overline\mathrm{CM} = \overline\mathrm{DM} \cdot \overline\mathrm{EM}.\]

Substituting: \[4ry\cos\theta = x^2.\]

Rearranging: \[y = \frac{x^2}{4r\cos\theta}.\]

For any given cone and parabola, r and θ are constants, but x and 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 coordinates of the points D and E, in a system in the pink plane with P as its origin. Since x is squared in the equation, the fact that D and E are on opposite sides of the 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 x and y shown in the equation. The parabolic curve is therefore the locus of points where the equation is satisfied, which makes it a Cartesian graph of the quadratic function in the equation.

It is proved in a preceding section that if a parabola has its vertex at the origin, and if it opens in the positive y direction, then its equation is y = ⁠x/4f⁠, where f is its focal length. Comparing this with the last equation above shows that the focal length of the parabola in the cone is r cos θ.

In the diagram above, the point V is the foot of the perpendicular from the vertex of the parabola to the axis of the cone. The point F is the foot of the perpendicular from the point V to the plane of the parabola. By symmetry, F is on the axis of symmetry of the parabola. Angle VPF is complementary to θ, and angle PVF is complementary to angle VPF, therefore angle PVF is θ. Since the length of PV is r, the distance of F from the vertex of the parabola is 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 F are therefore equally distant from the vertex, along the same line, which implies that they are the same point. Therefore, the point 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.

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 \(\pi\), 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) \(m_0\) of the cone, is a parabola (red curve in the diagram).

This generatrix \(m_0\) is the only generatrix of the cone that is parallel to plane \(\pi\). Otherwise, if there are two generatrices parallel to the intersecting plane, the intersection curve will be a hyperbola (or 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 a point).

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

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

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

It turns out that \(F\) is the focus of the parabola, and \(l\) is the directrix of the parabola.

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