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 (with ) 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
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. Extract of page 3. Designs were proposed in the early to mid-17th century by many mathematicians, including René Descartes, Marin Mersenne,Stargazer, p. 115. and James Gregory.Stargazer, pp. 123, 132. 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 where denotes Euclidean distance.
The point where this distance is minimal is the midpoint of the perpendicular from the focus to the directrix It is called the vertex, and its distance to both the focus and the directrix is the focal length of the parabola.
The line 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
In Cartesian coordinates, if the vertex is the origin and the directrix has the equation , then, by examining the case , the focus is on the positive -axis, with , where is the focal length.
The above geometric characterization implies that a point is on the parabola if and only if Solving for 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 . 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 , the equation of the parabola can be rewritten as
More generally, if the vertex is , the focus , and the directrix , one obtains the equation
Remarks: If 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 and , one obtains equations of the form . These parabolas open to the left (if ) or to the right (if ).
General position
If the focus is , and the directrix , then one obtains the equation
(the left side of the equation uses the Hesse normal form of a line to calculate the distance ).
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 or, equivalently, such that is the square of a linear polynomial.
As a graph of a function
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 the parabolas are opening to the top, and for are opening to the bottom (see picture). From the section above one obtains: The focus is , the focal length , the semi-latus rectum is , the vertex is , the directrix has the equation , the tangent at point has the equation .
For the parabola is the unit parabola with equation . Its focus is , the semi-latus rectum , and the directrix has the equation .
The general function of degree 2 is
Completing the square yields
which is the equation of a parabola with the axis (parallel to the y axis), the focal length , the semi-latus rectum , the vertex , the focus , the directrix , the point of the parabola intersecting the y axis has coordinates , the tangent at a point on the y axis has the equation .
Similarity to the unit parabola
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 with vertex can be transformed by the translation 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 can be transformed by a rigid motion to a parabola with an equation . Such a parabola can then be transformed by the uniform scaling into the unit parabola with equation . 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 onto the unit parabola, such as . But this mapping is not a similarity, and only shows that all parabolas are affinely equivalent (see ).
As a special conic section
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 can be represented by the equation
with the eccentricity. For the conic is a circle (osculating circle of the pencil), for an ellipse, for the parabola with equation for a hyperbola (see picture).
In polar coordinates
If , the parabola with equation (opening to the right) has the polar representation
where .
Its vertex is , and its focus is .
If one shifts the origin into the focus, that is, , one obtains the equation
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 (see picture): ( is the eccentricity).
Conic section and quadratic form Diagram, description, and definitions
The diagram represents a cone with its axis . 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 , 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 is a diameter. We will call its radius .
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 , which joins the points where the parabola intersects the circle. Another chord is the perpendicular bisector of and is consequently a diameter of the circle. These two chords and the parabola's axis of symmetry 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 .
Let us call the length of and of , and the length of .
Derivation of quadratic equation The lengths of and are:
Using the intersecting chords theorem on the chords and , we get
Substituting:
Rearranging:
For any given cone and parabola, and are constants, but and 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 is squared in the equation, the fact that D and E are on opposite sides of the 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 and 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.
Focal length