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.The parabola is a member of the family ofconic sections.
Inmathematics, aparabola is aplane curve which ismirror-symmetrical and is approximately U-shaped. It fits several superficially differentmathematical descriptions, which can all be proved to define exactly the same curves.
One description of a parabola involves apoint (thefocus) and aline (thedirectrix). The focus does not lie on the directrix. The parabola is thelocus of points in that plane that areequidistant from the directrix and the focus. Another description of a parabola is as aconic section, created from the intersection of a right circularconical surface and aplaneparallel to another plane that istangential to the conical surface.[a]
Thegraph of aquadratic function (with) is a parabola with its axis of symmetry coincident with they-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 thechord 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 geometricallysimilar.
Parabolas have the property that, if they are made of material thatreflectslight, 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 withsound and otherwaves. This reflective property is the basis of many practical uses of parabolas.
The earliest known work on conic sections was byMenaechmus in the 4th century BC. He discovered a way to solve the problem ofdoubling the cube using parabolas. (The solution, however, does not meet the requirements ofcompass-and-straightedge construction.) The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed byArchimedes by themethod of exhaustion in the 3rd century BC, in hisThe Quadrature of the Parabola. The name "parabola" is due toApollonius, 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.[1] The focus–directrix property of the parabola and other conic sections was mentioned in the works ofPappus.
Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity.
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, thefocus, equals the distance to a fixedline, thedirectrix. That is, if is the focus and is the directrix, the parabola is the set of all points such that where denotesEuclidean distance.
The point where this distance is minimal is the midpoint of the perpendicular from the focus to the directrix It is called thevertex, and its distance to both the focus and the directrix is thefocal length of the parabola.
The line is the uniqueaxis of symmetry of the parabola and called theaxis of the parabola.
Parabola with axis parallel toy-axis;p is thesemi-latus rectum
InCartesian 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 horizontalchord through the focus is on the line of equation (see picture in opening section); it is called thelatus rectum; one half of it is thesemi-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 getsThus, 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 aquadratic 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).
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 they 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 theuniform scaling into the unit parabola with equation. Thus, any parabola can be mapped to the unit parabola by a similarity.[6]
Asynthetic approach, using similar triangles, can also be used to establish this result.[7]
The general result is that two conic sections (necessarily of the same type) are similarif and only if they have the same eccentricity.[6] 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 the affine image of the unit parabola).
Thepencil ofconic sections with thex axis as axis of symmetry, one vertex at the origin (0, 0) and the same semi-latus rectum can be represented by the equationwith theeccentricity.
The diagram represents acone with its axisAV. The point A is itsapex. An inclinedcross-section of the cone, shown in pink, is inclined from the axis by the same angleθ, 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 appearselliptical when viewed obliquely, as is shown in the diagram. Its centre is V, andPK 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 achordDE, which joins the points where the parabolaintersects the circle. Another chordBC is theperpendicular bisector ofDE and is consequently a diameter of the circle. These two chords and the parabola's axis of symmetryPM all intersect at the point M.
All the labelled points, except D and E, arecoplanar. 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 ofDM and ofEMx, and the length ofPMy.
For any given cone and parabola,r andθ are constants, butx andy 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 asCartesian coordinates of the points D and E, in a system in the pink plane with P as its origin. Sincex is squared in the equation, the fact that D and E are on opposite sides of they 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 betweenx andy shown in the equation. The parabolic curve is therefore thelocus of points where the equation is satisfied, which makes it aCartesian graph of the quadratic function in the equation.
It is proved in apreceding section that if a parabola has its vertex at the origin, and if it opens in the positivey direction, then its equation isy =x2/4f, wheref is its focal length.[b] Comparing this with the last equation above shows that the focal length of the parabola in the cone isr cosθ.
In the diagram above, the point V is thefoot 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.[c] By symmetry, F is on the axis of symmetry of the parabola. Angle VPF iscomplementary toθ, and angle PVF is complementary to angle VPF, therefore angle PVF isθ. Since the length ofPV isr, the distance of F from the vertex of the parabola isr sinθ. 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.
Parabola (red): side projection view and top projection view of a cone with a Dandelin sphere
An alternative proof can be done usingDandelin spheres. It works without calculation and uses elementary geometric considerations only (see the derivation below).
The intersection of an upright cone by a plane, whose inclination from vertical is the same as ageneratrix (a.k.a. generator line, a line containing the apex and a point on the cone surface) of the cone, is a parabola (red curve in the diagram).
This generatrix is the only generatrix of the cone that is parallel to plane. Otherwise, if there are two generatrices parallel to the intersecting plane, the intersection curve will be ahyperbola (ordegenerate 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 anellipse or acircle (ora point).
Let plane be the plane that contains the vertical axis of the cone and line. The inclination of plane from vertical is the same as line means that, viewing from the side (that is, the plane is perpendicular to plane),.
In order to prove the directrix property of a parabola (see§ Definition as a locus of points above), one uses aDandelin sphere, which is a sphere that touches the cone along a circle and plane at point. The plane containing the circle intersects with plane at line. There is amirror symmetry in the system consisting of plane, Dandelin sphere and the cone (theplane of symmetry is).
Since the plane containing the circle is perpendicular to plane, and, their intersection line must also be perpendicular to plane. Since line is in plane,.
It turns out that is thefocus of the parabola, and is thedirectrix of the parabola.
Let be an arbitrary point of the intersection curve.
Thegeneratrix of the cone containing intersects circle at point.
The line segments and are tangential to the sphere, and hence are of equal length.
Generatrix intersects the circle at point. The line segments and are tangential to the sphere, and hence are of equal length.
Let line be the line parallel to and passing through point. Since, and point is in plane, line must be in plane. Since, we know that as well.
Let point bethe foot of the perpendicular from point to line, that is, is a segment of line, and hence.
Fromintercept theorem and we know that. Since, we know that, which means that the distance from to the focus is equal to the distance from to the directrix.
The reflective property states that if a parabola can reflect light, then light that enters it traveling parallel to the axis of symmetry is reflected toward the focus. This is derived fromgeometrical optics, based on the assumption that light travels in rays.
Consider the parabolay =x2. Since all parabolas are similar, this simple case represents all others.
The point E is an arbitrary point on the parabola. The focus is F, the vertex is A (the origin), and the lineFA is the axis of symmetry. The lineEC is parallel to the axis of symmetry, intersects thex axis at D and intersects the directrix at C. The point B is the midpoint of the line segmentFC.
The vertex A is equidistant from the focus F and from the directrix. Since C is on the directrix, they coordinates of F and C are equal in absolute value and opposite in sign. B is the midpoint ofFC. Itsx coordinate is half that of D, that is,x/2. The slope of the lineBE is the quotient of the lengths ofED andBD, which isx2/x/2 = 2x. But2x is also the slope (first derivative) of the parabola at E. Therefore, the lineBE is the tangent to the parabola at E.
The distancesEF andEC are equal because E is on the parabola, F is the focus and C is on the directrix. Therefore, since B is the midpoint ofFC, triangles △FEB and △CEB are congruent (three sides), which implies that the angles markedα 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 lineBE so it travels along the lineEF, as shown in red in the diagram (assuming that the lines can somehow reflect light). SinceBE 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.
The above proof and the accompanying diagram show that the tangentBE 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
Since triangles △FBE and △CBE are congruent,FB is perpendicular to the tangentBE. Since B is on thex 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[8] andpedal curve.
If light travels along the lineCE, 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 segmentFE.
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.PT is perpendicular to the directrix, and the lineMP bisects angle ∠FPT. Q is another point on the parabola, withQU perpendicular to the directrix. We know thatFP = PT andFQ = QU. Clearly,QT > QU, soQT > FQ. All points on the bisectorMP are equidistant from F and T, but Q is closer to F than to T. This means that Q is to the left ofMP, 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 ofMP. Therefore,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 lineBE to be the tangent to the parabola at E if the anglesα are equal. The reflective property follows as shown previously.
A parabola can be considered as the affine part of a non-degenerated projective conic with a point on the line of infinity, which is the tangent at. The 5-, 4- and 3- point degenerations ofPascal's theorem are properties of a conic dealing with at least one tangent. If one considers this tangent as theline at infinity and its point of contact as the point at infinity of they axis, one obtains three statements for a parabola.
The following properties of a parabola deal only with termsconnect,intersect,parallel, which are invariants ofsimilarities. So, it is sufficient to prove any property for theunit parabola with equation.
Any parabola can be described in a suitable coordinate system by an equation.
Let be four points of the parabola, and the intersection of the secant line with the line and let be the intersection of the secant line with the line (see picture). Then the secant line is parallel to line.(The lines and are parallel to the axis of the parabola.)
Proof: straightforward calculation for the unit parabola.
Application: The 4-points property of a parabola can be used for the construction of point, while and are given.
Remark: the 4-points property of a parabola is an affine version of the 5-point degeneration of Pascal's theorem.
Let be three points of the parabola with equation and the intersection of the secant line with the line and the intersection of the secant line with the line (see picture). Then the tangent at point is parallel to the line.(The lines and are parallel to the axis of the parabola.)
Proof: can be performed for the unit parabola. A short calculation shows: line has slope which is the slope of the tangent at point.
Application: The 3-points-1-tangent-property of a parabola can be used for the construction of the tangent at point, while 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.
Let be two points of the parabola with equation, and the intersection of the tangent at point with the line, and the intersection of the tangent at point with the line (see picture). Then the secant is parallel to the line.(The lines and are parallel to the axis of the parabola.)
Proof: straight forward calculation for the unit parabola.
Application: The 2-points–2-tangents property can be used for the construction of the tangent of a parabola at point, if and the tangent at 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 isnot related to Pascal's theorem.
The statements above presume the knowledge of the axis direction of the parabola, in order to construct the points. The following property determines the points by two given points and their tangents only, and the result is that the line is parallel to the axis of the parabola.
Let
be two points of the parabola, and be their tangents;
be the intersection of the tangents,
be the intersection of the parallel line to through with the parallel line to through (see picture).
Then the line is parallel to the axis of the parabola and has the equation
Proof: can be done (like the properties above) for the unit parabola.
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, seesection on parallel chords.
Remark: This property is an affine version of the theorem of twoperspective triangles of a non-degenerate conic.[10]
Related: Chord has two additional properties:
Its slope is the harmonic average of the slopes of tangents and.
It is parallel to the tangent at the intersection of with the parabola.
Steiner established the following procedure for the construction of a non-degenerate conic (seeSteiner conic):
Given twopencils of lines at two points (all lines containing and respectively) and a projective but not perspective mapping of onto, 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:
Consider the pencil at the vertex and the set of lines that are parallel to they axis.
Let be a point on the parabola, and,.
The line segment is divided inton equally spaced segments, and this division is projected (in the direction) onto the line segment (see figure). This projection gives rise to a projective mapping from pencil onto the pencil.
The intersection of the line and thei-th parallel to they axis is a point on the parabola.
Proof: straightforward calculation.
Remark: Steiner's generation is also available forellipses andhyperbolas.
Dual parabola and Bézier curve of degree 2 (right: curve point and division points for parameter)
Adual 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:
Let be given two point sets on two lines, and a projective but not perspective mapping 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 not on a line,
divides the line sections and each into equally spaced line segments and adds numbers as shown in the picture.
Then the lines are tangents of a parabola, hence elements of a dual parabola.
The parabola is aBézier curve of degree 2 with the control points.
A parabola with equation is uniquely determined by three points with differentx coordinates. The usual procedure to determine the coefficients is to insert the point coordinates into the equation. The result is a linear system of three equations, which can be solved byGaussian elimination orCramer's rule, for example. An alternative way uses theinscribed 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 the angle between two lines of equations is measured by
Four points with differentx coordinates (see picture) are on a parabola with equation if and only if the angles at and have the same measure, as defined above. That is,
(Proof: straightforward calculation: If the points are on a parabola, one may translate the coordinates for having the equation, then one has if the points are on the parabola.)
A consequence is that the equation (in) of the parabola determined by 3 points with differentx coordinates is (if twox coordinates are equal, there is no parabola with directrix parallel to thex axis, which passes through the points)Multiplying by the denominators that depend on one obtains the more standard form
In a suitable coordinate system any parabola can be described by an equation. The equation of the tangent at a point isOne obtains the functionon 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 to abijection between the points of and the lines with equations. The inverse mapping isThis relation is called thepole–polar relation of the parabola, where the point is thepole, and the corresponding line itspolar.
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:).
For a poleoutside the parabola the intersection points of its polar with the parabola are the touching points of the two tangents passing (see picture:).
For a pointwithin the parabola the polar has no point with the parabola in common (see picture: and).
The intersection point of two polar lines (see picture:) is the pole of the connecting line of their poles (see picture:).
Focus and directrix of the parabola are a pole–polar pair.
Remark: Pole–polar relations also exist for ellipses and hyperbolas.
Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q asf. 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 is2f, and (2) a tangent to the parabola at point T intersects the line of symmetry at a 45° angle.[13]: 26
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.
Let three tangents to a parabola form a triangle. ThenLambert's theorem states that the focus of the parabola lies on thecircumcircle of the triangle.[14][8]: 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.[15]
Suppose achord crosses a parabola perpendicular to its axis of symmetry. Let the length of the chord between the points where it intersects the parabola bec and the distance from the vertex of the parabola to the chord, measured along the axis of symmetry, bed. The focal length,f, of the parabola is given by
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 they axis. The parabola opens upward. It is shown elsewhere in this article that the equation of the parabola is4fy =x2, wheref is the focal length. At the positivex end of the chord,x =c/2 andy =d. Since this point is on the parabola, these coordinates must satisfy the equation above. Therefore, by substitution,. From this,.
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.[16][17] 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 byArchimedes in the 3rd century BCE. He used the areas of triangles, rather than that of the parallelogram.[d] SeeThe Quadrature of the Parabola.
If the chord has lengthb and is perpendicular to the parabola's axis of symmetry, and if the perpendicular distance from the parabola's vertex to the chord ish, the parallelogram is a rectangle, with sides ofb andh. The areaA of the parabolic segment enclosed by the parabola and the chord is therefore
This formula can be compared with the area of a triangle:1/2bh.
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.[e] Then, using the formula given inDistance 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
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 (seeAxis-direction of a parabola).[f]
If a point X is located on a parabola with focal lengthf, and ifp is theperpendicular distance from X to the axis of symmetry of the parabola, then the lengths ofarcs of the parabola that terminate at X can be calculated fromf andp as follows, assuming they are all expressed in the same units.[g]
This quantitys 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 is2s.
The perpendicular distancep can be given a positive or negative sign to indicate on which side of the axis of symmetry X is situated. Reversing the sign ofp reverses the signs ofh ands without changing their absolute values. If these quantities are signed,the length of the arc betweenany two points on the parabola is always shown by the difference between their values ofs. The calculation can be simplified by using the properties of logarithms:
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 they axis.
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.
The area of the parabolic sector.
Since triangles TSB and QBJ are similar,
Therefore, the area of the parabolic sector 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 isv, then the speed of J is equal to3v/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.
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'sPrincipia, 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 isv, then at the vertex V it is, and point J moves at a constant speed of.
The focal length of a parabola is half of itsradius of curvature at its vertex.
Proof
Image is inverted. AB isx axis. C is origin. O is center. A is(x,y). OA = OC =R. PA =x. CP =y. OP =(R −y). Other points and lines are irrelevant for this purpose.
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.
Consider a point(x,y) on a circle of radiusR and with center at the point(0,R). The circle passes through the origin. If the point is near the origin, thePythagorean theorem shows that
But if(x,y) is extremely close to the origin, since thex axis is a tangent to the circle,y is very small compared withx, soy2 is negligible compared with the other terms. Therefore, extremely close to the origin
1
Compare this with the parabola
2
which has its vertex at the origin, opens upward, and has focal lengthf (see preceding sections of this article).
Equations(1) and(2) are equivalent ifR = 2f. 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.
An affine transformation of the Euclidean plane has the form, where is a regular matrix (determinant is not 0), and is an arbitrary vector. If are the column vectors of the matrix, the unit parabola is mapped onto the parabolawhere
is apoint of the parabola,
is atangent vector at point,
isparallel to the axis of the parabola (axis of symmetry through the vertex).
In general, the two vectors are not perpendicular, and isnot the vertex, unless the affine transformation is asimilarity.
The tangent vector at the point is. At the vertex the tangent vector is orthogonal to. Hence the parameter of the vertex is the solution of the equationwhich isand thevertex is
Thefocal length can be determined by a suitable parameter transformation (which does not change the geometric shape of the parabola). The focal length isHence thefocus of the parabola is
The definition of a parabola in this section gives a parametric representation of an arbitrary parabola, even in space, if one allows to be vectors in space.
Simpson's rule: the graph of a function is replaced by an arc of a parabola
In one method ofnumerical 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
A parabola can be used as atrisectrix, that is it allows theexact trisection of an arbitrary angle with straightedge and compass. This is not in contradiction to the impossibility of an angle trisection withcompass-and-straightedge constructions alone, as the use of parabolas is not allowed in the classic rules for compass-and-straightedge constructions.
To trisect, place its leg on thex axis such that the vertex is in the coordinate system's origin. The coordinate system also contains the parabola. Theunit circle with radius 1 around the origin intersects the angle's other leg, and from this point of intersection draw the perpendicular onto they axis. The parallel toy axis through the midpoint of that perpendicular and the tangent on the unit circle in intersect in. The circle around with radius intersects the parabola at. The perpendicular from onto thex axis intersects the unit circle at, and is exactly one third of.
The correctness of this construction can be seen by showing that thex coordinate of is. Solving the equation system given by the circle around and the parabola leads to the cubic equation. Thetriple-angle formula then shows that is indeed a solution of that cubic equation.
This trisection goes back toRené Descartes, who described it in his bookLa Géométrie (1637).[18]
If one replaces the real numbers by an arbitraryfield, many geometric properties of the parabola are still valid:
A line intersects in at most two points.
At any point the line is the tangent.
Essentially new phenomena arise, if the field has characteristic 2 (that is,): the tangents are all parallel.
Inalgebraic geometry, the parabola is generalized by therational normal curves, which have coordinates(x,x2,x3, ...,xn); the standard parabola is the casen = 2, and the casen = 3 is known as thetwisted cubic. A further generalization is given by theVeronese variety, when there is more than one input variable.
The curvesy =xp for other values ofp are traditionally referred to as thehigher parabolas and were originally treated implicitly, in the formxp =kyq forp andq both positive integers, in which form they are seen to be algebraic curves. These correspond to the explicit formulay =xp/q for a positive fractional power ofx. Negative fractional powers correspond to the implicit equationxpyq =k and are traditionally referred to ashigher hyperbolas. Analytically,x can also be raised to an irrational power (for positive values ofx); the analytic properties are analogous to whenx is raised to rational powers, but the resulting curve is no longer algebraic and cannot be analyzed by algebraic geometry.
In nature, approximations of parabolas and paraboloids are found in many diverse situations. The best-known instance of the parabola in the history ofphysics is thetrajectory of a particle or body in motion under the influence of a uniformgravitational field withoutair resistance (for instance, a ball flying through the air, neglecting airfriction).
Theparabolic trajectory of projectiles was discovered experimentally in the early 17th century byGalileo, who performed experiments with balls rolling on inclined planes. He also later proved thismathematically in his bookDialogue Concerning Two New Sciences.[19][h] 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 thecenter 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.
Anotherhypothetical situation in which parabolas might arise, according to the theories of physics described in the 17th and 18th centuries bySir Isaac Newton, is intwo-body orbits, for example, the path of a small planetoid or other object under the influence of the gravitation of theSun.Parabolic orbits do not occur in nature; simple orbits most commonly resemblehyperbolas orellipses. The parabolic orbit is thedegenerate intermediate case between those two types of ideal orbit. An object following a parabolic orbit would travel at the exactescape velocity of the object it orbits; objects inelliptical orhyperbolic orbits travel at less or greater than escape velocity, respectively. Long-periodcomets 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 simplesuspension bridge. The curve of the chains of a suspension bridge is always an intermediate curve between a parabola and acatenary, 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.[20][21] Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise catenary-shaped cable is deformed toward a parabola (seeCatenary § 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 theparabolic reflector, which is a mirror or similar reflective device that concentrates light or other forms ofelectromagnetic radiation to a commonfocal 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 geometerArchimedes, who, according to a dubious legend,[22] constructed parabolic mirrors to defendSyracuse against theRoman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied totelescopes in the 17th century. Today, paraboloid reflectors can be commonly observed throughout much of the world inmicrowave and satellite-dish receiving and transmitting antennas.
Inparabolic microphones, 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, thecentrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind theliquid-mirror telescope.
Aircraft used to create aweightless state for purposes of experimentation, such asNASA's "Vomit Comet", follow a vertically parabolic trajectory for brief periods in order to trace the course of an object infree fall, which produces the same effect as zero gravity for most purposes.
Abouncing 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 andair resistance, causes the curve swept out to deviate slightly from the expected perfect parabola.
Parabolic trajectories of water in a fountain.
The path (in red) ofComet Kohoutek as it passed through the inner Solar system, showing its nearly parabolic shape. The blue orbit is the Earth's.
The supporting cables ofsuspension bridges follow a curve that is intermediate between a parabola and acatenary.
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. (SeeRotating furnace)
^The tangential plane just touches the conical surface along a line, which passes through the apex of the cone.
^As stated above in the lead, the focal length of a parabola is the distance between its vertex and focus.
^The point V is the centre of the smaller circular cross-section of the cone. The point F is in the (pink) plane of the parabola, and the lineVF is perpendicular to the plane of the parabola.
^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.
^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.
^A proof of this sentence can be inferred from the proof of theorthoptic property, above. It is shown there that the tangents to the parabolay =x2 at(p,p2) and(q,q2) intersect at a point whosex coordinate is the mean ofp andq. Thus if there is a chord between these two points, the intersection point of the tangents has the samex coordinate as the midpoint of the chord.
^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.
^abKumpel, P. G. (1975), "Do similar figures always have the same shape?",The Mathematics Teacher,68 (8):626–628,doi:10.5951/MT.68.8.0626,ISSN0025-5769.
^Shriki, Atara; David, Hamatal (2011), "Similarity of Parabolas – A Geometrical Perspective",Learning and Teaching Mathematics,11:29–34.
^Middleton, W. E. Knowles (December 1961). "Archimedes, Kircher, Buffon, and the Burning-Mirrors".Isis.52 (4). Published by: The University of Chicago Press on behalf of The History of Science Society:533–543.doi:10.1086/349498.JSTOR228646.S2CID145385010.
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