Parabola

History

The earliest known work on conic sections was by Menaechmus in the fourth 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 via the method of exhaustion in the third 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 conics is due to 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 seventeenth 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 of a parabola as locus of points

A parabola can be defined geometrically as a set of points (locus of points) in the Euclidean plane:

A parabola is a set of points, such that for any point P of the set the distance | P F | to a fixed point F , the focus, is equal to the distance | P l | to a fixed line l , the directrix:

{ P ∣ | P F | = | P l | }

The midpoint V of the perpendicular from the focus F onto the directrix l is called vertex and the line F V the axis of symmetry of the parabola.

Axis of symmetry parallel to the y-axis

If one introduces cartesian coordinates, such that F = ( 0 , f )   , f > 0 , and the directrix has the equation y = − f one gets for a point P = ( x , y ) from | P F | 2 = | P l | 2 the equation x 2 + ( y − f ) 2 = ( y + f ) 2 . Solving for y yields

y = 1 4 f x 2 .

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

The length of the horizontal chord through the focus (s. picture) is called latus rectum, one half of it semi latus rectum and designated by p . From the picture one gets

p = 2 f .

Latus rectum is similar defined for an ellipse and hyperbola resp. For any case p is the radius of the osculating circle at the vertex. For a parabola p is the distance of the focus from the directrix, too. Using parameter p , the equation of the parabola can be rewritten as

x 2 = 2 p y .

More general, 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 gets the equation

y = 1 4 f ( x − v 1 ) 2 + v 2 = 1 4 f x 2 − v 1 2 f x + v 1 2 4 f + v 2 .

Remark:

1. In the case of f < 0 the parabola has a downwards opening.
2. The presumption axis parallel to the y-axis allows to consider a parabola as the graph of a polynomial of degree 2 and vice versa: The graph of an arbitrary polynomial of degree 2 is a parabola (see next section).
3. If one changes x and y , one gets equations y 2 = 2 p x of parabolas, which are open to the left ( if p < 0 ) or right (if p > 0 ).

General case

If the focus is F = ( f 1 , f 2 ) and the directrix a x + b y + c = 0 one gets the equation

( a x + b y + 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 | P l | .)

A parametric representation of a parabola in general position is contained in a section below.

Parabola as graph of a function

The previous section shows: 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  with  a ≠ 0 .

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

The focus is ( 0 , 1 4 a ) ,

the focal length 1 4 a , the semi latus rectum is p = 1 2 a ,

the directrix has the equation y = − 1 4 a and

the tangent at point ( x 0 , a x 0 2 ) has the equation y = 2 a x 0 x − a x 0 2 .

For a = 1 the parabola is the unit parabola with equation y = x 2 . Its focus is ( 0 , 1 4 ) , the semi latus rectum p = 1 2 and the directrix has the equation y = − 1 4 .

The general function of degree 2 is

f ( x ) = a x 2 + b x + c  with  a , b , c ∈ R , a ≠ 0 .

Completing the square yields

f ( x ) = a ( x + b 2 a ) 2 + 4 a c − b 2 4 a ,

which is the equation of a parabola with

the axis x = − b 2 a (parallel to the y-axis),

the focal length 1 4 a , the semi latus rectum p = 1 2 a ,

the vertex V = ( − b 2 a , 4 a c − b 2 4 a ) ,

the focus F = ( − b 2 a , 4 a c − b 2 + 1 4 a ) ,

the directrix y = 4 a c − b 2 − 1 4 a .

Any parabola is similar to the unit parabola y=x²

Any parabola P has a Vertex V = ( v 1 , v 2 ) and can be transformed by the translation ( x , y ) → ( x − v 1 , y − v 2 ) and a suitable rotation around the origin such that the transformed parabola has the origin as vertex and the y-axis as axis of symmetry. A translation and a rotation preserve any length and any angle. Hence the parabola P is similar (even congruent) to a parabola with an equation y = a x 2 . Additionaly such a parabola can be transformed by the uniform scaling ( x , y ) → ( a x , a y ) into the unit parabola with equation y = x 2 . A uniform scaling preserves angles, hence is a similarity, too.

Remark:

1. This property is special for parabolas and is not true for ellipses / unit circle and hyperbolas / unit hyperbola, resp..
2. There exist other simple affine transformations, which map the parabola y = a x 2 onto the unit parabola. For example ( x , y ) → ( x , y a ) . But this mapping is not a similarity !

Parabola 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 p can be represented by the equation

y 2 = 2 p x + ( e 2 − 1 ) x 2 ,   e ≥ 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 = 2 p x and

for e > 1 a hyperbola (s. picture).

Parabola in polar coordinates

A parabola with equation y 2 = 2 p x (opening to the right !) has the polar coordinate representation:

r = 2 p cos ⁡ φ sin 2 ⁡ φ  with  φ ∈ [ − π 2 , π 2 ] ∖ { 0 } .

( r 2 = x 2 + y 2 ,   x = r cos ⁡ φ .)
Its vertex is V = ( 0 , 0 ) and its focus is F = ( p 2 , 0 ) .

If one shifts the origin into the focus, i.e. F = ( 0 , 0 ) , one gets the equation

r = p 1 − cos ⁡ φ  with  φ ≠ 2 π k .

Remark 1: Inverting this polar form shows: 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 ) (s. picture):

r = p 1 − e cos ⁡ φ , ( e : eccentricity).

Diagram, description, and definitions

The diagram represents a cone with its axis vertical. The point A is its apex. An inclined cross-section of the cone, shown in pink, is inclined from the vertical 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 horizontal cross-section 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 horizontal, 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 the paragraph "Position of the focus".

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

Derivation of quadratic equation

The lengths of BM and CM are:

B M ¯ = 2 y sin ⁡ θ  (triangle BPM is isosceles.) C M ¯ = 2 r  (PMCK is a parallelogram.)

Using the intersecting chords theorem on the chords BC and DE, we get:

B M ¯ ⋅ C M ¯ = D M ¯ ⋅ E M ¯

Substituting:

4 r y sin ⁡ θ = x 2

Rearranging:

y = x 2 4 r sin ⁡ θ

For any given cone and parabola, r and θ are constants, but x and y are variables which 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.

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.

Focal length

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 = x²/4f, 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 sin θ.

Position of the focus

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.

Alternative proof with Dandelin spheres

An alternative proof can be done using Dandelin spheres. It works without calculation and uses elementary geometric considerations, only. (see German article on Parabel)

Proof of the reflective property

The reflective property states that, if a parabola can reflect light, then light which enters it travelling parallel to the axis of symmetry is reflected to the focus. This is derived from the wave nature of light in the paragraph "description of final diagram", which describes a diagram just above it, at the end of this article. This derivation is valid, but may not be satisfying to readers who would prefer a mathematical approach. In the following proof, the fact that every point on the parabola is equidistant from the focus and from the directrix is taken as axiomatic.

Consider the parabola y = x². Since all parabolas are similar, this simple case represents all others. The right-hand side of the diagram shows part of this parabola.

Construction and definitions

The point E is an arbitrary point on the parabola, with coordinates (x, x²). The focus is F, the vertex is A (the origin), and the line FA (the y-axis) is the axis of symmetry. The line EC is parallel to the axis of symmetry, and intersects the x-axis at D. The point C is located on the directrix (which is not shown, to minimize clutter). The point B is the midpoint of the line segment FC.

Deductions

Measured along the axis of symmetry, the vertex, A, is equidistant from the focus, F, and from the directrix. Correspondingly, since C is on the directrix, the y-coordinates of F and C are equal in absolute value and opposite in sign. B is the midpoint of FC, so its y-coordinate is zero, so it lies on the x-axis. Its x-coordinate is half that of E, D, and C, i.e. x/2. The slope of the line BE is the quotient of the lengths of ED and BD, which is x²/x/2, which comes to 2x. But 2x is also the slope (first derivative) of the parabola at E. Therefore, the line BE is the tangent to the parabola at E.

The distances EF and EC are equal because E is on the parabola, F is the focus and C is on the directrix. Therefore, since B is the midpoint of FC, triangles △FEB and △CEB are congruent (three sides), which implies that the angles marked α are congruent. (The angle above E is vertically opposite angle ∠BEC.) This means that a ray of light which enters the parabola and arrives at E travelling parallel to the axis of symmetry will be reflected by the line BE so it travels along the line EF, as shown in red in the diagram (assuming that the lines can somehow reflect light). Since BE is the tangent to the parabola at E, the same reflection will be done by an infinitesimal arc of the parabola at E. Therefore, light that enters the parabola and arrives at E travelling parallel to the axis of symmetry of the parabola is reflected by the parabola toward its focus.

The point E has no special characteristics. This conclusion about reflected light applies to all points on the parabola, as is shown on the left side of the diagram. This is the reflective property.

Other consequences

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

Tangent bisection property

The above proof, and the accompanying diagram, show that the tangent BE bisects the angle ∠FEC. In other words, the tangent to the parabola at any point bisects the angle between the lines joining the point to the focus, and perpendicularly to the directrix.

Intersection of a tangent and perpendicular from focus

Since triangles △FBE and △CBE are congruent, FB is perpendicular to the tangent BE. Since B is on the x-axis, which is the tangent to the parabola at its vertex, it follows that the point of intersection between any tangent to a parabola and the perpendicular from the focus to that tangent lies on the line that is tangential to the parabola at its vertex. See animated diagram. and pedal curve.

Reflection of light striking the convex side

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

Alternative proofs

The above proofs of the reflective and tangent bisection properties use a line of calculus. For readers who are not comfortable with calculus, the following alternative 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 line MP bisects angle ∠FPT. Q is another point on the parabola, with QU perpendicular to the directrix. We know that FP = PT and FQ = QU. Clearly, QT > QU, so QT > FQ. All points on the bisector MP are equidistant from F and T, but Q is closer to F than to T. This means that Q is to the left of MP, i.e. 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 MP. 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 line BE to be the tangent to the parabola at E if the angles α are equal. The reflective property follows as shown previously.

Properties of a parabola related to Pascal's theorem

A parabola can be considered as the affine part of a non degenerated projective conic with a point Y ∞ on the line of infinity g ∞ , which is the tangent at Y ∞ . 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 line of infinity and its point of contact as the point of infinity of the y-axis, one gets 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 y = x 2 .

4-points-property of a parabola

Any parabola can be described in a suitable coordinate system by an equation y = a x 2 .

Let P 1 = ( x 1 , y 1 ) , P 2 = ( x 2 , y 2 ) , P 3 = ( x 3 , y 3 ) , P 4 = ( x 4 , y 4 ) be four points of the parabola y = a x 2 and Q 2 the intersection of the secant line P 1 P 4 with the line x = x 2 and let be Q 1 the intersection of the secant line P 2 P 3 with the line x = x 1 (s. picture), then the secant line P 3 P 4 is parallel to line Q 1 Q 2 .

(The lines x = x 1 and x = x 2 are parallel to the axis of the parabola.)

Proof: straight forward calculation for the unit parabola y = x 2 .

Application: The 4-points-property of a parabola can be used for the construction of point P 4 , while P 1 , P 2 , P 3 and Q 2 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 of a parabola

Let be P 0 = ( x 0 , y 0 ) , P 1 = ( x 1 , y 1 ) , P 2 = ( x 2 , y 2 ) three points of the parabola with equation y = a x 2 and Q 2 the intersection of the secant line P 0 P 1 with the line x = x 2 and Q 1 the intersection of the secant line P 0 P 2 with the line x = x 1 (s. picture), then the tangent at point P 0 is parallel to the line Q 1 Q 2 .

(The lines x = x 1 and x = x 2 are parallel to the axis of the parabola.)

Proof: can be performed for the unit parabola y = x 2 . A short calculation shows: line Q 1 Q 2 has slope 2 x 0 which is the slope of the tangent at point P 0 .

Application: The 3-points-1-tangent-property of a parabola can be used for the construction of the tangent at point P 0 , while P 1 , P 2 , P 0 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 of a parabola

Let be P 1 = ( x 1 , y 1 ) , P 2 = ( x 2 , y 2 ) two points of the parabola with equation y = a x 2 and Q 2 the intersection of the tangent at point P 1 with the line x = x 2 and Q 1 the intersection of the tangent at point P 2 with the line x = x 1 (s. picture) then the secant P 1 P 2 is parallel to the line Q 1 Q 2 .

(The x = x 1 and x = x 2 are parallel to the axis of the parabola.)

Proof: straight forward calculation for the unit parabola y = x 2 .

Application: The 2-points-2-tangents-property can be used for the construction of the tangent of a parabola at point P 2 while P 1 , P 2 and the tangent at P 1 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 deals with 2 points and 2 tangents, too, but is not related to Pascal's theorem !

Axis-direction of a parabola

The statements above presume the knowledge of the axis-direction of the parabola, in order to construct the points Q 1 , Q 2 . The following property determines the points Q 1 , Q 2 by 2 given points and their tangents, only, and the result is: the line Q 1 Q 2 is parallel to the axis of the parabola.

Let be

(1) P 1 = ( x 1 , y 1 ) , P 2 = ( x 2 , y 2 ) two points of the parabola y = a x 2 and t 1 , t 2 their tangents, and (2) Q 1 the intersection of the tangents t 1 , t 2 , (3) Q 2 the intersection of the parallel line to t 1 through P 2 with the parallel line to t 2 through P 1 (s.  picture) then the line Q 1 Q 2 is parallel to the axis of the parabola and has the equation x = x 1 + x 2 2 .

Proof: can be done (like the properties above) for the unit parabola y = x 2 .

Application: This property can be used to determine the direction of the axis of a parabola, if 2 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.

Parabola

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

Given two pencils B ( U ) , B ( V ) of lines at two points U , V (all lines containing U and V resp.) and a projective but not perspective mapping π of B ( U ) onto B ( V ) . Then the intersection points of corresponding lines form a non-degenerate projective conic section.

This procedure can be used for a simple construction of points of the parabola y = a x 2 :

One considers the pencil at the vertex S ( 0 , 0 ) and the set of lines Π y , which are parallel to the y-axis.

1. Let be P = ( x 0 , y 0 ) a point of the parabola and A = ( 0 , y 0 ) , B = ( x 0 , 0 ) ,
2. The line section B P ¯ is divided into n equally spaced sections and this division is projected parallely (direction is A B ) onto the line section A P ¯ (s. picture). The parallel projection gives rise to a projective mapping π from pencil S onto the pencil Π y .
3. The intersection of the line S B i and the i-th parallel to the y-axis is a point of the parabola.

Proof: straight forward calculation.

Remark: Steiner's generation is available for ellipses and hyperbolas, too.

Dual parabola

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:

Let be given two point sets on two lines u , v and a projective but not perspective mapping π 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

1. three points P 0 , P 1 , P 2 not on a line,
2. divides the line sections P 0 P 1 ¯ and P 1 P 2 ¯ each into n equally spaced line segments and adds numbers as shown in the picture.
3. Then the lines P 0 P 1 , P 1 P 2 , ( 1 , 1 ) , ( 2 , 2 ) , … are tangents of a parabola, hence elements of a dual parabola.
4. The parabola is a Bezier curve of degree 2 with the control points P 0 , P 1 , P 2 .

The proof is a consequence of the de Casteljau algorithm for a Bezier curve of degree 2.

Inscribed angles for parabolas y=ax²+bx+c and the 3-point-form

A parabola with equation y = a x 2 + b x + c , a ≠ 0 is uniquely determined by three points ( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) with different x-coordinates. The usual procedure to determine the coefficients a , b , c is to insert the point coordinates into the equation. The result is a linear system of 3 equations, which can be solved by the Gauss algorithm. An alternative way uses the inscribed angle theorem for parabolas:

In order to measure an angle between two lines with equations y = m 1 x + d 1 ,   y = m 2 x + d 2 in this context one uses the difference m 1 − m 2 of their slopes.

Two lines are parallel, if m 1 = m 2 , hence the measure result is 0 .

Analogous to the inscribed angle theorem for circles one gets the

Inscribed angle theorem for parabolas:

For four points P i = ( x i , y i ) ,   i = 1 , 2 , 3 , 4 ,   x i ≠ x k , i ≠ k (s. picture) the following statement is true: The four points are on a parabola with equation y = a x 2 + b x + c if and only if the angles at P 3 and P 4 are equal in the sence of the measurement above. That means if y 4 − y 1 x 4 − x 1 − y 4 − y 2 x 4 − x 2 = y 3 − y 1 x 3 − x 1 − y 3 − y 2 x 3 − x 2 .

(Proof: straight forward calculation. If the points are on a parabola, one can assume the parabola's equation is y = a x 2 .)

A consequence of the inscribed angle theorem for parabolas is the

3-point-form of a parabola's equation:

The equation of the parabola determined by 3 points P i = ( x i , y i ) ,   i = 1 , 2 , 3 ,   x i ≠ x k , i ≠ k is the solution of the equation y − y 1 x − x 1 − y − y 2 x − x 2 = y 3 − y 1 x 3 − x 1 − y 3 − y 2 x 3 − x 2 for y .

Pole-polar relation of a parabola

In a suitable coordinatesystem any parabola can be described by an equation y = a x 2 . The equation of the tangent at a point P 0 = ( x 0 , y 0 ) , y 0 = a x 0 2 is

y = 2 a x 0 ( x − x 0 ) + y 0 = 2 a x 0 x − a x 0 2 = 2 a x 0 x − y 0 .

One gets the function

( x 0 , y 0 ) → y = 2 a x 0 x − y 0

on the set of points of the parabola onto the set of tangents.

Obviously this function can be extended onto the set of all points of R 2 to a bijection between the points of R 2 and the lines with equations y = m x + d ,   m , d ∈ R . The inverse mapping is

line y = m x + d → point ( m 2 a , − d ) .

This relation is called the pole-polar relation of the parabola, where the point is the pole and the corresponding line its polar.

By calculation one checks the following properties of the pole-polar relation of the parabola:

For a point (pole) on the parabola the polar is the tangent at this point (s. picture: P 1 ,   p 1 ).

For a pole P outside the parabola the intersection points of its polar with the parabola are the touching points of the two tangents passing P (s. picture: P 2 ,   p 2 ).

For a point within the parabola the polar has no point with the parabola in common. (s. picture: P 3 ,   p 3 and P 4 ,   p 4 ).

The intersection point of two polar lines (for example: p 3 , p 4 ) is the pole of the connecting line of theit poles (in example: P 3 , P 4 ).

focus and directrix of the parabola are a pole-polar pair.

Remark: Pole-polar relations exist for ellipses and hyperbolas, too.

Two tangent properties related to the latus rectum

Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then (1) the distance from F to T is 2f, and (2) a tangent to the parabola at point T intersects the line of symmetry at a 45° angle.

Orthoptic property

If two tangents to a parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect on the directrix are perpendicular.

Proof

Without loss of generality, consider the parabola y = ax², a ≠ 0. Suppose that two tangents contact this parabola at the points (p, ap²) and (q, aq²). Their slopes are 2ap and 2aq respectively. Thus the equation of the first tangent is of the form y = 2apx + C, where C is a constant. In order to make the line pass through (p, ap²), the value of C must be −ap², so the equation of this tangent is y = 2apx − ap². Likewise, the equation of the other tangent is y = 2aqx − aq². At the intersection point of the two tangents, 2apx − ap² = 2aqx − aq². Thus 2x(p − q) = p² − q². Factoring the difference of squares, cancelling, and dividing by 2 gives x = p + q/2. Substituting this into one of the equations of the tangents gives an expression for the y-coordinate of the intersection point: y = 2ap(p + q/2) − ap². Simplifying this gives y = apq.

We now use the fact that these tangents are perpendicular. The product of the slopes of perpendicular lines is −1, assuming that both of the slopes are finite. The slopes of our tangents are 2ap and 2aq, so (2ap)(2aq) = −1, so pq = −1/4a². Thus the y-coordinate of the intersection point of the tangents is given by y = −1/4a. This is also the equation of the directrix of this parabola, so the two perpendicular tangents intersect on the directrix.

Lambert's theorem

Let three tangents to a parabola form a triangle. Then Lambert's theorem states that the focus of the parabola lies on the circumcircle of the triangle.

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.

Focal length calculated from parameters of a chord

Suppose a chord crosses a parabola perpendicular to its axis of symmetry. Let the length of the chord between the points where it intersects the parabola be c and the distance from the vertex of the parabola to the chord, measured along the axis of symmetry, be d. The focal length, f, of the parabola is given by:

f = c 2 16 d

Proof

Suppose a system of Cartesian coordinates is used such that the vertex of the parabola is at the origin, and the axis of symmetry is the y-axis. The parabola opens upward. It is shown elsewhere in this article that the equation of the parabola is 4fy = x², where f is the focal length. At the positive x end of the chord, x = c/2 and y = d. Since this point is on the parabola, these coordinates must satisfy the equation above. Therefore, by substitution, 4fd = (c/2)2
. From this, f = c²/16d.

Length of an arc of a parabola

If a point X is located on a parabola which has focal length f, and if p is the perpendicular distance from X to the axis of symmetry of the parabola, then the lengths of arcs of the parabola which terminate at X can be calculated from f and p as follows, assuming they are all expressed in the same units.

h = p 2 q = f 2 + h 2 s = h q f + f ln ⁡ ( h + q f )

This quantity, s, is the length of the arc between X and the vertex of the parabola.

The length of the arc between X and the symmetrically opposite point on the other side of the parabola is 2s.

The perpendicular distance, p, can be given a positive or negative sign to indicate on which side of the axis of symmetry X is situated. Reversing the sign of p reverses the signs of h and s without changing their absolute values. If these quantities are signed, the length of the arc between any two points on the parabola is always shown by the difference between their values of s. The calculation can be simplified by using the properties of logarithms:

s 1 − s 2 = h 1 q 1 − h 2 q 2 f + f ln ⁡ h 1 + q 1 h 2 + q 2

This can be useful, for example, in calculating the size of the material needed to make a parabolic reflector or parabolic trough.

This calculation can be used for a parabola in any orientation. It is not restricted to the situation where the axis of symmetry is parallel to the y-axis.

Focal length and radius of curvature at the vertex

The focal length of a parabola is half of its radius of curvature at its vertex.

Proof

Consider a point (x, y) on a circle of radius R and with centre at the point (0, R). The circle passes through the origin. If the point is near the origin, the Pythagorean theorem shows that:

x 2 + ( R − y ) 2 = R 2 ∴ x 2 + R 2 − 2 R y + y 2 = R 2 ∴ x 2 + y 2 = 2 R y

But if (x, y) is extremely close to the origin, since the x-axis is a tangent to the circle, y is very small compared with x, so y² is negligible compared with the other terms. Therefore, extremely close to the origin:

x 2 = 2 R y  (Equation 1)

Compare this with the parabola:

x 2 = 4 f y  (Equation 2)

which has its vertex at the origin, opens upward, and has focal length f. (See preceding sections of this article.)

Equations 1 and 2 are equivalent if R = 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 which is a small segment of a sphere behaves approximately like a parabolic mirror, focusing parallel light to a point which is midway between the centre and the surface of the sphere.

Parabola as the affine image of the unit parabola y=x²

Another definition of a parabola uses affine transformations:

Any parabola is the affine image of the unit parabola with equation y = x 2 .

An affine transformation of the Euclidean plane has the form x → → f → 0 + A x → , where A is a regular matrix (determinant is not 0) and f → 0 is an arbitrary vector. If f → 1 , f → 2 are the column vectors of the matrix A , the unit parabola ( t , t 2 ) , t ∈ R , is mapped onto the parabola

f → 0 is a point of the parabola and f → 1 is a tangent vector at point f → 0 . f → 2 is parallel to the axis of the parabola (axis of symmetry through the vertex).

In general the two vectors f → 1 , f → 2 are not perpendicular and f → 0 is not the vertex, unless the affine transformation is a similarity.

The tangent vector at the point p → ( t ) is p → ′ ( t ) = f → 1 + 2 t f → 2 . At the vertex the tangent vector is orthogonal to f → 2 . Hence the parameter t 0 of the vertex is the solution of the equation p → ′ ( t ) ⋅ f → 2 = f → 1 ⋅ f → 2 + 2 t f → 2 2 = 0 , which is t 0 = − f → 1 ⋅ f → 2 2 f → 2 2 and

p → ( t 0 ) = f → 0 − f → 1 ⋅ f → 2 2 f → 2 2 f → 1 + ( f → 1 ⋅ f → 2 ) 2 4 ( f → 2 2 ) 2 f → 2 is the vertex.

The focal length can be determined by a suitable parameter transformation (which does not change the geometric shape of the parabola). The focal length is

f = f → 1 2 ⋅ f → 2 2 − ( f → 1 ⋅ f → 2 ) 2 4 | f → 2 | 3 .

Hence

F :   f → 0 − f → 1 ⋅ f → 2 2 f → 2 2 f → 1 + f → 1 2 ⋅ f → 2 2 4 ( f → 2 2 ) 2 f → 2 is the focus of the parabola.

Remark: The advantage of this definition is, one gets a simple parametric representation of an arbitrary parabola, even in the space, if the vectors f → 0 , f → 1 , f → 2 are vectors of the Euclidean space.

Parabola as quadratic Bézier curve

A quadratic Bézier curve is a curve c → ( t ) defined by three points P 0 : p → 0 , P 1 : p → 1 and P 2 : p → 2 , its control points:

c → ( t )   =   ∑ i = 0 2 ( 2 i ) t i ( 1 − t ) 2 − i p → i   =   ( 1 − t ) 2 p → 0 + 2 t ( 1 − t ) p → 1 + t 2 p → 2   =   ( p → 0 − 2 p → 1 + p → 2 ) t 2 + ( − 2 p → 0 + 2 p → 1 ) t + p → 0  ,  t ∈ [ 0 , 1 ]

This curve is an arc of a parabola (s. section: parabola as affine image of the unit parabola).

Parabola and numerical integration

For numerical integration one replaces the graph of a function by arcs of parabolas and integrates the parabola arcs. A parabola is determined by three points. The formula for one arc is

∫ a b f ( x ) d x ≈ b − a 6 ⋅ ( f ( a ) + 4 f ( a + b 2 ) + f ( b ) ) .

The method is called Simpson's rule.

Parabolas as plane sections of quadrics

The following quadrics contain parabolas as plane sections:

Elliptical Cone

Parabolic cylinder

Elliptical paraboloid

Hyperbolic paraboloid

Hyperboloid of one sheet

Hyperboloid of two sheets

Mathematical generalizations

If one replaces the real numbers by an arbitrary field, many geometric properties of the parabola y = x 2 are still valid: 1) a line intersects in at most two points. 2) At any point ( x 0 , x 0 2 ) the line y = 2 x 0 x − x 0 2 is the tangent.... Essentially new phenomena arise, if the field has characteristic 2 (i.e. 1 + 1 = 0 ) : the tangents are all parallel.

In algebraic geometry, the parabola is generalized by the rational normal curves, which have coordinates (x, x², x³,…,xⁿ); the standard parabola is the case n =2, and the case n = 3 is known as the twisted cubic. A further generalization is given by the Veronese variety, when there is more than one input variable.

In the theory of quadratic forms, the parabola is the graph of the quadratic form x² (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x² + y² (or scalings) and the hyperbolic paraboloid is the graph of the indefinite quadratic form x² − y². Generalizations to more variables yield further such objects.

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

Parabolas in the physical world

In nature, approximations of parabolas and paraboloids are found in many diverse situations. The best-known instance of the parabola in the history of physics is the trajectory of a particle or body in motion under the influence of a uniform gravitational field without air resistance (for instance, a ball flying through the air, neglecting air friction).

The parabolic trajectory of projectiles was discovered experimentally by Galileo in the early 17th century, who performed experiments with balls rolling on inclined planes. He also later proved this mathematically in his book Dialogue Concerning Two New Sciences. For objects extended in space, such as a diver jumping from a diving board, the object itself follows a complex motion as it rotates, but the center of mass of the object nevertheless forms a parabola. As in all cases in the physical world, the trajectory is always an approximation of a parabola. The presence of air resistance, for example, always distorts the shape, although at low speeds, the shape is a good approximation of a parabola. At higher speeds, such as in ballistics, the shape is highly distorted and does not resemble a parabola.

Another hypothetical situation in which parabolas might arise, according to the theories of physics described in the 17th and 18th centuries by Sir Isaac Newton, is in two-body orbits; for example the path of a small planetoid or other object under the influence of the gravitation of the Sun. Parabolic orbits do not occur in nature; simple orbits most commonly resemble hyperbolas or ellipses. The parabolic orbit is the degenerate intermediate case between those two types of ideal orbit. An object following a parabolic orbit would travel at the exact escape velocity of the object it orbits; objects in elliptical or hyperbolic orbits travel at less or greater than escape velocity, respectively. Long-period comets travel close to the Sun's escape velocity while they are moving through the inner solar system, so their paths are close to being parabolic.

Approximations of parabolas are also found in the shape of the main cables on a simple suspension bridge. The curve of the chains of a suspension bridge is always an intermediate curve between a parabola and a catenary, but in practice the curve is generally nearer to a parabola, and in calculations the second degree parabola is used. Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise catenary-shaped cable is deformed toward a parabola. 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, e.g. bending, forces. Similarly, the structures of parabolic arches are purely in compression.

Paraboloids arise in several physical situations as well. The best-known instance is the parabolic reflector, which is a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point, or conversely, collimates light from a point source at the focus into a parallel beam. The principle of the parabolic reflector may have been discovered in the 3rd century BC by the geometer Archimedes, who, according to a legend of debatable veracity, constructed parabolic mirrors to defend Syracuse against the Roman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to telescopes in the 17th century. Today, paraboloid reflectors can be commonly observed throughout much of the world in microwave and satellite-dish receiving and transmitting antennas.

In parabolic microphones, a parabolic reflector that reflects sound, but not necessarily electromagnetic radiation, is used to focus sound onto a microphone, giving it highly directional performance.

Paraboloids are also observed in the surface of a liquid confined to a container and rotated around the central axis. In this case, the centrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind the liquid mirror telescope.

Aircraft used to create a weightless state for purposes of experimentation, such as NASA's "Vomit Comet", follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall, which produces the same effect as zero gravity for most purposes.

In the United States, vertical curves in roads are usually parabolic by design.

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