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Tuesday, May 19, 2015

Parabola

From Wikipedia, the free encyclopedia

A parabola (/pəˈræbələ/; plural parabolas or parabolae, adjective parabolic, from Greek: παραβολή) is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, but which can be in any orientation in its plane. It fits any of several superficially different mathematical descriptions which can all be proved to define curves of exactly the same shape.

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 both 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 which is parallel to another plane which is tangential to the conical surface.[a] A third description is algebraic. A parabola is a graph of a quadratic function, such as y=x^2.

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 on the axis of symmetry that intersects the parabola is called the "vertex", and it is the point where the curvature is greatest. 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 which 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 which 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 forms of energy. 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 to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas.

Strictly, the adjective parabolic should be applied only to things that are shaped as a parabola, which is a two-dimensional shape. However, as shown in the last paragraph, the same adjective is commonly used for three-dimensional objects, such as parabolic reflectors, which are really paraboloids. Sometimes, the noun parabola is also used to refer to these objects. Though not perfectly correct, this usage is generally understood.


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.

Introductory images

  • Parabolic Cartesian graph of the function y=6x2+4x-8
  • A parabola obtained as the intersection of a cone with a (red) plane parallel to a (checkered) plane which is tangential to the cone's surface.
  • Parabolic curve showing directrix (L) and focus (F). The distance from any point on the parabola to the focus (PnF) equals the perpendicular distance from the same point on the parabola to the directrix (PnQn).
  • For description, see text below.

Description of final image

Parabolic curve showing chord (L), focus (F), and vertex (V). L is an arbitrary chord of the parabola perpendicular to its axis of symmetry, which passes through V and F. (The ends of the chord are not shown here.) The lengths of all paths Qn - Pn - F are the same, equalling the distance between the chord L and the directrix. (See previous image.) This is similar to saying that a parabola is an ellipse, but with one focal point at infinity. It also directly implies, by the wave nature of light, that parallel light arriving along the lines Qn - Pn will be reflected to converge at F. A linear wavefront along L is concentrated, after reflection, to the one point where all parts of it have travelled equal distances and are in phase, namely F. No consideration of angles is required

History


Parabolic compass designed by Leonardo da Vinci

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 imposed by 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.[1] 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.[2] Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne,[3] and James Gregory.[4] 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.[5]

Equation in Cartesian coordinates

Let the directrix be the line x = −p and let the focus be the point (p, 0). If (x, y) is a point on the parabola then, by definition of a parabola, it is the same distance from the directrix as the focus; in other words:
|x+p|=\sqrt{(x-p)^2+y^2}
Squaring both sides and simplifying produces
y^2 = 4px\
as the equation of the parabola. By interchanging the roles of x and y one obtains the corresponding equation of a parabola with a vertical axis as
x^2 = 4py \
The equation can be generalized to allow the vertex to be at a point other than the origin by defining the vertex as the point (h, k). The equation of a parabola with a vertical axis then becomes
(x-h)^{2}=4p(y-k) \,
The last equation can be rewritten
y=ax^2+bx+c\,
so the graph of any function which is a polynomial of degree 2 in x is a parabola with a vertical axis.

More generally, a parabola is a curve in the Cartesian plane defined by an irreducible equation — one that does not factor as a product of two not necessarily distinct linear equations — of the general conic form
A x^{2} + B xy + C y^{2} + D x + E y + F = 0 \,
with the parabola restriction that
B^{2} = 4 AC \,
where all of the coefficients are real and where A and C are not both zero. The equation is irreducible if and only if the determinant of the 3×3 matrix
\begin{bmatrix} A & B/2 & D/2 \\ B/2 & C & E/2 \\ D/2 & E/2 & F \end{bmatrix}
is non-zero: that is, if (AC − B2/4)F + BED/4 − CD2/4 − AE2/4 ≠ 0. The reducible case, also called the degenerate case, gives a pair of parallel lines, possibly real, possibly imaginary, and possibly coinciding with each other.[6]

Conic section and quadratic form

Cone with cross-sections (To enlarge, click on diagram. To shrink, go to previous page.)
Cone with cross-sections (To enlarge, click on diagram. To shrink, go to previous page.)
The diagram represents a cone with its axis vertical.[b] The point A is its apex. A horizontal cross-section of the cone passes through the points B, E, C, and D. This cross-section is circular, but appears elliptical when viewed obliquely, as is shown in the diagram. 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. The cone also has another horizontal cross-section, which passes through the vertex, P, of the parabola, and is also circular, with a radius which we will call r. Its centre is V, and PK is a diameter. The chord BC is a diameter of the lower circle, and passes through the point M, which is the midpoint of the chord ED. Let us call the lengths of the line segments EM and DM x, and the length of PM y.

Thus:
BM=2y\sin{\theta}.   (The triangle BPM is isosceles.)
CM=2r.   (PMCK is a parallelogram.)
Using the intersecting chords theorem on the chords BC and DE, we get:
EM \cdot DM=BM \cdot CM
Substituting:
x^2=4ry\sin{\theta}
Rearranging:
y=\frac{x^2}{4r\sin{\theta}}
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 is a simple quadratic one which describes how x and y are related to each other, and therefore defines the shape of the parabolic curve. This shows that the definition of a parabola as a conic section implies its definition as the graph of a quadratic function. Both definitions produce curves of exactly the same shape.

Focal length

It is proved below that if a parabola has an equation of the form y=ax^2 where a is a positive constant, then a=\frac{1}{4f} where f is its focal length. Comparing this with the last equation above shows that the focal length of the above parabola is r\sin{\theta}.

Position of the focus

In the diagram, 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 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 position of the focus is at F.

Other geometric definitions

A parabola may also be characterized as a conic section with an eccentricity of 1. As a consequence of this, all parabolas are similar, meaning that while they can be different sizes, they are all the same shape. Another consequence is that the universal parabolic constant is the same for all parabolas.[7] A parabola can also be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction. In this sense, a parabola may be considered an ellipse that has one focus at infinity. The parabola is an inverse transform of a cardioid.

A parabola has a single axis of reflective symmetry, which passes through its focus and is perpendicular to its directrix. The point of intersection of this axis and the parabola is called the vertex. A parabola spun about this axis in three dimensions traces out a shape known as a paraboloid of revolution.

The parabola is found in numerous situations in the physical world (see below).

Equations

Cartesian

In the following equations h and k are the coordinates of the vertex (h,k) of the parabola and p is the distance from the vertex to the focus and the vertex to the directrix.

Vertical axis of symmetry

(x - h)^2 = 4p(y - k) \,
y =\frac{(x-h)^2}{4p}+k\,
y = ax^2 + bx + c \,
where
a = \frac{1}{4p}; \ \ b = \frac{-h}{2p}; \ \ c = \frac{h^2}{4p} + k; \ \
h = \frac{-b}{2a}; \ \ k = \frac{4ac - b^2}{4a}.
Parametric form:
x(t) = 2pt + h; \ \ y(t) = pt^2 + k \,

Horizontal axis of symmetry

(y - k)^2 = 4p(x - h) \,
x =\frac{(y - k)^2}{4p} + h;\ \,
x = ay^2 + by + c \,
where
a = \frac{1}{4p}; \ \ b = \frac{-k}{2p}; \ \ c = \frac{k^2}{4p} + h; \ \
h = \frac{4ac - b^2}{4a}; \ \ k = \frac{-b}{2a}.
Parametric form:
x(t) = pt^2 + h; \ \ y(t) = 2pt + k \,

General parabola

The general form for a parabola is
(\alpha x+\beta y)^2 + \gamma x + \delta y + \epsilon = 0 \,
This result is derived from the general conic equation given below:
Ax^2 +Bxy + Cy^2 + Dx + Ey + F = 0 \,
and the fact that, for a parabola,
B^2=4AC \,.
The equation for a general parabola with a focus point F(u, v), and a directrix in the form
ax+by+c=0 \,
is
\frac{\left(ax+by+c\right)^2}{{a}^{2}+{b}^{2}}=\left(x-u\right)^2+\left(y-v\right)^2 \,

Latus rectum, semilatus rectum, and polar coordinates

In Polar coordinate system, a parabola with the focus at the origin and the directrix parallel to the y-axis, is given by the equation
r (1 + \cos \theta) = l \,
where l is the semilatus rectum: the distance from the focus to the parabola itself, measured along a line perpendicular to the axis of symmetry. Note that this equals the perpendicular distance from the focus to the directrix, and is twice the focal length, which is the distance from the focus to the vertex of the parabola.
The latus rectum is the chord that passes through the focus and is perpendicular to the axis of symmetry. It has a length of 2l.

Dimensions of parabolas with axes of symmetry parallel to the y-axis

These parabolas have equations of the form y=ax^2+bx+c . By interchanging x and y the parabolas' axes of symmetry become parallel to the x-axis.
Some features of a parabola
Some features of a parabola

Coordinates of the vertex

The x-coordinate at the vertex can be found by completing the square to put the equation y=ax^2+bx+c in vertex form, or by differentiating the original equation, setting the resulting \frac{dy}{dx}=2ax+b equal to zero (a critical point), and solving for x. Both methods yield: x=\frac{-b}{2a}.
Substituting this into the original equation yields:
y=a\left (-\frac{b}{2a}\right )^2 + b \left ( -\frac{b}{2a} \right ) + c
=\frac{ab^2}{4a^2} -\frac{b^2}{2a} + c = c - \frac{b^2}{4a}
These terms can be combined over a common denominator:
y= \frac{4ac-b^2}{4a}=-\frac{b^2-4ac}{4a}=-\frac{D}{4a}, where D=(b^2-4ac) is the discriminant.
Thus, the vertex is at the point \left (-\frac{b}{2a},-\frac{D}{4a}\right ).

Coordinates of the focus

Since the axis of symmetry of this parabola is parallel with the y-axis, the x-coordinates of the focus and the vertex are equal. The coordinates of the vertex are calculated in the preceding section. The x-coordinate of the focus is therefore also -\frac{b}{2a}.

To find the y-coordinate of the focus, consider the point, P, located on the parabola where the slope is 1, so the tangent to the parabola at P is inclined at 45 degrees to the axis of symmetry. Using the reflective property of a parabola, we know that light which is initially travelling parallel to the axis of symmetry is reflected at P toward the focus. The 45-degree inclination causes the light to be turned 90 degrees by the reflection, so it travels from P to the focus along a line that is perpendicular to the axis of symmetry and to the y-axis. This means that the y-coordinate of P must equal that of the focus.

By differentiating the equation of the parabola and setting the slope to 1, we find the x-coordinate of P:
y=ax^2+bx+c,
\frac{dy}{dx}=2ax+b=1
\therefore x=\frac{1-b}{2a}
Substituting this value of x in the equation of the parabola, we find the y-coordinate of P, and also of the focus:
y=a\left(\frac{1-b}{2a}\right)^2+b\left(\frac{1-b}{2a}\right)+c
=a\left(\frac{1-2b+b^2}{4a^2}\right)+\left(\frac{b-b^2}{2a}\right)+c
=\left(\frac{1-2b+b^2}{4a}\right)+\left(\frac{2b-2b^2}{4a}\right)+c
=\frac{1-b^2}{4a}+c=\frac{1-(b^2-4ac)}{4a}=\frac{1-D}{4a}
where D=(b^2-4ac) is the discriminant, as used in the "Coordinates of the vertex" section.

The focus is therefore the point:
\left(-\frac{b}{2a},\frac{1-D}{4a}\right)

Axis of symmetry, focal length, latus rectum, and directrix

The above coordinates of the focus of a parabola of the form:
y=ax^2+bx+c
can be compared with the coordinates of its vertex, which are derived in the section "Coordinates of the vertex", above, and are:
\left(\frac{-b}{2a},\frac{-D}{4a}\right)
where D=b^2-4ac.

The axis of symmetry is the line which passes through both the focus and the vertex. In this case, it is vertical, with equation:
x=-\frac{b}{2a}.
The focal length of the parabola is the difference between the y-coordinates of the focus and the vertex:
f=\left(\frac{1-D}{4a}\right)-\left(\frac{-D}{4a}\right)
=\frac{1}{4a}
It is sometimes useful to invert this equation and use it in the form: a=\frac{1}{4f}. See the section "Conic section and quadratic form", above.

The point where the slope of the parabola is 1 lies at one end of the latus rectum. The length of the semilatus rectum (half of the latus rectum) is the difference between the x-coordinates of this point, which is considered as P in the above derivation of the coordinates of the focus, and of the focus itself. Thus, the length of the semilatus rectum is:
\frac{1-b}{2a}+\frac{b}{2a}
=\frac{1}{2a}
=2f, where f is the focal length.
The total length of the latus rectum is therefore four times the focal length.

Measured along the axis of symmetry, the vertex is the midpoint between the focus and the directrix. Therefore, the equation of the directrix is:
y=-\frac{D}{4a}-\frac{1}{4a}=-\frac{1+D}{4a}

Proof of the reflective property


Reflective property of a parabola

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 caption to a diagram near the top 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=x^2. 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,x^2). 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. \frac{{x}}{{2}}. The slope of the line BE is the quotient of the lengths of ED and BD, which is \frac{x^2}{\left(\frac{x}{2}\right)}, 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 \alpha 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


Perpendicular from focus to tangent

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.[8]

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


Parabola and tangent

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 \alpha are equal. The reflective property follows as shown previously.

Tangent properties

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.[9]:p.26

Orthoptic property


Perpendicular tangents intersect on the directrix

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=ax^2, \ a\ne 0. Suppose that two tangents contact this parabola at the points (p,ap^2) and (q,aq^2). 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,ap^2), the value of C must be -ap^2, so the equation of this tangent is y=2apx-ap^2. Likewise, the equation of the other tangent is y=2aqx-aq^2. At the intersection point of the two tangents, 2apx-ap^2=2aqx-aq^2. Thus 2x(p-q)=p^2-q^2. Factoring the difference of squares, cancelling, and dividing by 2 gives x=\frac{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\left(\frac{p+q}{2}\right)-ap^2. 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=-\frac{1}{4a^2}. Thus the y-coordinate of the intersection point of the tangents is given by y=-\frac{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.[8]:Corollary 20 [10]

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.[11]

Facts related to chords

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=\frac{c^2}{16d}
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).

Area enclosed between a parabola and a chord

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.
Parabola and line including chord.
The area enclosed between a parabola and a chord (see diagram) is two-thirds of the area of a parallelogram which surrounds it. One side of the parallelogram is the chord, and the opposite side is a tangent to the parabola.[12][13] 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 by Archimedes in the 3rd Century BCE. He used the areas of triangles, rather than that of the parallelogram.[d] See the article "The Quadrature of the Parabola".

If the chord has length b, and is perpendicular to the parabola's axis of symmetry, and if the perpendicular distance from the parabola's vertex to the chord is h, the parallelogram is a rectangle, with sides of b and h. The area, A, of the parabolic segment enclosed by the parabola and the chord is therefore:
A=\frac{2}{3}bh
This formula can be compared with the area of a triangle: \frac{1}{2}bh.

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 with 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 in the article "Distance 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 \textstyle\frac{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 which is 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.[f]

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.[g]
h=\frac{p}{2}
q=\sqrt{f^2+h^2}
s=\frac{hq}{f}+f\ln\left(\frac{h+q}{f}\right)
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 = \frac{h_1 q_1 - h_2 q_2}{f} +f \ln \left(\frac{h_1 + q_1}{h_2 + q_2}\right)
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
  • Image is inverted. A"B" is x-axis. C is origin. O is centre. 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.
  • Concave mirror.svg
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
\therefore x^2+R^2-2Ry+y^2=R^2
\therefore x^2+y^2=2Ry.

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 y^2 is negligible compared with the other terms. Therefore, extremely close to the origin:
x^2=2Ry......(Equation 1)
Compare this with the parabola:
x^2=4fy......(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.

Mathematical generalizations

In algebraic geometry, the parabola is generalized by the rational normal curves, which have coordinates (x,x^2,x^3,\dots,x^n); 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 x^2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x^2+y^2 (or scalings) and the hyperbolic paraboloid is the graph of the indefinite quadratic form x^2-y^2. 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 baseball 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.[14][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 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.[15][16] 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,[17] 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.

Gallery

  • A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and air 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) of Comet Kohoutek as it passed through the inner solar system, showing its nearly parabolic shape. The blue orbit is the Earth's
  • Hercilio Luz Bridge, Florianópolis, Brazil. The supporting cables of suspension bridges follow a curve which is intermediate between a parabola and a catenary.[15]
  • The Rainbow Bridge across the Niagara River, connecting Canada (left) to the United States (right). The parabolic arch is in compression, and carries the weight of the road.
  • Parabolic arches used in architecture
  • 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. (See Rotating furnace)
  • Parabolic microphone with optically transparent plastic reflector, used to overhear referee conversations at an American college football game.
  • Array of parabolic troughs to collect solar energy
  • Edison's searchlight, mounted on a cart. The light had a parabolic reflector.
  • Physicist Stephen Hawking in an aircraft flying a parabolic trajectory to produce zero-gravity
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Hyperbolic growth


From Wikipedia, the free encyclopedia


The reciprocal function, exhibiting hyperbolic growth.

When a quantity grows towards a singularity under a finite variation (a "finite-time singularity") it is said to undergo hyperbolic growth.[1] More precisely, the reciprocal function 1/x has a hyperbola as a graph, and has a singularity at 0, meaning that the limit as x \to 0 is infinite: any similar graph is said to exhibit hyperbolic growth.

Description

If the output of a function is inversely proportional to its input, or inversely proportional to the difference from a given value x_0, the function will exhibit hyperbolic growth, with a singularity at x_0.

In the real world hyperbolic growth is created by certain non-linear positive feedback mechanisms.[2]

Comparisons with other growth

Like exponential growth and logistic growth, hyperbolic growth is highly nonlinear, but differs in important respects. These functions can be confused, as exponential growth, hyperbolic growth, and the first half of logistic growth are convex functions; however their asymptotic behavior (behavior as input gets large) differs dramatically:
  • logistic growth is constrained (has a finite limit, even as time goes to infinity),
  • exponential growth grows to infinity as time goes to infinity (but is always finite for finite time),
  • hyperbolic growth has a singularity in finite time (grows to infinity at a finite time).

Applications

Population

Certain mathematical models suggest that until the early 1970s the world population underwent hyperbolic growth (see, e.g., Introduction to Social Macrodynamics by Andrey Korotayev et al.). It was also shown that until the 1970s the hyperbolic growth of the world population was accompanied by quadratic-hyperbolic growth of the world GDP, and developed a number of mathematical models describing both this phenomenon, and the World System withdrawal from the blow-up regime observed in the recent decades. The hyperbolic growth of the world population and quadratic-hyperbolic growth of the world GDP observed till the 1970s have been correlated by Andrey Korotayev and his colleagues to a non-linear second order positive feedback between the demographic growth and technological development, described by a chain of causation: technological growth leads to more carrying capacity of land for people, which leads to more people, which leads to more inventors, which in turn leads to yet more technological growth, and on and on.[3] Other models suggest exponential growth, logistic growth, or other functions.

Queuing theory

Another example of hyperbolic growth can be found in queueing theory: the average waiting time of randomly arriving customers grows hyperbolically as a function of the average load ratio of the server. The singularity in this case occurs when the average amount of work arriving to the server equals the server's processing capacity. If the processing needs exceed the server's capacity, then there is no well-defined average waiting time, as the queue can grow without bound. A practical implication of this particular example is that for highly loaded queuing systems the average waiting time can be extremely sensitive to the processing capacity.

Enzyme kinetics

A further practical example of hyperbolic growth can be found in enzyme kinetics. When the rate of reaction (termed velocity) between an enzyme and substrate is plotted against various concentrations of the substrate, a hyperbolic plot is obtained for many simpler systems. When this happens, the enzyme is said to follow Michaelis-Menten kinetics.

Mathematical example

The function
x(t)=\frac{1}{t_c-t}
exhibits hyperbolic growth with a singularity at time t_c: in the limit as t \to t_c, the function goes to infinity.

More generally, the function
x(t)=\frac{K}{t_c-t}
exhibits hyperbolic growth, where K is a scale factor.

Note that this algebraic function can be regarded as analytical solution for the function's differential:[4]
\frac{dx}{dt}=\frac{K}{(t_c-t)^2}=\frac{x^2}{K}
This means that with hyperbolic growth the absolute growth rate of the variable x in the moment t is proportional to the square of the value of x in the moment t.

Respectively, the quadratic-hyperbolic function looks as follows:
x(t)=\frac{K}{(t_c-t)^2}.
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Vortex Bladeless aims for lower-cost wind energy approach

17 hours ago by Nancy Owano weblog
Original link:  http://phys.org/news/2015-05-vortex-bladeless-aims-lower-cost-energy.html


Vortex Bladeless aims for lower-cost wind energy approach

A technology leap forward in wind energy? Or, as the company in charge calls it, a "new paradigm" of wind power, lowering costs, requiring no training, using fewer supplies? They believe they have a great idea and they aim to bring it to market. They will start crowdfunding next month. The company is Vortex Bladeless.

David Yanez, co-founder, Vortex Bladeless, said it all began with a bridge disaster. The bridge, he said, started swaying and oscillating in heavy winds. This was the Tacoma Narrows Bridge collapse of 1940, and the event occurred under 40-mile-per-hour (64 km/h) wind conditions. The collapse would in later years continue to be a topic among engineers and scientists discussing the aeroelastic flutter and motivating their research in aeroelastics.

The structure, said Yanez, was caught up in aeroelastic coupling. Yanez and team worked on recreating similar conditions to lead to their development of a bladeless wind turbine. Instead of turning, the turbine oscillates, producing movement and displacement, said Yanez. "The system is based on the same principles as an alternator—electromagnetic induction." They multiply that movement and speed magnetically—without any gear assemblies or ball bearings. They turn the mechanical energy of the structure into electricity.

Writing in Treehugger, Derek Markham, commented : "The Vortex wind generator represents a fairly radical break with conventional wind turbine design, in that it has no spinning blades (or any moving parts to wear out at all), and looks like nothing more than a giant straw that oscillates in the wind. It works not by spinning in the wind, but by taking advantage of a phenomenon called vorticity, or the Kármán street, which is a 'repeating pattern of swirling vortices.'"

Raul Martin, Vortex Bladeless co-founder, said, "Compare our invention to a conventional wind turbine with similar energy generation—ours would cost significantly less," around 50 percent or 47 percent less. The company site said that Vortex saves 53 percent in manufacturing costs and 51 percent in operating costs compared to conventional .

"Because there is no contact between moving parts," said the Vortex site, "there is no friction. Therefore no lubricant is required."

Graham in Treehugger said another advantage was that "the devices can be used to generate more power in less space, because not only is the wind wake narrower than a traditional turbine, but installing them closer together can actually be beneficial to the technology, based on tunnel testing."

The next step? Martin said the step will be to develop a small 4-kilowatt turbine for small businesses, distributed energy grids or individual homes.

Explore further: Dutch company launches new-generation urban wind turbines

More information: vortexbladeless.com/home.php
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Feature: Solving the mystery of dog domestication

Original link:  http://news.sciencemag.org/archaeology/2015/04/feature-solving-mystery-dog-domestication


David is the Online News Editor of Science.
 
 
By David Grimm

Dogs were the first thing humans domesticated—before any plant, before any other animal. Yet scientists have argued for years over where and when they arose. Some studies suggest that canines evolved in Europe, others Asia, with time frames ranging from 15,000 to more than 30,000 years ago. Now, an unprecedented collaboration of archaeologists and geneticists has brought the warring camps together for the first time. The group is analyzing thousands of bones from around the world, employing new techniques, and trying to put aside years of bad blood and bruised egos. If it succeeds, it will uncover the history of man's oldest friend—and solve one of the greatest mysteries of domestication.

To read the full story, see the 17 April issue of Science.
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Exponential growth


From Wikipedia, the free encyclopedia


The graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth.
  Exponential growth
  Linear growth
  Cubic growth

Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value. Exponential decay occurs in the same way when the growth rate is negative. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay (the function values form a geometric progression).

The formula for exponential growth of a variable x at the (positive or negative) growth rate r, as time t goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is
x_t = x_0(1+r)^t
where x0 is the value of x at time 0. For example, with a growth rate of r = 5% = 0.05, going from any integer value of time to the next integer causes x at the second time to be 1.05 times (i.e., 5% larger than) what it was at the previous time.

Examples


Bacteria exhibit exponential growth under optimal conditions.
  • Biology
    • The number of microorganisms in a culture will increase exponentially until an essential nutrient is exhausted. Typically the first organism splits into two daughter organisms, who then each split to form four, who split to form eight, and so on.
    • A virus (for example SARS, or smallpox) typically will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people.
    • Human population, if the number of births and deaths per person per year were to remain at current levels (but also see logistic growth). For example, according to the United States Census Bureau, over the last 100 years (1910 to 2010), the population of the United States of America is exponentially increasing at an average rate of one and a half percent a year (1.5%). This means that the doubling time of the American population (depending on the yearly growth in population) is approximately 50 years.[1]
    • Many responses of living beings to stimuli, including human perception, are logarithmic responses, which are the inverse of exponential responses; the loudness and frequency of sound are perceived logarithmically, even with very faint stimulus, within the limits of perception. This is the reason that exponentially increasing the brightness of visual stimuli is perceived by humans as a linear increase, rather than an exponential increase. This has survival value. Generally it is important for the organisms to respond to stimuli in a wide range of levels, from very low levels, to very high levels, while the accuracy of the estimation of differences at high levels of stimulus is much less important for survival.
    • Genetic complexity of life on Earth has doubled every 376 million years. Extrapolating this exponential growth backwards indicates life began 9.7 billion years ago, potentially predating the Earth by 5.2 billion years.[2][3]
  • Physics
    • Avalanche breakdown within a dielectric material. A free electron becomes sufficiently accelerated by an externally applied electrical field that it frees up additional electrons as it collides with atoms or molecules of the dielectric media. These secondary electrons also are accelerated, creating larger numbers of free electrons. The resulting exponential growth of electrons and ions may rapidly lead to complete dielectric breakdown of the material.
    • Nuclear chain reaction (the concept behind nuclear reactors and nuclear weapons). Each uranium nucleus that undergoes fission produces multiple neutrons, each of which can be absorbed by adjacent uranium atoms, causing them to fission in turn. If the probability of neutron absorption exceeds the probability of neutron escape (a function of the shape and mass of the uranium), k > 0 and so the production rate of neutrons and induced uranium fissions increases exponentially, in an uncontrolled reaction. "Due to the exponential rate of increase, at any point in the chain reaction 99% of the energy will have been released in the last 4.6 generations. It is a reasonable approximation to think of the first 53 generations as a latency period leading up to the actual explosion, which only takes 3–4 generations."[4]
    • Positive feedback within the linear range of electrical or electroacoustic amplification can result in the exponential growth of the amplified signal, although resonance effects may favor some component frequencies of the signal over others.
    • Heat transfer experiments yield results whose best fit line are exponential decay curves.
  • Economics
    • Economic growth is expressed in percentage terms, implying exponential growth. For example, U.S. GDP per capita has grown at an exponential rate of approximately two percent since World War 2.[citation needed]
  • Finance
  • Computer technology
    • Processing power of computers. See also Moore's law and technological singularity. (Under exponential growth, there are no singularities. The singularity here is a metaphor.)
    • In computational complexity theory, computer algorithms of exponential complexity require an exponentially increasing amount of resources (e.g. time, computer memory) for only a constant increase in problem size. So for an algorithm of time complexity 2x, if a problem of size x = 10 requires 10 seconds to complete, and a problem of size x = 11 requires 20 seconds, then a problem of size x = 12 will require 40 seconds. This kind of algorithm typically becomes unusable at very small problem sizes, often between 30 and 100 items (most computer algorithms need to be able to solve much larger problems, up to tens of thousands or even millions of items in reasonable times, something that would be physically impossible with an exponential algorithm). Also, the effects of Moore's Law do not help the situation much because doubling processor speed merely allows you to increase the problem size by a constant. E.g. if a slow processor can solve problems of size x in time t, then a processor twice as fast could only solve problems of size x+constant in the same time t. So exponentially complex algorithms are most often impractical, and the search for more efficient algorithms is one of the central goals of computer science today.
    • Internet traffic growth.[citation needed]

Basic formula

A quantity x depends exponentially on time t if
x(t)=a\cdot b^{t/\tau}\,
where the constant a is the initial value of x,
x(0)=a\, ,
the constant b is a positive growth factor, and τ is the time constant—the time required for x to increase by one factor of b:
x(t+\tau)=a \cdot b^{\frac{t+\tau}{\tau}} = a \cdot b^{\frac{t}{\tau}} \cdot b^{\frac{\tau}{\tau}} = x(t)\cdot b\, .
If τ > 0 and b > 1, then x has exponential growth. If τ < 0 and b > 1, or τ > 0 and 0 < b < 1, then x has exponential decay.

Example: If a species of bacteria doubles every ten minutes, starting out with only one bacterium, how many bacteria would be present after one hour? The question implies a = 1, b = 2 and τ = 10 min.
x(t)=a\cdot b^{t/\tau}=1\cdot 2^{(60\text{ min})/(10\text{ min})}
x(1\text{ hr})= 1 \cdot 2^6 =64.
After one hour, or six ten-minute intervals, there would be sixty-four bacteria.

Many pairs (b, τ) of a dimensionless non-negative number b and an amount of time τ (a physical quantity which can be expressed as the product of a number of units and a unit of time) represent the same growth rate, with τ proportional to log b. For any fixed b not equal to 1 (e.g. e or 2), the growth rate is given by the non-zero time τ. For any non-zero time τ the growth rate is given by the dimensionless positive number b.

Thus the law of exponential growth can be written in different but mathematically equivalent forms, by using a different base. The most common forms are the following:
x(t) = x_0\cdot e^{kt} = x_0\cdot e^{t/\tau} = x_0 \cdot 2^{t/T} = x_0\cdot \left( 1 + \frac{r}{100} \right)^{t/p},
where x0 expresses the initial quantity x(0).

Parameters (negative in the case of exponential decay):
The quantities k, τ, and T, and for a given p also r, have a one-to-one connection given by the following equation (which can be derived by taking the natural logarithm of the above):
k = \frac{1}{\tau} = \frac{\ln 2}{T} = \frac{\ln \left( 1 + \frac{r}{100} \right)}{p}\,
where k = 0 corresponds to r = 0 and to τ and T being infinite.

If p is the unit of time the quotient t/p is simply the number of units of time. Using the notation t for the (dimensionless) number of units of time rather than the time itself, t/p can be replaced by t, but for uniformity this has been avoided here. In this case the division by p in the last formula is not a numerical division either, but converts a dimensionless number to the correct quantity including unit.

A popular approximated method for calculating the doubling time from the growth rate is the rule of 70, i.e. T \simeq 70 / r.

Reformulation as log-linear growth

If a variable x exhibits exponential growth according to x(t)=x_0(1+r)^t, then the log (to any base) of x grows linearly over time, as can be seen by taking logarithms of both sides of the exponential growth equation:
\log x(t) = \log x_0 + t \cdot \log (1+r).
This allows an exponentially growing variable to be modeled with a log-linear model. For example, if one wishes to empirically estimate the growth rate from intertemporal data on x, one can linearly regress log x on t.

Differential equation

The exponential function \scriptstyle x(t)=x(0) e^{kt} satisfies the linear differential equation:
\!\, \frac{dx}{dt} = kx
saying that the growth rate of x at time t is proportional to the value of x(t), and it has the initial value
x(0).\,
The differential equation is solved by direct integration:
\frac{dx}{dt} = kx
\Rightarrow \frac{dx}{x} = k\, dt
\Rightarrow \int_{x(0)}^{x(t)} \frac{dx}{x} = k \int_0^t \, dt
\Rightarrow \ln \frac{x(t)}{x(0)} = kt .
so that
\Rightarrow x(t) = x(0) e^{kt}\,

For a nonlinear variation of this growth model see logistic function.

Difference equation

The difference equation
x_t = a \cdot x_{t-1}
has solution
x_t = x_0 \cdot a^t,
showing that x experiences exponential growth.

Other growth rates

In the long run, exponential growth of any kind will overtake linear growth of any kind (the basis of the Malthusian catastrophe) as well as any polynomial growth, i.e., for all α:
\lim_{t\rightarrow\infty} {t^\alpha \over ae^t} =0.
There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). See Degree of a polynomial#The degree computed from the function values.

Growth rates may also be faster than exponential.

In the above differential equation, if k < 0, then the quantity experiences exponential decay.

Limitations of models

Exponential growth models of physical phenomena only apply within limited regions, as unbounded growth is not physically realistic. Although growth may initially be exponential, the modelled phenomena will eventually enter a region in which previously ignored negative feedback factors become significant (leading to a logistic growth model) or other underlying assumptions of the exponential growth model, such as continuity or instantaneous feedback, break down.

Further information: Limits to Growth, Malthusian catastrophe, Apparent infection rate

Exponential stories

Rice on a chessboard

According to an old legend, vizier Sissa Ben Dahir presented an Indian King Sharim with a beautiful, hand-made chessboard. The king asked what he would like in return for his gift and the courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third etc. The king readily agreed and asked for the rice to be brought. All went well at first, but the requirement for 2 n − 1 grains on the nth square demanded over a million grains on the 21st square, more than a million million (aka trillion) on the 41st and there simply was not enough rice in the whole world for the final squares. (From Swirski, 2006)[5]
For variation of this see second half of the chessboard in reference to the point where an exponentially growing factor begins to have a significant economic impact on an organization's overall business strategy.

Water lily

French children are told a story in which they imagine having a pond with water lily leaves floating on the surface. The lily population doubles in size every day and if left unchecked will smother the pond in 30 days, killing all the other living things in the water. Day after day the plant seems small and so it is decided to leave it to grow until it half-covers the pond, before cutting it back. They are then asked on what day half-coverage will occur. This is revealed to be the 29th day, and then there will be just one day to save the pond. (From Meadows et al. 1972)[5]
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