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Circumference

Circumference

The circumference is the distance around a closed curve. Circumference is a kind of perimeter.

Circle

The circumference of a circle can be calculated from its diameter using the formula:

c = \pi d

Or, substituting the radius for the diameter:

c = 2r\pi

Where r is the radius and d is the diameter of the circle, and π (the Greek letter pi) is the constant 3.141 592 6...

Ellipse

The circumference of an ellipse is more problematical, as the exact solution requires finding the complete elliptic integral of the second kind. This can be achieved either via numerical integration (the best type being Gaussian quadrature) or by one of many binomial series expansions. Where

a,b

are the ellipse's semi-major and semi-minor axes, respectively, and

e\,\!

is the ellipse's eccentricity,

O\!z = \arcsin\!\left\=\arccos\!\left\\quad (\mbox\ modular\ angle\mbox\ angular\ eccentricity\ );\,\!

\operatorname\left[0,90^\circ\right]= \mbox's\mbox;

Pr=a\times\operatorname\left[0,90^\circ\right] \quad(\mbox);\,\!

c=2\pi\times Pr.\,\!

There are many different approximations for

\operatorname\left[0,90^\circ\right]

, with varying degrees of sophistication and corresponding accuracy. In comparing the different approximations, the

tan\!\left\^2\,\!

based series expansion is used to find the actual value:

Muir-1883

:Probably the most accurate to its given simplicity is Thomas Muir's: ::

Pr \approx \left[\frac\right]^\frac =a\left[\frac\right]^\frac,\,\!

::::

\approx a\times cos\!\left\^2\left[1+\fractan\!\left\^4\right];\,\!

Ramanujan-1914 (#1,#2)

:Srinivasa Ramanujan introduced two different approximations, both from 1914: ::

1.\ Pr \approx \frac\left[3(a+b) - \sqrt\right];\,\!

::::

=\fraca\left[6\cos\!\left\^2 sqrt\right];\,\!

2.\ Pr \approx\frac\left[a+b\right]\left[1+\frac\right];\,\!

:::

=a\times
cos\!\left\^2\left[1+\frac
\right];\,\!

:The second equation is by far the better of the two, and may be the most accurate approximation known. Letting a = 10000 and b = a×cos, results with different ellipticities can be found and compared:

External link

- [http://home.att.net/~numericana/answer/ellipse.htm#elliptic Numericana - Circumference of an ellipse] Category:Geometry ja:円周 th:เส้นรอบวง

Perimeter

The perimeter is the distance around a given two-dimensional object. The word perimeter is a Greek root meaning measure around, or literally "around measure." Perimeter can be calculated for any two dimensional object, with the right formula.

Practical Uses

Perimeter and area play a great role in today's world. Perimeter is used in calculating the border of an object such as a yard or flowerbed when a fence or other border is being installed around the edges. Area is used when all the area inside of a perimeter is being covered with something, such as a yard being covered with sod or fertilizer.

Formulas

As a general rule, the perimeter of a polygon can always be calculated by adding all the length of the sides together. Circles: For circles the equation is P= 2 π r, where r is the radius and π is the mathematical constant, or about 3.14. (An equivalent formula is P= π d, where d is the diameter). The equation for area is A= πr². R is the radius and pi is about 3.14. Pi comes from the equation c/d, which is the circumference of a circle divided by the diameter. Triangles: For triangles the equation for perimeter is P=Side1 + Side2 + Side3. The area can be calculated by the equation A=.5bh. B is the length of the base and H is the height of the triangle. Quadrilaterals: For quadrilaterals the equation for perimeter is P=Side1 + Side2 + Side3 + Side4. More area formulas can be found here.

Diameter

For the authentication, authorisation, and accounting protocol, see DIAMETER. In geometry, a diameter (Greek words diairo = divide and metro = measure) of a circle is any straight line segment that passes through the center and whose endpoints are on the circular boundary, or, in more modern usage, the length of such a line segment. When using the word in the more modern sense, one speaks of the diameter rather than a diameter, because all diameters of a circle have the same length. This length is twice the radius. The diameter of a circle is also the longest chord that the circle has. The diameter of a connected graph is the distance between the two vertices which are furthest from each other. The distance between two vertices a and b is the length of the shortest path connecting them (for the length of a path, see Graph theory). The two definitions given above are special cases of a more general definition. The diameter of a subset of a metric space is the least upper bound of the distances between pairs of points in the subset. So, if A is the subset, the diameter is :sup .

Diameter symbol

least upper bound The symbol or variable for diameter is similar in size and design to ø, the lowercase letter o with stroke. Unicode provides character number 8960 (hexadecimal 2300) for the symbol, which can be encoded in HTML webpages as ⌀ or ⌀. Proper display of this character, however, is unlikely in most situations, as most fonts do not have it included. (Your browser displays ⌀ and ⌀ in the current font.) In most situations the letter ø is acceptable, obtained in Microsoft Windows by holding the [Alt] key down while entering 0 2 4 8 on the numeric keypad. It is important not to confuse a diameter symbol (ø) with the empty set symbol, similar to the uppercase Ø. Diameter is also sometimes called phi (pronounced the same as "fie"), although this seems to come from the fact that Ø and ø look like Φ and φ, the letter phi in the Greek alphabet. See also: angular diameter, hydraulic diameter Category:Elementary geometry Category:Length ja:径

Pi

The mathematical constant π is the ratio of a circle's circumference (Greek περιφέρεια, periphery) to its diameter and is commonly used in mathematics, physics, and engineering. The name of the Greek letter π is pi (pronounced pie), and this spelling can be used in typographical contexts where the Greek letter is not available. π is also known as Archimedes' constant (not to be confused with Archimedes' number) and Ludolph's number. In Euclidean plane geometry, π may be defined either as the ratio of a circle's circumference to its diameter, or as the ratio of a circle's area to the area of a square whose side is the radius. Advanced textbooks define π analytically using trigonometric functions, for example as the smallest positive x for which sin(x) = 0, or as twice the smallest positive x for which cos(x) = 0. All these definitions are equivalent. The numerical value of π truncated to 50 decimal places is: :3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 Although this precision is more than sufficient for use in engineering and science, much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, in addition to supercomputer calculations that have determined over 1 trillion digits of π, no pattern in the digits has ever been found. Digits of π are available from multiple resources on the Internet, and a regular personal computer can compute billions of digits with available software.

Properties

π is an irrational number; that is, it cannot be written as the ratio of two integers, as was proven in 1761 by Johann Heinrich Lambert. π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with ruler and compass are constructible numbers, it is impossible to square the circle, that is, it is impossible to construct, using ruler and compass alone, a square whose area is equal to the area of a given circle.

Formulae involving π

Geometry

\pi

appears in many formulae in geometry involving circles and spheres. (All of these are a consequence of the first one, as the area of a circle can be written as A = ∫(2πr)dr ("sum of annuli of infinitesimal width"), and others concern a surface or solid of revolution.) Also, the angle measure of 180° (degrees) is equal to π radians.

Analysis

Many formulae in analysis contain π, including infinite series (and infinite product) representations, integrals, and so-called special functions.
- François Viète, 1593 (proof): :

\frac2\pi=
\frac2
\frac2
\frac2\ldots

- Leibniz' formula (proof): :

\frac - \frac + \frac - \frac + \frac - \cdots = \frac

:This commonly cited infinite series is usually written as above, but is more technically expressed as: :

\sum_^ \frac = \frac

- Wallis product (proof): :

\frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdots = \frac

\prod_^ \frac = \prod_^ \frac \cdot \frac = \frac

- Bailey-Borwein-Plouffe algorithm (See Bailey, 1997 and [http://www.nersc.gov/~dhbailey/ Bailey web page]) :

\pi=\sum_^\infty\frac\left [ \frac  - \frac  - \frac  - \frac \right ]

- An integral formula from calculus (see also Error function and Normal distribution): :

\int_^ e^\,dx = \sqrt

- Basel problem, first solved by Euler (see also Riemann zeta function): :

\zeta(2) = \frac + \frac + \frac + \frac + \cdots = \frac

\zeta(4)= \frac + \frac + \frac + \frac + \cdots = \frac

:and generally,

\zeta(2n)

is a rational multiple of

\pi^

for positive integer n
- Gamma function evaluated at 1/2: :

\Gamma\left(\right)=\sqrt

- Stirling's approximation: :

n! \sim \sqrt \left(\frac\right)^n

- Euler's identity (called by Richard Feynman "the most remarkable formula in mathematics"): :

e^ + 1 = 0\;

- Property of Euler's totient function (see also Farey sequence): :

\sum_^ \phi (k) \sim 3 n^2 / \pi^2

- Area of one quarter of the unit circle: :

\int_0^1 \sqrt\,dx =

- An application of the residue theorem :

\oint\frac=2\pi i ,

:where the path of integration is a circle around the origin, traversed in the standard (anti-clockwise) direction.

Continued fractions

π has many continued fractions representations, including: :

\frac = 1 + \frac

(Other representations are available at [http://functions.wolfram.com/Constants/Pi/10/ The Wolfram Functions Site].)

Number theory

Some results from number theory:
- The probability that two randomly chosen integers are coprime is 6/π².
- The probability that a randomly chosen integer is square-free is 6/π².
- The average number of ways to write a positive integer as the sum of two perfect squares (order matters) is π/4.
- The product of (1-1/p²) over the primes, p, is 6/π².

\prod_ \left(1-\frac   \right) = \frac

Here, "probability", "average", and "random" are taken in a limiting sense, e.g. we consider the probability for the set of integers , and then take the limit as N approaches infinity. The fact (note the order to which the number approaches an integer) that :

e^ = 262537412640768743.99999999999925007...

or equivalently, :

e^ = 640320^3+743.99999999999925007...

can be explained by the theory of complex multiplication.

Dynamical systems and ergodic theory

Consider the recurrence relation :

x_ = 4 x_i (1 - x_i) \,

Then for almost every initial value x₀ in the unit interval [0,1], :

\lim_ \frac \sum_^ \sqrt = \frac

This recurrence relation is the logistic map with parameter r = 4, known from dynamical systems theory. See also: ergodic theory.

Physics

The number π appears routinely in equations describing fundamental principles of the universe, due in no small part to its relationship to the nature of the circle and, correspondingly, spherical coordinate systems.
- The cosmological constant: :

\Lambda =  \rho

- Heisenberg's uncertainty principle: :

\Delta x \Delta p \ge \frac

- Einstein's field equation of general relativity: :

R_ -  + \Lambda g_ =  T_

- Coulomb's law for the electric force: :

F = \frac

- Magnetic permeability of free space: :

\mu_0 = 4 \pi \times 10^\,\mathrm\,

Probability and statistics

In probability and statistics, there are many distributions whose formulae contain π, including:
- probability density function (pdf) for the normal distribution with mean μ and standard deviation σ: :

f(x) = \,e^

- pdf for the (standard) Cauchy distribution: :

f(x) = \frac

Note that since

\int_^ f(x)\,dx = 1

, for any pdf f(x), the above formulae can be used to produce other integral formulae for π. An interesting empirical approximation of π is based on Buffon's needle problem. Consider dropping a needle of length L repeatedly on a surface containing parallel lines drawn S units apart (with S > L). If the needle is dropped n times and x of those times it comes to rest crossing a line (x > 0), then one may approximate π using: :

\pi \approx \frac

Another approximation of π is to [http://www.statisticool.com/pi.htm throw points randomly] into a quarter of a circle with radius 1 that is inscribed in a square of length 1. Pi, the area of a unit circle, is then approximated as 4
- (points in the quarter circle)/(total points).

History of π

Main article: History of Pi. π has been known in some form since antiquity. References to measurements of a circular basin in the Bible give a corresponding value of 3 for π: "And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about." — 1 Kings 7:23; KJV. Nehemiah, a late antique Jewish rabbi and mathematician explained this apparent lack of precision in π, by considering the thickness of the basin, and assuming that the thirty cubits was the inner circumference, while the ten cubits was the diameter of the outside of the basin.

Numerical approximations of π

Due to the transcendental nature of π, there are no closed expressions for the number in terms of algebraic numbers and functions. Therefore numerical calculations must use approximations of π. For many purposes, 3.14 or 22/7 is close enough, although engineers often use 3.1416 (5 significant figures) or 3.14159 (6 significant figures) for more accuracy. The approximations 22/7 and 355/113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction expansion of π. An Egyptian scribe named Ahmes wrote the oldest known text to give an approximate value for π. The Rhind Mathematical Papyrus dates from the Egyptian Second Intermediate Period—though Ahmes stated that he copied a Middle Kingdom papyrus—and describes the value in such a way that the result obtained comes out to 256 divided by 81 or 3.160. The Chinese mathematician Liu Hui computed π to 3.141014 (good to three decimal places) in AD 263 and suggested that 3.14 was a good approximation. The Indian mathematician and astronomer Aryabhata gave an accurate approximation for π. He wrote "Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words (4+100)×8 + 62000 is the circumference of a circle with diameter 20000. This provides a value of π = 62832/20000 = 3.1416, correct when rounded off to four decimal places. The Chinese mathematician and astronomer Zu Chongzhi computed π to 3.1415926 to 3.1415927 and gave two approximations of π 355/113 and 22/7 in the 5th century. The Iranian mathematician and astronomer, Ghyath ad-din Jamshid Kashani, 1350-1439, computed π to 9 digits in the base of 60, which is equivalent to 16 decimal digits as: :2 π = 6.2831853071795865 The German mathematician Ludolph van Ceulen (circa 1600) computed the first 35 decimals. He was so proud of this accomplishment that he had them inscribed on his tombstone. The Slovene mathematician Jurij Vega in 1789 calculated the first 140 decimal places for π of which the first 137 were correct and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct. Vega improved John Machin's formula from 1706 and his method is still mentioned today. None of the formulae given above can serve as an efficient way of approximating π. For fast calculations, one may use formulae such as Machin's: :

\frac = 4 \arctan\frac - \arctan\frac

together with the Taylor series expansion of the function arctan(x). This formula is most easily verified using polar coordinates of complex numbers, starting with :

(5+i)^4\cdot(-239+i)=-114244-114244i.

Formulae of this kind are known as Machin-like formulae. Extremely long decimal expansions of π are typically computed with the Gauss-Legendre algorithm and Borwein's algorithm; the Salamin-Brent algorithm which was invented in 1976 has also been used in the past. The first one million digits of π and 1/π are available from Project Gutenberg (see external links below). The current record (December 2002) by Yasumasa Kanada of Tokyo University stands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64-node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulae were used for this: :

\frac = 12 \arctan\frac + 32 \arctan\frac - 5 \arctan\frac + 12 \arctan\frac

:K. Takano (1982). :

\frac = 44 \arctan\frac + 7 \arctan\frac - 12 \arctan\frac + 24 \arctan\frac

:F. C. W. Störmer (1896). These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers and (obviously) for establishing new π calculation records. In 1997, David H. Bailey, Peter Borwein and Simon Plouffe published a paper (Bailey, 1997) on a new formula for π as an infinite series: :

\pi = \sum_^ \frac
\left( \frac - \frac - \frac - \frac\right)

This formula permits one to easily compute the k^th binary or hexadecimal digit of π, without having to compute the preceding k − 1 digits. [http://www.nersc.gov/~dhbailey/ Bailey's website] contains the derivation as well as implementations in various programming languages. The PiHex project computed 64-bits around the quadrillionth bit of π (which turns out to be 0). Other formulae that have been used to compute estimates of π include: :

\frac=
\sum_^\infty\frac=
1+\frac\left(1+\frac\left(1+\frac\left(1+\frac(1+...)\right)\right)\right)

:Newton. :

\frac = \frac \sum^\infty_ \frac

:Ramanujan. This converges extraordinarily rapidly. Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate π. :

\frac = 12 \sum^\infty_ \frac

:David Chudnovsky and Gregory Chudnovsky. :

= 20 \arctan\frac + 8 \arctan\frac

:Euler.

Miscellaneous formulae

In base 60, π can be approximated to eight significant figures as :

3 + \frac + \frac + \frac

In addition, the following expressions can be used to estimate π
- accurate to 9 digits: :

(63/25)((17+15\sqrt 5)/(7+15\sqrt5))

- accurate to 17 digits: :

3 + \frac

- accurate to 3 digits: :

\sqrt + \sqrt

:Karl Popper conjectured that Plato knew this expression; that he believed it to be exactly π; and that this is responsible for some of Plato's confidence in the omnicompetence of mathematical geometry — and Plato's repeated discussion of right triangles which are either isosceles or halves of equilateral triangles.

Less accurate approximations

In 1897, a physician and amateur mathematician from Indiana named Edward J. Goodwin believed that the transcendental value of π was wrong. He proposed a bill to Indiana Representative T. I. Record which expressed the "new mathematical truth" in several ways: :The ratio of the diameter of a circle to its circumference is 5/4 to 4. (π = 3.2) :The ratio of the length of a 90 degree arc to the length of a segment connecting the arc's two endpoints is 8 to 7. (π ≈ 3.23...) :The area of a circle equals the area of a square whose side is 1/4 the circumference of the circle. (π = 4) :It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square on one side. (π ≈ 9.24 if rectangle is emended to triangle; if not, as above.) The bill also recites Goodwin's previous accomplishments: "his solutions of the trisection of the angle, doubling the cube [and the value of π] having been already accepted as contributions to science by the American Mathematical Monthly....And be it remembered that these noted problems had been long since given up by scientific bodies as unsolvable mysteries and above man's ability to comprehend." These false claims are typical of a mathematical crank. The claims trisection of an angle and the doubling of the cube are particularly widespread in crank literature. The Indiana Assembly referred the bill to the Committee on Swamp Lands, which Petr Beckmann has seen as symbolic. It was transferred to the Committee on Education, which reported favorably, and the bill passed unanimously. One argument used was that Goodwin had copyrighted his discovery, and proposed to let the State use it in the public schools for free. As this debate concluded, Professor C. A. Waldo arrived in Indianapolis to secure the annual appropriation for the Indiana Academy of Sciences. An assemblyman handed him the bill, offering to introduce him to the genius who wrote it. He declined, saying that he already knew as many crazy people as he cared to. The Indiana Senate had not yet finally passed the bill (which they had referred to the Committee on Temperance), and Professor Waldo coached enough Senators overnight that they postponed the bill indefinitely. [http://faqs.jmas.co.jp/FAQs/sci-math-faq/indianabill source]

Open questions

The most pressing open question about π is whether it is a normal number -- whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely "randomly". This must be true in any base, not just in base 10. Current knowledge in this direction is very weak; e.g., it is not even known which of the digits 0,…,9 occur infinitely often in the decimal expansion of π. Bailey and Crandall showed in 2000 that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulae imply that the normality in base 2 of π and various other constants can be reduced to a plausible conjecture of chaos theory. See Bailey's above mentioned web site for details. It is also unknown whether π and e are algebraically independent. However it is known that at least one of πe and π + e is transcendental (q.v.). John Harrison, (1693–1776) (of Longitude fame), devised a meantone temperament musical tuning system derived from π. This Lucy Tuning system (due to the unique mathematical properties of π), can map all musical intervals, harmony and harmonics. This suggests that musical harmonics beat, and that using π could provide a more precise model for the analysis of both musical and other harmonics in vibrating systems.

The nature of π

In non-Euclidean geometry the sum of the angles of a triangle may be more or less than π radians, and the ratio of a circle's circumference to its diameter may also differ from π. This does not change the definition of π, but it does affect many formulae in which π appears. So, in particular, π is not affected by the shape of the universe; it is not a physical constant but a mathematical constant defined independently of any physical measurements. Nonetheless, it occurs often in physics. For example, consider Coulomb's law :

F = \frac \frac

. Here, 4πr² is just the surface area of sphere of radius r. In this form, it is a convenient way of describing the inverse square relationship of the force at a distance r from a point source. It would of course be possible to describe this law in other, but less convenient ways, or in some cases more convenient. If Planck charge is used, it can be written as :

F = \frac

and thus eliminate the need for π.

Fictional references

- Contact -- Carl Sagan's science fiction work. Sagan contemplates the possibility of finding a signature embedded in the base-11 expansion of Pi by the creators of the universe.
- Eon -- science fiction novel by Greg Bear. The protagonists measure the amount of space curvature using a device that computes π. Only in completely flat space/time will a circle have a circumference, diameter ratio of 3.14159... .
- Going Postal -- fantasy novel by Terry Pratchett. Famous inventor Bloody Stupid Johnson invents an organ/mail sorter that contains a wheel for which pi is exactly 3. This "New Pie" starts a chain of events that leads to the failure of the Ankh-Morpork Post Office (and possibly the destruction of the Universe all in one go.)
- π (film) -- On the relationship between numbers and nature: finding one without being a numerologist.
- The Simpsons -- "Pi is exactly 3!" was an announcement used by Professor Frink to gain the full attention of a hall full of scientists.
- Time's Eye -- science fiction by Arthur C. Clarke and Stephen Baxter. In a world restructured by alien forces, a spherical device is observed whose circumference to diameter ratio appears to be an exact integer 3 across all planes. It is the first book in The Time Odyssey series.

π culture

There is an entire field of humorous yet serious study that involves the use of mnemonic techniques to remember the digits of π, which is known as piphilology. See Pi mnemonics for examples. March 14 (3/14 in US date format) marks Pi Day which is celebrated by many lovers of π. On July 22, Pi Approximation Day is celebrated (22/7 - in European date format - is a popular approximation of π). In the early hours of Saturday 2 July, 2005, a Japanese mental health counsellor, Akira Haraguchi, 59, managed to recite π's first 83,431 decimal places from memory, thus breaking the standing world record [http://news.bbc.co.uk/1/hi/world/asia-pacific/4644103.stm]. 355/113 (~3.1415929) is sometimes jokingly referred to as "not π, but an incredible simulation!" Singer Kate Bush's recently released album "Aerial" contains a song titled "π," in which she sings π to over one hundred decimal places. Fans have discovered that she got some of them wrong, however, and actually misses twenty-two numbers. Fans are calling Bush's version "Kate's π."

References

-
- Petr Beckmann, A History of Pi

External links

Digit resources

- [http://www.gutenberg.net/etext/50 Project Gutenberg E-Text containing a million digits of Pi]
- [http://3.141592653589793238462643383279502884197169399375105820974944592.com/ Pi to a million places]
- [http://www.solidz.com/pi/ Archives of Pi calculated to 1,000,000 or 10,000,000 places.]
- [http://www.pisearch.de.vu Search π] – search and print π's digits (up to 3.2 billion places)
- [http://www.super-computing.org/pi-decimal_current.html Statistics about the first 1.2 trillion digits of Pi]
- [http://3.14.maxg.org/ A banner of approximately 220 million digits of pi]
- [http://3.141592653589793238462643383279502884197169399375105820974944592.com/ Pi to 1 million decimal places]

Calculation

- [http://projectpi.sourceforge.net/ Calculating Pi: The open source project for calculating Pi.]
- [http://backpi.sourceforge.net Background Pi: An open source project for calculating Pi over many computers. (Inspired by "Calulating Pi", Above)]
- [http://numbers.computation.free.fr/Constants/PiProgram/pifast.html PiFast: a fast program for calculating Pi with a large number of digits]
- [http://oldweb.cecm.sfu.ca/projects/pihex/index.html PiHex Project]
- [http://files.extremeoverclocking.com/file.php?f=36 Super Pi: Another program to calculate Pi to the 33.55 millionth digit. Also used a benchmark]
- [http://www.pislice.com/ PiSlice: A distributed computing project to calculate Pi]
- Calculating the digits of π using generalised continued fractions - open source Python code

General

- [http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Pi_through_the_ages.html J J O'Connor and E F Robertson: A history of Pi. Mac Tutor project]
- [http://machination.mysite.freeserve.com/ A collection of Machin-type formulae for Pi]
- [http://www.lrz-muenchen.de/~hr/numb/pi-irr.html A proof that Pi Is Irrational]
- [http://www.joyofpi.com/pifacts.html PiFacts-Record Broken]
- [http://www.joyofpi.com/thebook.html The Joy of Pi-About the Book]
- [http://mathworld.wolfram.com/PiFormulas.html From the Wolfram Mathematics site lots of formulae for π]
- [http://www.pisymphony.com/gpage.html Pi Symphony : An orchestral work by Lars Erickson based on the digits of pi and 'e'.]
- [http://planetmath.org/encyclopedia/Pi.html PlanetMath: Pi]
- [http://groups.yahoo.com/group/pi-hacks The pi-hacks Yahoo! Group]
- [http://mathforum.org/isaac/problems/pi1.html Finding the value of Pi]
- [http://cf.geocities.com/ilanpi/pi-exists.html Proof that Pi exists]
- [http://pi314.at/ Friends of Pi Club (German and English)]
- [http://www.cut-the-knot.org/pythagoras/NatureOfPi.shtml Determination of Pi] at cut-the-knot
- [http://www.lucytune.co.uk LucyTuning - musical tuning derived from Pi]
- [http://dse.webonastick.com/pi/ The Pi Is Rational Page]

Mnemonics

- [http://users.aol.com/s6sj7gt/mikerav.htm One of the more popular mnemonic devices for remembering pi]
- [http://www.cilea.it/~bottoni/www-cilea/F90/piph.htm Andreas P. Hatzipolakis: PiPhilology. A site with hundreds of examples of π mnemonics]
- [http://www.startfromhere.freeserve.co.uk/nudesci/abc/pi.htm Pi memorised as poetry]
- [http://www.archivestowearpantsto.com/tracks/0052_i_am_the_first_fifty_digits_of_pi.mp3 First fifty digits of Pi, memorised as a humorous song]
- [http://www.geocities.com/sviveknayak/riddles.htm Phrase to easily remember upto 8 decimal places of the value of Pi (See Item #3 on page)]
- [http://brianbondy.com/other/pi.aspx Free software to help memorise Pi] Category:Transcendental numbers 3.1416 als:Π ko:원주율 ja:円周率 simple:Pi th:ไพ

Ellipse

Elliptical redirects here, for the exercise machine, see Elliptical trainer. Elliptical trainer In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). An ellipse is a type of conic section: if a cone is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse. For a short elementary proof of this, see Dandelin spheres. Algebraically, an ellipse is a curve in the Cartesian plane defined by an equation of the form :

A x^2 + B xy + C y^2 + D x + E y + F = 0

such that

B^2 < 4 AC

, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists. An ellipse can be drawn with two pins, a loop of string, and a pencil. The pins are placed at the foci and the pins and pencil are enclosed inside the string. The pencil is placed on the paper inside the string, so the string is taut. The string will form a triangle. If the pencil is moved around so that the string stays taut, the sum of the distances from the pencil to the pins will remain constant, satisfying the definition of an ellipse. triangle The line segment which passes through the foci and terminates on the ellipse is called the major axis. The major axis is along the longest segment that passes through the ellipse. The line which passes through the center (halfway between the foci), at right angles to the major axis, is called the minor axis. A semimajor axis is one half the major axis: the line segment from the center, through a focus, and to the edge of the ellipse. Likewise, the semiminor axis is one half the minor axis. If the two foci coincide, then the ellipse is a circle; in other words, a circle is a special case of an ellipse, one where the eccentricity is zero. An ellipse centred at the origin can be viewed as the image of the unit circle under a linear map associated with a symmetric matrix

A = PDP^T

, D being a diagonal matrix with the eigenvalues of A, both of which are real positive, along the main diagonal, and P being a real unitary matrix having as columns the eigenvectors of A. Then the axes of the ellipse will lie along the eigenvectors of A, and the squares of the lengths of the axes are the inverses of the eigenvalues.

Parametrisation

The size of an ellipse is determined by two constants, conventionally denoted a and b. The constant a equals the length of the semimajor axis; the constant b equals the length of the semiminor axis. length An ellipse centered at the origin of an x-y coordinate system with its major axis along the x-axis is defined by the equation :

\frac + \frac = 1

The derivation of this formula is quite instructive and not overly difficult. The following diagram shows an ellipse demonstrating the Pythagoras equation a² = b² + c² as a special case of the non-parametric equation above (x=0, y=b). center The same ellipse is also represented by the parametric equations: :

x = a\,\cos t

y = b\,\sin t

0 \leq t < 2\pi

which use the trigonometric functions sine and cosine. If an ellipse is not centered at the origin of an x-y coordinate system, but again has its major axis along the x-axis, it may be specified by the equation :

\frac + \frac = 1

where (h,k) is the center. A Gauss-mapped form: :

\left(\frac,\frac\right)

has normal

(\cos\phi,\sin\phi)

Eccentricity

The shape of an ellipse is usually expressed by a number called the eccentricity of the ellipse, conventionally denoted e (not to be confused with the mathematical constant e). The eccentricity is related to a and b by the statement :

e = \sqrt

or where

c

(the linear eccentricity of the ellipse) equals the distance from the center to either focus :

e = \frac

The eccentricity is a positive number less than 1, or 0 in the case of a circle. The greater the eccentricity is, the larger the ratio of a to b is, and therefore the more elongated the ellipse is. The ellipse shown in the image below has an eccentricity of approximately 0.8733. The distance between the foci is 2ae.

Semi-latus rectum and polar coordinates

The semi-latus rectum of an ellipse, usually denoted

l\,\!

(lowercase L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis. It is related to

a\,\!

and

b\,\!

(the ellipse's semi-axes) by the formula

al=b^2\,\!

or, if using the eccentricity,

l=a(1-e^2)\,\!

. perpendicular In polar coordinates, an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation :

r (1 + e \cos \theta) = l \,\!

An ellipse can also be thought of as a projection of a circle: a circle on a plane at angle φ to the horizontal projected vertically onto a horizontal plane gives an ellipse of eccentricity sin φ, provided φ is not 90°.

Area

The area enclosed by an ellipse is

\pi ab\,\!

, where

\pi

is Archimedes' constant.

Circumference

The circumference of an ellipse is 4aE(e), where the function E is the complete elliptic integral of the second kind. The exact infinite series is: :

c = 2\pi a \left[\right]\!\,

A good approximation is Ramanujan's: :

c \approx \pi \left[3(a+b) - \sqrt\right]\!\,

which can also be written as: :

c \approx \pi a \left[ 3 (1+\sqrt) - \sqrt \right] \!\,

More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral. The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions.

Stretching and Projection

An ellipse may be uniformly stretched along any axis, in or out of the plane of the ellipse, and it will still be an ellipse. The stretched ellipse will have different properties (perhaps changed eccentricity and semi-major axis length, for instance), but it will still be an ellipse (or a degenerate ellipse: a circle or a point). Similarly, any oblique projection onto a plane results in a conic section. If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse.

Reflection property

Assume an elliptic mirror with a light source at one of the foci. Then all rays are reflected to a single point — the second focus. Since no other curve has such a property, it can be used as an alternative definition of an ellipse.

Ellipses in physics

Indian astronomer Aryabhata discovered that the orbits of the planets around the sun are ellipses in 499, which he described in his book, the Aryabhatiya [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aryabhata_I.html]. In the 17th century, Johannes Kepler explained that the orbits along which the planets travel around the Sun are ellipses, which is Kepler's first law. Later, Isaac Newton explained this as a corollary of his law of universal gravitation. More generally, in the gravitational two-body problem, if the two bodies are bound to each other (i.e., the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus. The general solution for a harmonic oscillator in two or more dimensions is also an ellipse, but this time with the origin of the force located at the center of the ellipse. Albert Einstein also used the ellipse to prove his theory of relativity by using an elliptical shaped mass. Einstein's contributions to modern physics may not have been discovered if it were not for ellipses.

Ellipses in computer graphics

Drawing an ellipse is a common graphics primitive in standard display libraries, such as the QuickDraw and GDI interfaces on the Macintosh and Windows systems. Often such libraries are limited and can only draw an ellipse with either the major axis or the minor axis horizontal. Jack Bresenham at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. An efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken (IEEE CG&A, Sept. 1984). A more challenging task is to perform these drawing operations with antialiasing, to create a smooth-looking curve. The curve drawing algorithms of Xiaolin Wu (SIGGRAPH 91) are an example.

Numerical integration

In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term quadrature is more or less a synonym for numerical integration, especially as applied to one-dimensional integrals. Two-dimensional integration is sometimes described as cubature, although this term is much less frequently used and the meaning of quadrature is understood for higher dimensional integration as well. The basic problem considered by numerical integration is to compute an approximate solution to a definite integral: :

\int_a^b f(x)\, dx

This problem can also be stated as an initial value problem for an ordinary differential equation, as follows. :

y'(x) = f(x), \quad y(a) = 0

Finding y(b) is equivalent to computing the integral. Methods developed for ordinary differential equations, such as the Runge-Kutta method, can be applied to the restated problem. In the remainder of this article, we shall discuss methods developed specifically for the problem stated as a definite integral.

Reasons for numerical integration

There are several reasons for carrying out numerical integration. The integrand f may be known only at certain points, such as obtained by sampling. Some embedded systems and other computer applications may need numerical integration for this reason. A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative. An example of such an integrand is exp(-t²). It may be possible to find an antiderivative symbolically, but it may be easier to compute a numerical approximation than to compute the antiderivative. That may be the case if the antiderivative is given as an infinite series or product, or if its evaluation requires a special function which is not available.

Methods for one-dimensional integrals

Numerical integration methods can generally be described as combining evaluations of the integrand to get an approximation to the integral. An important part of the analysis of any numerical integration method is to study the behavior of the approximation error as a function of the number of integrand evaluations. A method which yields a small error for a small number of evaluations is usually considered superior. Reducing the number of evaluations of the integrand reduces the number of arithmetic operations involved, and therefore reduces the total round-off error. Also, each evaluation takes time, and the integrand may be arbitrarily complicated. It should be noted, however, that a 'brute force' kind of numerical integration can always be done, in a very simplistic way, by evaluating the integrand with very small increments.

Quadrature rules based on interpolating functions

A large class of quadrature rules can be derived by constructing interpolating functions which are easy to integrate. Typically these interpolating functions are polynomials. The simplest method of this type is to let the interpolating function be a constant function (a polynomial of order zero) which passes through the point ((a+b)/2, f((a+b)/2)). This is called the midpoint rule or rectangle rule. :

\int_a^b f(x)\,dx \approx (b-a) \, f\left(\frac\right).

The interpolating function may be an affine function (a polynomial of degree 1) which passes through the points (a, f(a)) and (b, f(b)). This is called the trapezoidal rule. :

\int_a^b f(x)\,dx \approx (b-a) \, \frac.

For either one of these rules, we can make a more accurate approximation by breaking up the interval [a, b] into some number n of subintervals, computing an approximation for each subinterval, then adding up all the results. This is called a composite rule, extended rule, or iterated rule. For example, the composite trapezoidal rule can be stated as :

\int_a^b f(x)\,dx \approx \frac \left(  + \sum_^ f \left( a+k \frac \right) \right)

where the subintervals have the form [k h, (k+1) h], with h = (b-a)/n and k = 0, 1, 2, ..., n-1. Interpolation with polynomials evaluated at equally-spaced points in [a, b] yields the Newton-Cotes formulas, of which the rectangle rule and the trapezoidal rule are examples. Simpson's rule, which is based on a polynomial of order 2, is also a Newton-Cotes formula. If we allow the intervals between interpolation points to vary, we find another group of quadrature formulas, called Gaussian quadrature formulas. A Gaussian quadrature rule is typically more accurate than a Newton-Cotes rule which requires the same number of function evaluations, if the integrand is smooth (i.e., if it has many derivatives.)

Adaptive algorithms

If f does not have many derivatives at all points, or if the derivatives become large, then Gaussian quadrature is often insufficient. In this case, an algorithm similar to the following will perform better: // This algorithm calculates the definite integral of a function // from 0 to 1, adaptively, by choosing smaller steps near // problematic points. // Set initial_h to the initial step size. x:=0 h:=initial_h accumulator:=0 WHILE x<1.0 DO IF x+h>1.0 THEN h=1.0-x END IF IF error from quadrature over [x,x+h] for f is too large THEN Make h smaller ELSE accumulator:=accumulator + quadrature of f over [x,x+h] x:=x+h IF error from quadrature over [x,x+h] is very small THEN Make h larger END IF END IF END WHILE RETURN accumulator Some details of the algorithm require careful thought. For many cases, estimating the error from quadrature over an interval for a function f isn't obvious. One popular solution is to use two different rules of quadrature, and use their difference as an estimate of the error from quadrature. The other problem is deciding what "too large" or "very small" signify. A possible criterion for "too large" is that the quadrature error should not be larger than th where t, a real number, is the tolerance we wish to set for global error. Then again, if h is already tiny, it may not be worthwhile to make it even smaller even if the quadrature error is apparently large. This type of error analysis is usually called "a posteriori" since we compute the error after having computed the approximation. Heuristics for adaptive quadrature are discussed by Forsythe et al. (Section 5.4).

Extrapolation methods

The accuracy of a quadrature rule of the Newton-Cotes type is generally a function of the number of evaluation points. The result is usually more accurate as number of evaluation points increases, or, equivalently, as the width of the step size between the points decreases. It is natural to ask what the result would be if the step size were allowed to approach zero. This can be answered by extrapolating the result from two or more nonzero step sizes (see Richardson extrapolation). The extrapolation function may be a polynomial or rational function. Extrapolation methods are described in more detail by Stoer and Bulirsch (Section 3.4).

Conservative (a priori) error estimation

Let f have a bounded first derivative over [a,b]. The mean value theorem for f, where

x < b

, gives :

(x - a) f'(y_x) = f(x) - f(a)\,

for some y_x in [a,x] depending on x. If we integrate in x from a to b on both sides and the take absolute values, we obtain :

\left| \int_a^b f(x)\,dx - (b - a) f(a) \right|
  = \left| \int_a^b (x - a) f'(y_x)\, dx \right|

We can further approximate the integral on the right-hand side by bringing the absolute value into the integrand, and replacing the term in f' by an upper bound: :

\left| \int_a^b f(x)\,dx - (b - a) f(a) \right| \leq  \sup_ \left| f'(x) \right|

(
- ) (See supremum.) Hence, if we approximate the integral ∫_a^bf(x)dx by the quadrature rule (b-a)f(a) our error is no greater than the right hand side of (
- ). We can convert this into an error analysis for the Riemann sum (
- ), giving an upper bound of :

\sup_ \left| f'(x) \right|

for the error term of that particular approximation. (Note that this is precisely the error we calculated for the example

f(x) = x

.) Using more derivatives, and by tweaking the quadrature, we can do a similar error analysis using a Taylor series (using a partial sum with remainder term) for f. This error analysis gives a strict upper bound on the error, if the derivatives of f are available. This integration method can be combined with interval arithmetic to produce computer proofs and verified calculations.

Multidimensional integrals

The quadrature rules discussed so far are all designed to compute one-dimensional integrals. To compute integrals in multiple dimensions, one approach is to phrase the multiple integral as repeated one-dimensional integrals by appealing to Fubini's theorem. This approach requires the function evaluations to grow exponentially as the number of dimensions increases. Two methods are known to overcome this so-called curse of dimension.

Monte Carlo

Monte Carlo methods and quasi-Monte Carlo methods are easy to apply to multi-dimensional integrals, and may yield greater accuracy for the same number of function evaluations than repeated integrations using one-dimensional methods. A large class of useful Monte Carlo methods are the so-called Markov chain Monte Carlo algorithms, which include the Metropolis-Hastings algorithm and Gibbs sampling.

Sparse grids

Sparse grids were originally developed by Smolyak for the quadrature of high dimensional functions. The method is always based on a one dimensional quadrature rule, but performs a more sophisticated combination of univariate results.

Software for numerical integration

Numerical integration is one of the most intensively studied problems in numerical analysis. Of the many software implementations we list a few here.
- QUADPACK (part of SLATEC): description [http://www.netlib.org/slatec/src/qpdoc.f], source code [http://www.netlib.org/slatec/src]. QUADPACK is a collection of algorithms, in Fortran, for numerical integration based on Gaussian quadrature.
- [http://www.gnu.org/software/gsl/ GSL]: The GNU Scientific Library (GSL) is a numerical library written in C which provides a wide range of mathematical routines, like Monte Carlo integration. Numerical integration algorithms are found in GAMS class [http://gams.nist.gov/serve.cgi/Class/H2 H2].

References

- George E. Forsythe, Michael A. Malcolm, and Cleve B. Moler. Computer Methods for Mathematical Computations. Englewood Cliffs, NJ: Prentice-Hall, 1977. (See Chapter 5.)
- William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Chapter 4.)
- Josef Stoer and Roland Bulirsch. Introduction to Numerical Analysis. New York: Springer-Verlag, 1980. (See Chapter 3.) Category:Numerical analysis

Gaussian quadrature

In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1, by a suitable choice of the n points x_i and n weights w_i. The domain of integration for such a rule is conventionally taken as [-1, 1], so the rule is stated as :

\int_^1 f(x)\,dx \approx \sum_^n w_i f(x_i)

It can be shown (see Press, et al., or Stoer and Bulirsch) that the evaluation points are just the roots of a polynomial belonging to a class of orthogonal polynomials.

Rules for the basic problem

For the integration problem stated above, the associated polynomials are Legendre polynomials. Some low-order rules for solving the integration problem are listed below.

Number of points, n	Weights, w_i	Points, x_i
1	2	0
2	1, 1	$-\sqrt, \sqrt$
3	5/9, 8/9, 5/9	$-\sqrt, 0, \sqrt$

Change of interval for Gaussian quadrature

An integral over [a, b] must be changed into an integral over [-1, 1] before applying the Gaussian quadrature rule. This change of interval can be done in the following way: :

\int_a^b f(t)\,dt = \frac \int_^1 f\left(\fracx 
+ \frac\right)\,dx

After applying the Gaussian quadrature rule, the following approximation is obtained: :

\frac \sum_^n w_i f\left(\fracx_i + \frac\right)

Other forms of Gaussian quadrature

The integration problem can be expressed in a slightly more general way by introducing a weight function ω into the integrand, and allowing an interval other than [-1, 1]. That is, the problem is to calculate :

\int_a^b \omega(x)\,f(x)\,dx

for some choices of a, b, and ω. For a = -1, b = 1, and ω(x) = 1, the problem is the same as that considered above. Other choices lead to other integration rules. Some of these are tabulated below. Equation numbers are given for Abramowitz and Stegun (A&S).

Interval	ω(x)	Orthogonal polynomials	A&S
[-1, 1]	$1\,$	Legendre polynomials	Eq. 25.4.29
[-1, 1]	$\frac$	Chebyshev polynomials	Eq. 25.4.38
$[0, \infty)$	$e^\,$	Laguerre polynomials	Eq. 25.4.45
$(-\infty, \infty)$	$e^$	Hermite polynomials	Eq. 25.4.46

Error estimates

The error of a Gaussian quadrature rule can be stated as follows (theorem 3.6.24 in Stoer and Bulirsch). For an integrand which has 2n continuous derivatives, :

\int_a^b \omega(x)\,f(x)\,dx - \sum_^n w_i\,f(x_i)
 = \frac \, \|p_n\|^2

for some ξ in (a, b), where p_n is the orthogonal polynomial of order n. Stoer and Bulirsch remark that this error estimate is inconvenient in practice, since it may be difficult to estimate the 2nth derivative, and furthermore the actual error may be much less than a bound established by the derivative. Another approach is to use two Gaussian quadrature rules of different orders, and to estimate the error as the difference between the two results. For this purpose, Gauss-Kronrod rules can be useful.
Gauss-Kronrod rules
If the interval [a, b] is subdivided, the evaluation points of the new subintervals generally do not coincide with the previous evaluation points, and thus the integrand must be evaluated at every point. Gauss-Kronrod rules are Gaussian quadrature rules that are modified to make some of the evaluation points coincide after subdivision. The difference between the results before and after subdivision can be taken as an estimate of the error of approximation, so such an approach can increase the accuracy achieved for a given number of function evaluations. The algorithms in QUADPACK (see below) are based on Gauss-Kronrod rules.
References

- Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Section 25.4.)
- Robert Piessens, Elise de Doncker-Kapenga, C.W. Überhuber, D.K. Kahaner. QUADPACK, A subroutine package for automatic integration. Springer Verlag, 1983. (Reference guide for QUADPACK.)
- William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Section 4.5.)
- Josef Stoer and Roland Bulirsch. Introduction to Numerical Analysis. New York: Springer-Verlag, 1980. (See Section 3.6.)
External links

- QUADPACK (part of SLATEC): description [http://www.netlib.org/slatec/src/qpdoc.f], source code [http://www.netlib.org/slatec/src]. QUADPACK is a collection of algorithms, in Fortran, for numerical integration based on Gauss-Kronrod rules. SLATEC (at Netlib) is a large public domain library for numerical computing. Category:Numerical analysis

Semi-major axis

In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolas.

Ellipse

The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the edge of the ellipse. It is related to the semi-minor axis

b\,\!

through the eccentricity

e\,\!

and the semi-latus rectum

\ell\,\!

, as follows: :

b = a \sqrt\,\!

\ell=a(1-e^2)\,\!

. :

a\ell=b^2\,\!

. A parabola can 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, keeping

\ell\,\!

fixed. Thus

a\,\!

and

b\,\!

tend to infinity,

a\,\!

faster than

b\,\!

. The semi-major axis is the mean value of the smallest and largest distances from one focus to the points on the ellipse. Now consider the equation in polar coordinates, with one focus at the origin and the other on the positive x-axis, :

r(1-e\cos\theta)=l\,\!

The mean value of

r=\,\!

and

r=\,\!

, is

a=\,\!

Hyperbola

The semi-major axis of a hyperbola is one half of the distance between the two branches; if this is in the x-direction the equation is:

\frac - \frac = 1

In terms of the semi-latus rectum and the eccentricity we have

a=

Astronomy

Orbital period

In astrodynamics the orbital period

T\,

of a small body orbiting a central body in a circular or elliptical orbit is: :

T = 2\pi\sqrt

where: :

a\,

is the length of the orbit's semi-major axis :

\mu

is the standard gravitational parameter Note that for all ellipses with a given semi-major axis, the orbital period is the same, regardless of eccentricity. In astronomy, the semi-major axis is one of the most important orbital elements of an orbit, along with its orbital period. For solar system objects, the semi-major axis is related to the period of the orbit by Kepler's third law (originally empirically derived), :

P^2=a^3\,

where P is the period in years, and a is the semimajor axis in astronomical units. This form turns out to be a simplification of the general form for the two-body problem, as determined by Newton: :

P^2= \fraca^3\,

where G is the gravitational constant, and M is the mass of the central body, and m is the mass of the orbiting body. Typically, the central body's mass is so much greater than the orbiting body's, that m may be ignored. Making that assumption and using typical astronomy units results in the simpler form Kepler discovered. Remarkably, the orbiting body's path around the barycentre and its path relative to its primary are both ellipses. The semi-major axis used in astronomy is always the primary-to-secondary distance; thus, the orbital parameters of the planets are given in heliocentric terms. The difference between the primocentric and "absolute" orbits may best be illustrated by looking at the Earth-Moon system. The mass ratio in this case is 81.30059. The Earth-Moon characteristic distance, the semi-major axis of the geocentric lunar orbit, is 384,400 km. The barycentric lunar orbit, on the other hand, has a semi-major axis of 379,700 km, the Earth's counter-orbit taking up the difference, 4,700 km. The Moon's average barycentric orbital speed is 1.010 km/s, whilst the Earth's is 0.012 km/s. The total of these speeds gives the geocentric lunar average orbital speed, 1.022 km/s; the same value may be obtained by considering just the geocentric semi-major axis value.

Average distance

It is often said that the semi-major axis is the "average" distance between the primary (the focus of the ellipse) and the orbiting body. This is not quite accurate, as it depends over what the average is taken.
- averaging the distance over the eccentric anomaly (q.v.) indeed results in the semi-major axis.
- averaging over the true anomaly (the true orbital angle, measured at the focus) results, oddly enough, in the semi-minor axis

b = a \sqrt\,\!

.
- averaging over the mean anomaly (the fraction of the orbital period that has elapsed since pericentre, expressed as an angle), finally, gives the time-average (which is what "average" usually means to the layman):

a (1 + \frac)\,\!

. The time-average of the inverse of the radius,

r^\,\!

, is

a^\,\!

Energy; calculation of semi-major axis from state vectors

In astrodynamics semi-major axis

a \,

can be calculated from orbital state vectors:

a = \,

for an elliptical orbit and

a = \,

for a hyperbolic trajectory and

\epsilon =  -

(specific orbital energy) and

\mu = GM \,

(standard gravitational parameter), where:
-

v\,

is orbital velocity from velocity vector of an orbiting object,
-

\mathbf\,

is cartesian position vector of an orbiting object in coordinates of a reference frame with respect to which the elements of the orbit are to be calculated (e.g. geocentric equatorial for an orbit around Earth, or heliocentric ecliptic for an orbit around the Sun),
-

G \,

is the gravitational constant,
-

M \,

the mass of the central body. Note that for a given central body and total specific energy, the semi-major axis is always the same, regardless of eccentricity. Conversely, for a given central body and semi-major axis, the total specific energy is always the same.

Example

The International Space Station has an orbital period of 91.74 minutes, hence the semi-major axis is 6738 km [http://www.google.com/search?num=100&hl=en&lr=&newwindow=1&safe=off&q=%28%2891.74
- 60%2F2%2Fpi%29%5E2
- 398600%29%5E%281%2F3%29]. Every minute more corresponds to ca. 50 km more: the extra 300 km of orbit length takes 40 seconds, the lower speed accounts for an additional 20 seconds.

References

- [http://orca.phys.uvic.ca/~tatum/celmechs/celm9.pdf Jeremy B. Tatum, Celestial Mechanics, Chapter 9 - The Two Body Problem in Two Dimensions (2004)]
- [http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000071000011001198000001&idtype=cvips&gifs=yes Darren M. Williams, Average distance between a star and planet in an eccentric orbit, American Journal of Physics, November 2003, Volume 71, Issue 11, pp. 1198-1200] Category:Conic sections Category:Astrodynamics Category:Celestial mechanics

Eccentricity (mathematics)

(This page refers to eccentricity in mathematics. For other uses, see the disambiguation page eccentricity.) In mathematics, eccentricity is a parameter associated with every conic section, see Conic section#Eccentricity. It can be thought of as a measure of how much the conic section deviates from being circular. In particular,
- The eccentricity of a circle is zero.
- The eccentricity of an ellipse is greater than zero and less than 1.
- The eccentricity of a parabola is 1.
- The eccentricity of a hyperbola is greater than 1 and less than infinity.
- The eccentricity of a straight line is infinity. It is given by: :

e = \sqrt

Where a is the length of the semimajor axis of the section, b the length of the semiminor axis, and k is equal to +1 for an ellipse, 0 for a parabola, and -1 for a hyperbola. It is also called the first eccentricity when necessary to distinguish it from the second eccentricity, e', which is sometimes used for algebraic convenience. The second eccentricity is defined as: :

e' = \sqrt

And is related to the first eccentricity by the equation: :

1 = (1 - e^2)(1 + e'^2)\,\!

Ellipse

semiminor axis For any ellipse, where the length of the semi-major axis is a, and where the same of the semi-minor axis is b, the eccentricity is given by: :

e = \sqrt

The eccentricity is the ratio of the distance between the foci (

F_1

and

F_2

) to the major axis; i.e.

\left ( \frac \right )

. The term linear eccentricity is used for

Hyperbola

For any hyperbola, where the length of the semi-major axis is a, and where the same of the semi-minor axis is b, eccentricity is given by: :

e = \sqrt

Surfaces

The eccentricity of a surface is the eccentricity of a designated section of the surface. For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane).

External links

- [http://mathworld.wolfram.com/Eccentricity.html MathWorld: Eccentricity] Category:Conic sections als:Exzentrizität (Mathematik)

Approximation

An approximation is an inexact representation of something that is still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws. Approximations may be used because incomplete information prevents use of exact representations. Alternately, even when the exact representation is known, it may be preferable to use an approximation that simplifies analysis without too great a cost in accuracy. For instance, physicists often approximate the shape of the Earth as a sphere even though more accurate representations are possible, because many physical behaviours — e.g. gravity — are much easier to calculate for a sphere than for less regular shapes. The type of approximation used depends on the available information, the degree of accuracy required, the sensitivity of the problem to this data, and the savings (usually in time and effort) that can be achieved by approximation.

Science

The scientific method is carried out with a constant interaction between scientific laws (theory) and empirical measurements, which are constantly compared to one another. Approximation also refers to using a simpler process. This model is used to make predictions easier. The most common versions of philosophy of science accept that empirical measurements are always approximations — they do not perfectly represent what is being measured. The history of science indicates that the scientific laws commonly felt to be true at any time in history are only approximations to some deeper set of laws. Each time a newer set of laws is proposed, it is required that in the limiting situations in which the older set of laws were tested against experiments, the newer laws are nearly identical to the older laws, to within the measurement uncertainties of the older measurements. This is the correspondence principle.

Mathematics

Numerical approximations sometimes result from using a small number of significant digits. Approximation theory is a branch of mathematics, a quantitative part of functional analysis. Diophantine approximation deals with approximation to real numbers by rational numbers. The symbol "≈" means "approximately equal to".

Thomas Muir (mathematician)

Sir Thomas Muir (25 August 1844-21 March 1934) was a Scottish mathematician, remembered as an authority on determinants. He was born in Stonebyres in Lanarkshire, and brought up in Biggar. At the University of Glasgow he changed his studies from classics to mathematics after advice from the future Lord Kelvin. After graduating he held positions at St. Andrews University and Glasgow University. From 1874 to 1892 he taught at Glasgow High School. In 1882 he published Treatise on the theory of determinants; then in 1890 he published a History of determinants. From 1892 he was in South Africa working in education, and then in administration at the University of the Cape. He was knighted in 1910. From 1906 onwards he published a five-volume expansion of his history of determinants, the final part (1929) taking the theory to 1920. A further book followed in 1930. His name now attaches to a duality theorem on relations between minors. In more abstract language, it is a general result on the equations defining Grassmannians as algebraic varieties. Muir, Thomas Muir, Thomas Muir, Thomas

Category:Geometry

Geometry is the branch of mathematics dealing with spatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry. Such axioms are insusceptible to proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions. Category:Mathematics zh-min-nan:Category:Kí-hô-ha̍k ko:분류:기하학 ja:Category:幾何学

Whydahs

See text See also Whydah Gally for the pirate ship and Ouidah for the town in Benin. The Indigobirds and whydahs, are small passerine birds native to Africa. These are finch-like species which usually have black or indigo predominating in their plumage. The birds named as "whydahs" have long or very long tails. All are brood parasites, which lay their eggs in the nests of estrildid finch species; most indigobirds use fire-finches as hosts, whereas the paradise whydahs chose pytilias. Unlike the cuckoo, the host's eggs are not destroyed. Typically, 2-4 eggs are laid in with the those already present. The eggs of both the host and the victim are white, although the indigobird's are slightly larger. Many of the indigo-plumaged species named as "indigobirds" are very similar in appearance, with the males difficult to separate in the field, and the young and females near impossible. The best guide is often the estrildid finch with which they are associating, since each indigobird parasitises a different host species. Thus the Village Indigobird is usually found with Red-billed Fire-finches. Indigobirds and whydahs imitate their host's song, which the males learn in the nest. Although females do not sing, they also learn to recognise the song, and chose males with the same song, thus perpetuating the link between each species of indigobird and firefinch. Similarly, the nestling indigobirds mimic the unique gape pattern of the fledglings of the host species. The matching with the host is the driving force behind speciation in this family, but the close gemetic and morphological similarities among species suggest that they are of recent origin.
- Family: Viduidae
- Village Indigobird, Vidua chalybeata
- Jambandu Indigobird, Vidua raricola
- Baka Indigobird, Vidua larvaticola
- Jos Plateau Indigobird, Vidua maryae
- Quailfinch Indigobird, Vidua nigeriae
- Variable Indigobird, Vidua funerea
- Green Indigobird, Vidua codringtoni
- Purple Indigobird, Vidua purpurascens
- Pale-winged Indigobird, Vidua wilsoni
- Cameroon Indigobird, Vidua camerunensis
- Steel-blue Whydah, Vidua hypocherina
- Straw-tailed Whydah, Vidua fischeri
- Shaft-tailed Whydah, Vidua regia
- Pin-tailed Whydah, Vidua macroura
- Togo Paradise-Whydah, Vidua togoensis
- Long-tailed Paradise-Whydah, Vidua interjecta
- Eastern Paradise-Whydah, Vidua paradisaea
- Northern Paradise-Whydah, Vidua orientalis
- Broad-tailed Paradise-Whydah, Vidua obtusa Category:Brood parasites

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Circumference

Circle

Ellipse

Muir-1883

Ramanujan-1914 (#1,#2)

External link

Perimeter

Practical Uses

Formulas

See Also

Diameter

Diameter symbol

Pi

Properties

Formulae involving π

Geometry

Analysis

Continued fractions

Number theory

Dynamical systems and ergodic theory

Physics

Probability and statistics

History of π

Numerical approximations of π

Miscellaneous formulae

Less accurate approximations

Open questions

The nature of π

Fictional references

π culture

See also

References

External links

Digit resources

Calculation

General

Mnemonics

Ellipse

Parametrisation

Eccentricity

Semi-latus rectum and polar coordinates

Area

Circumference

Stretching and Projection

Reflection property

Ellipses in physics

Ellipses in computer graphics

See also

Numerical integration

Reasons for numerical integration

Methods for one-dimensional integrals

Quadrature rules based on interpolating functions

Adaptive algorithms

Extrapolation methods

Conservative (a priori) error estimation

Multidimensional integrals

Monte Carlo

Sparse grids

Software for numerical integration

References

Gaussian quadrature

Rules for the basic problem

Change of interval for Gaussian quadrature

Other forms of Gaussian quadrature

Error estimates

Gauss-Kronrod rules

References

External links

Semi-major axis

Ellipse

Hyperbola

Astronomy

Orbital period

Average distance

Energy; calculation of semi-major axis from state vectors

Example

References

Eccentricity (mathematics)

Ellipse

Hyperbola

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