"Cìrcunfirenza (giometrìa)" : Diffirenzi ntrê virsioni

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==External links==
==External links==
{{Wiktionarypar|circumference}}
{{Wiktionarypar|circumference}}
* [http://home.att.net/~numericana/answer/ellipse.htm#elliptic Numericana - Circumference of an ellipse]
* [http://www.numericana.com/answer/ellipse.htm#elliptic Numericana - Circumference of an ellipse]
*[http://www.mathopenref.com/circumference.html Circumference of a circle] With interactive applet and animation
*[http://www.mathopenref.com/circumference.html Circumference of a circle] With interactive applet and animation



Virsioni dû 03:15, 6 maiu 2010

Circumference = π × diameter

La circunfirenza eni la distanza attornu a na curva chiusa.
La circumfirenza eni na speci di pirìmitru.

Circunfirenza di nu cìrculu

La circunfirenza di nu cìrculu eni la lunghizza attornu a iddu.
La circunfirenza di nu cìrculu pò ssiri calculata di lu sò diàmmitru usannu la fòrmula:

O, sustituennu lu diàmmitru a lu raiu:

unni r eni lu raiu, d eni lu diàmmmitru di lu cìrculu e (a littra greca pipi) eni difinutu comu lu rapportu ntra la circunfirenza di lu cìrculu cu lu sò diàmmmitru (lu valuri numèricu eni 3.141 592 653 589 793...).

If desired, the above circumference formula can be derived without reference to the definition of π by using some integral calculus, as follows:

The upper half of a circle centered at the origin is the graph of the function where x runs from -r to +r. The circumference (c) of the entire circle can be represented as twice the sum of the lengths of the infinitesimal arcs that make up this half circle. The length of a single infinitesimal part of the arc can be calculated using the Pythagorean formula for the length of the hypotenuse of a rectangular triangle with side lengths dx and f'(x)dx, which gives us

Thus the circle circumference can be calculated as dara:)

= =

The antiderivative needed to solve this definite integral is the arcsine function:

Pi (π) is the ratio of the circumference of a circle to its diameter.

Circumference of an ellipse

The circumference of an ellipse is more problematic, 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) orr by one of many binomial series expansions.

Where are the ellipse's semi-major and semi-minor axes, respectively, and is the ellipse's angular eccentricity,

There are many different approximations for the divided difference, with varying degrees of sophistication and corresponding accuracy.

In comparing the different approximations, the based series expansion is used to find the actual value:

Muir-1883

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

Ramanujan-1914 (#1,#2)

Srinivasa Ramanujan introduced two different approximations, both from 1914


The second equation is demonstratively 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:

b Pr Ramanujan-#2 Ramanujan-#1 Muir
9975  9987.50391 11393   9987.50391 11393   9987.50391 11393   9987.50391 11389
9966  9983.00723 73047  9983.00723 73047  9983.00723 73047  9983.00723 73034
9950  9975.01566 41666  9975.01566 41666  9975.01566 41666  9975.01566 41604
9900  9950.06281 41695  9950.06281 41695  9950.06281 41695  9950.06281 40704
9000  9506.58008 71725  9506.58008 71725  9506.58008 67774  9506.57894 84209
8000  9027.79927 77219  9027.79927 77219  9027.79924 43886  9027.77786 62561
7500  8794.70009 24247  8794.70009 24240  8794.69994 52888  8794.64324 65132
6667  8417.02535 37669  8417.02535 37460  8417.02428 62059  8416.81780 56370
5000  7709.82212 59502  7709.82212 24348  7709.80054 22510  7708.38853 77837
3333  7090.18347 61693  7090.18324 21686  7089.94281 35586  7083.80287 96714
2500  6826.49114 72168  6826.48944 11189  6825.75998 22882  6814.20222 31205
1000  6468.01579 36089  6467.94103 84016  6462.57005 00576  6431.72229 28418
 100  6367.94576 97209  6366.42397 74408  6346.16560 81001  6303.80428 66621
  10  6366.22253 29150  6363.81341 42880  6340.31989 06242  6299.73805 61141
   1  6366.19804 50617  6363.65301 06191  6339.80266 34498  6299.60944 92105
iota  6366.19772 36758  6363.63636 36364  6339.74596 21556  6299.60524 94744

External links

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