Cunfruntu tra virsioni di "Cìrcunfirenza (giometrìa)"

travagghiu in corsu
(sbagghiai e quinni canciai li culligamenti a li wikipedie stranieri: eranu a "Cerchiu" e ora sunu a "Circunfirenza")
(travagghiu in corsu)
[[Image:CirclePi -eq blackC simpleover d.svg|200px|right|thumb|220px|Circumference = [[Pi|π]] × diameter]]
A '''circunfirenza''' eni a distanza attonnu a na [[curva]] ciusa.<br>
Na '''cìrcunfirenza''' è lu locu dî [[puntu|punti]] equidistanti di nu puntu fissu chiamatu ''centru''.<br>
A circumfirenza eni na specie di [[pirimitru]].
Lu locu di li punta ca si trovanu intra â cìrcunfirenza si ciama '''[[Cìrculu (giometrìa)‎|cìrculu]]'''.<br>
Ntâ nu sistema di rifirmentu cartisianu, siddu lu centru havi coordinati <math>(a,b)</math> e lu raiu è <math>R</math>, l'[[equazzioni]] [[Algibra|algèbbrica]] chi rapprisenta na cìrcunfirenza eni:
 
==Circunfirenza di nu circulu==
<math>(x-a)^2+(y-b)^2=R^2</math>
a circunfirenza di nu circulu eni a lunghizza attonna a iddu.<br>
A circumfirenza di nu [[circulu]] po ssiri calucatu da lu so [[diamitru]] usannu a fommula:
 
:<math>c=\pi\cdot{d}.\,\!</math>
[[Category:Matimàtica]]
 
[[Catigurìa:Giometrìa]]
O, sustituiennu lu diamitru a lu [[raiu]]:
[[Catigurìa:Àlgibbra]]
 
:<math>c=2\pi\cdot{r}=\pi\cdot{2r},\,\!</math>
 
unni ''r'' eni lu [[raiu]], ''d'' eni lu diammitru di lu circulu e (a [[Pi (littira)|littira greca pipi]]) eniis [[pi|difinutu]] comu lu rapportu ntra a circunfirenza di lu circulu cu lu so diammitru (lu valuri numericu 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 <math>f(x) = \sqrt{r^2-x^2},</math> 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 Theorem | Pythagorean formula]] for the length of the hypotenuse of a rectangular triangle with side lengths dx and f'(x)dx, which gives us <math>\sqrt{(dx)^2+(f'(x)dx)^2} = \left( \sqrt{1+f'(x)^2} \right) dx.</math>
 
Thus the circle circumference can be calculated as dara:)
 
<math>c = 2 \int_{-r}^r \sqrt{1+f'(x)^2}dx</math> = <math>2 \int_{-r}^r \sqrt{1+\frac{x^2}{r^2-x^2}}dx</math> = <math>2 \int_{-r}^r \sqrt{\frac{1}{1-\frac{{x}^2}{{r}^2}}}dx</math>
 
The [[antiderivative]] needed to solve this definite integral is the [[arcsine]] function:
 
<math>c = 2r \big[ arcsin(\frac{x}{r}) \big]_{-r}^{r} = 2r \big[ arcsin(1)-arcsin(-1) \big] = 2r(\tfrac{\pi}{2}-(-\tfrac{\pi}{2})) = 2\pi r.</math>
 
Pi (π) is the ratio of the circumference of a circle to its diameter.
 
== Circumference of an ellipse ==<!-- This section is linked from [[Spherical Earth]] -->
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 <math>a,b</math> are the ellipse's [[semi-major axis|semi-major]] and [[semi-minor axis|semi-minor]] axes, respectively, and <math>o\!\varepsilon\,\!</math> is the ellipse's [[angular eccentricity]],
 
<math>o\!\varepsilon=\arccos\!\left(\frac{b}{a}\right)=2\arctan\!\left(\!\sqrt{\frac{a-b}{a+b}}\,\right);\,\!</math>
 
<math>\begin{align}\mbox{E2}\left[0,90^\circ\right]&= \mbox{Integral}'s\mbox{ divided difference};\\ Pr&=a\times\mbox{E2}\left[0,90^\circ\right] \quad(\mbox{perimetric radius});\\
c&=2\pi\times Pr.\end{align}\,\!</math>
 
There are many different [[approximation]]s for the <math>\mbox{E2}\left[0,90^\circ\right]</math> [[Difference quotient|divided difference]], with varying degrees of sophistication and corresponding accuracy.
 
In comparing the different approximations, the <math>\tan\!\left(\frac{o\!\varepsilon}{2}\right)^2\,\!</math> based series expansion is used to find the actual value:
 
<math>\begin{align}\mbox{E2}\left[0,90^\circ\right]
&=\cos\!\left(\frac{o\!\varepsilon}{2}\right)^2 \frac{1}{UT}\sum_{TN=1}^{UT=\infty}{.5\choose{}TN}^2\tan\!\left(\frac{o\!\varepsilon}{2}\right)^{4TN},\\
&=\cos\!\left(\frac{o\!\varepsilon}{2}\right)^2\Bigg(1+\frac{1}{4}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^4
+\frac{1}{64}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^8\\ &\qquad\qquad\qquad\;\,+\frac{1}{256}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^{12}
+\frac{25}{16384}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^{16}
+...\Bigg);\end{align}\,\!</math>
 
=== Muir-1883 ===
:Probably the most accurate to its given simplicity is [[Thomas Muir (mathematician)|Thomas Muir's]]:
::<math>\begin{align}Pr
&\approx\left(\frac{a^{1.5}+b^{1.6}}{2}\right)^\frac{1}{1.5}=a\left(\frac{1+\cos\!\left(o\!\varepsilon\right)^{1.5}}{2}\right)^\frac{1}{1.5},\\
&\quad\approx{a}\times\cos\!\left(\frac{o\!\varepsilon}{2}\right)^2\left(1+\frac{1}{4}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^4\right);\end{align}\,\!</math>
 
=== Ramanujan-1914 (#1,#2) ===
:[[Srinivasa Ramanujan]] introduced ''two'' different approximations, both from 1914
::<math>\begin{align}1.\;Pr&\approx\pi\Big(3(a+b)-\sqrt{\big(3a+b\big)\big(a+3b\big)}\Big),\\
&\quad=\pi{a}\bigg(6\cos\!\left(\frac{o\!\varepsilon}{2}\right)^2\sqrt{\big(3+\cos\!\left(o\!\varepsilon\right)\big)\big(1+3\cos\!\left(o\!\varepsilon\right)\big)}\bigg);\end{align}\,\!</math>
 
::<math>\begin{align}2.\;Pr&\approx\frac{1}{2}\Big(a+b\Big)\Bigg(1+\frac{3\big(\frac{a-b}{a+b}\big)^2}{10+\sqrt{4-3\big(\frac{a-b}{a+b}\big)^2}}\Bigg);\\
&\quad=a\times\cos\!\left(\frac{o\!\varepsilon}{2}\right)^2\Bigg(1+\frac{3\tan\!\big(\frac{o\!\varepsilon}{2}\big)^4}{10+\sqrt{4-3\tan\!\big(\frac{o\!\varepsilon}{2}\big)^4}}\Bigg);\end{align}\,\!</math>
 
 
: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{''oε''}, results with different ellipticities can be found and compared:
 
{|class="wikitable" HII
|-
! b !! Pr !! Ramanujan-#2 !! Ramanujan-#1 !! Muir
|-
|9975
||&nbsp;'''9987.50391&nbsp;11393'''&nbsp;
||&nbsp;'''9987.50391&nbsp;11393'''&nbsp;
||&nbsp;'''9987.50391&nbsp;11393'''&nbsp;
||&nbsp;'''9987.50391&nbsp;113'''89
|-
|9966
||&nbsp;'''9983.00723 73047'''
||&nbsp;'''9983.00723 73047'''
||&nbsp;'''9983.00723 73047'''
||&nbsp;'''9983.00723 730'''34
|-
|9950
||&nbsp;'''9975.01566 41666'''
||&nbsp;'''9975.01566 41666'''
||&nbsp;'''9975.01566 41666'''
||&nbsp;'''9975.01566 416'''04
|-
|9900
||&nbsp;'''9950.06281 41695'''
||&nbsp;'''9950.06281 41695'''
||&nbsp;'''9950.06281 41695'''
||&nbsp;'''9950.06281 4'''0704
|-
|9000
||&nbsp;'''9506.58008 71725'''
||&nbsp;'''9506.58008 71725'''
||&nbsp;'''9506.58008''' 67774
||&nbsp;'''9506.5'''7894 84209
|-
|8000
||&nbsp;'''9027.79927 77219'''
||&nbsp;'''9027.79927 77219'''
||&nbsp;'''9027.7992'''4 43886
||&nbsp;'''9027.7'''7786 62561
|-
|7500
||&nbsp;'''8794.70009 24247'''
||&nbsp;'''8794.70009 2424'''0
||&nbsp;'''8794'''.69994 52888
||&nbsp;'''8794'''.64324 65132
|-
|6667
||&nbsp;'''8417.02535 37669'''
||&nbsp;'''8417.02535 37'''460
||&nbsp;'''8417.02'''428 62059
||&nbsp;'''841'''6.81780 56370
|-
|5000
||&nbsp;'''7709.82212 59502'''
||&nbsp;'''7709.82212''' 24348
||&nbsp;'''7709.8'''0054 22510
||&nbsp;'''770'''8.38853 77837
|-
|3333
||&nbsp;'''7090.18347 61693'''
||&nbsp;'''7090.183'''24 21686
||&nbsp;'''70'''89.94281 35586
||&nbsp;'''70'''83.80287 96714
|-
|2500
||&nbsp;'''6826.49114 72168'''
||&nbsp;'''6826.4'''8944 11189
||&nbsp;'''682'''5.75998 22882
||&nbsp;'''68'''14.20222 31205
|-
|1000
||&nbsp;'''6468.01579 36089'''
||&nbsp;'''646'''7.94103 84016
||&nbsp;'''646'''2.57005 00576
||&nbsp;'''64'''31.72229 28418
|-
|&nbsp;100
||&nbsp;'''6367.94576 97209'''
||&nbsp;'''636'''6.42397 74408
||&nbsp;'''63'''46.16560 81001
||&nbsp;'''63'''03.80428 66621
|-
|&nbsp;&nbsp;10
||&nbsp;'''6366.22253 29150'''
||&nbsp;'''636'''3.81341 42880
||&nbsp;'''63'''40.31989 06242
||&nbsp;'''6'''299.73805 61141
|-
|&nbsp;&nbsp;&nbsp;1
||&nbsp;'''6366.19804 50617'''
||&nbsp;'''636'''3.65301 06191
||&nbsp;'''63'''39.80266 34498
||&nbsp;'''6'''299.60944 92105
|-
|iota
||&nbsp;'''6366.19772 36758'''
||&nbsp;'''636'''3.63636 36364
||&nbsp;'''63'''39.74596 21556
||&nbsp;'''6'''299.60524 94744
|}
 
==External links==
{{Wiktionarypar|circumference}}
* [http://home.att.net/~numericana/answer/ellipse.htm#elliptic Numericana - Circumference of an ellipse]
*[http://www.mathopenref.com/circumference.html Circumference of a circle] With interactive applet and animation
 
[[Category:Geometry]]
 
<!-- interwiki -->
 
 
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