Cunfruntu tra virsioni di "Cìrcunfirenza (giometrìa)"

nuddu riassuntu dû canciamentu
 
unni ''r'' eni lu [[raiu]], ''d'' eni lu diàmmmitru di lu cìrculu e (a [[Pi (littra)|littra greca pipi]]) eni [[pi|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 <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://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
 
[[Category:Geometry]]
 
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[[Catigurìa:Giomitrìa]]