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× diameter]]

CIRCLE

The circumference of a Circle can be calculated from its Diameter using the formula:

:c=\pi\cdot{d}.\,\!

Or, substituting the radius for the diameter:

:c=2\pi\cdot{r}=\pi\cdot{2r},\,\!

where ''r'' is the Radius and ''d'' is the diameter of the circle, and π (the Greek letter Pi ) is the Constant 3.141 592 653 589 793...


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 ) or by one of many Binomial Series expansions.

Where a,b are the ellipse's Semi-major and Semi-minor axes, respectively, and o\! arepsilon\,\! is the ellipse's Angular Eccentricity ,

o\! arepsilon=\arccos\!\left( rac{b}{a} ight)=2\arctan\!\left(\!\sqrt{ rac{a-b}{a+b}}\, ight);\,\!

\begin{align}\mbox{E2}\left[0,90^\circ ight]&= \mbox{Integral}'s\mbox{ divided difference};\
Pr&=a imes\mbox{E2}\left[0,90^\circ ight] \quad(\mbox{perimetric radius});\
c&=2\pi imes Pr.\end{align}\,\!

There are many different Approximation s for the \mbox{E2}\left[0,90^\circ ight] Divided Difference , with varying degrees of sophistication and corresponding accuracy.

In comparing the different approximations, the an\!\left( rac{o\! arepsilon}{2} ight)^2\,\! based series expansion is used to find the actual value:

\begin{align}\mbox{E2}\left[0,90^\circ ight]
&=\cos\!\left( rac{o\! arepsilon}{2} ight)^2 rac{1}{UT}\sum_{TN=1}^{UT=\infty}{.5\choose{}TN}^2 an\!\left( rac{o\! arepsilon}{2} ight)^{4TN},\
&=\cos\!\left( rac{o\! arepsilon}{2} ight)^2\Bigg(1+ rac{1}{4} an\!\left( rac{o\! arepsilon}{2} ight)^4
+ rac{1}{64} an\!\left( rac{o\! arepsilon}{2} ight)^8\ &\qquad\qquad\qquad\;\,+ rac{1}{256} an\!\left( rac{o\! arepsilon}{2} ight)^{12}
+ rac{25}{16384} an\!\left( rac{o\! arepsilon}{2} ight)^{16}
+...\Bigg);\end{align}\,\!


Muir-1883

:Probably the most accurate to its given simplicity is Thomas Muir's :
::\begin{align}Pr
&\approx\left( rac{a^{1.5}+b^{1.5}}{2} ight)^ rac{1}{1.5}=a\left( rac{1+\cos\!\left(o\! arepsilon ight)^{1.5}}{2} ight)^ rac{1}{1.5},\
&\quad\approx{a} imes\cos\!\left( rac{o\! arepsilon}{2} ight)^2\left(1+ rac{1}{4} an\!\left( rac{o\! arepsilon}{2} ight)^4 ight);\end{align}\,\!


Ramanujan-1914 (#1,#2)

: Srinivasa Ramanujan introduced ''two'' different approximations, both from 1914
::\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( rac{o\! arepsilon}{2} ight)^2\sqrt{\big(3+\cos\!\left(o\! arepsilon ight)\big)\big(1+3\cos\!\left(o\! arepsilon ight)\big)}\bigg);\end{align}\,\!

::\begin{align}2.\;Pr&\approx rac{1}{2}\Big(a+b\Big)\Bigg(1+ rac{3\big( rac{a-b}{a+b}\big)^2}{10+\sqrt{4-3\big( rac{a-b}{a+b}\big)^2}}\Bigg);\
&\quad=a imes\cos\!\left( rac{o\! arepsilon}{2} ight)^2\Bigg(1+ rac{3 an\!\big( rac{o\! arepsilon}{2}\big)^4}{10+\sqrt{4-3 an\!\big( rac{o\! arepsilon}{2}\big)^4}}\Bigg);\end{align}\,\!

: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:


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