In place of a separate Ellipse primitive, Mathematica generalizes the Circle primitive by (optionally) replacing the radius slot by the pair of semiaxes.
In[]:=
circumference[Circle[center_,{a_:1,b_}]]:=4bEllipticE[1-a^2/b^2]
In[]:=
circumference[Circle[_,{1,1}]]
Out[]=
2π
In[]:=
circumferenceCircle_,1,√2
Out[]=
4
2
EllipticE
1
2

In[]:=
Equal[%,FunctionExpand@%,N@%]
Out[]=
4
2
EllipticE
1
2

2
3/2
π
2
Gamma
3
4

+2
2
π
2
Gamma
3
4

7.6404
The circumference of the
1×√2
ellipse is comfortably
>2π
, and free of EllipticE. Unfortunately, this seems to be the only such case known to Mathematica, and I haven’t found the tables of special values of EllipticE. But such values are apparently innumerable. We can find some with Mathematica’s AskConstants facility. Defaulting the (in this case) semiminor axis to 1, with semimajor axis
b=√2(√3-1)
, we get
In[]:=
circumference[Circle[_,{1,Sqrt[2](Sqrt[3]-1)}]]//FunctionExpand//FullSimplify
Out[]=
4
2
(-1+
3
)EllipticE
1
2
-
3
4

So let’s see if EllipticE[1/2 - Sqrt[3]/4] can be”nicely” free of EllipticE:
tim[xp_]:=(Print[#[[1]],",",Length[#[[2]]]];#[[2]])&@AbsoluteTiming[xp](*timismyidentityfunctionthatinterjectselapsedtimeandlength.*)
Propose[N[%[[4]],33]]//tim
{{{(Sqrt[π]((2Gamma[5/6])/Gamma[1/3]+((3+Sqrt[3])Gamma[4/3])/Gamma[5/6]))/(43^(1/4)),Excellent Agreement≃48.,Entropy10≃25.7970587526855,Excellent Margin≃22.2029412473145}}}
Bingo! So, with
In[]:=
b:=√2√3-1
ellipse has circumference
(
3
-1)
2π
2Γ
5
6

Γ
1
3

+
(3+
3
)Γ
4
3

Γ
5
6

4
3
In[]:=
4bEllipticE[1-
-2
b
]
Out[]=
4
2
(-1+
3
)EllipticE1-
1
2
2
(-1+
3
)

In[]:=
N@%
Out[]=
6.39449
In[]:=
ArcLength@Circle[{0,0},{a,b}]
Out[]=
4bEllipticE1-
2
a
2
b

Even though EllipticE is simply a definite integral, it is more “special” than than, say, the Gamma function. Incidentally, 𝚪(¼) is the rightful claimant to the symbol 𝜏. It would be lamebrained to waste it on 6.28!

CITE THIS NOTEBOOK

Which ellipses have "nice" circumferences? …or when can we eschew EllipticE?​
by Bill Gosper​
Wolfram Community, STAFF PICKS, February 26, 2024
​https://community.wolfram.com/groups/-/m/t/3130830