Re: [math-fun] Two ellipse circumferences
Please stick with ASCII, as it's impossible to read these UTF8 characters (see below). What about time around an elliptical orbit? Any closed form solutions for -- e.g., approximations to #days from solstice? At 11:09 AM 10/17/2018, Bill Gosper wrote:
(Mail to some kids.)
Beginners may find it discouraging that the circumference of an ellipse requires this unfamiliar EllipticE function, but it is actually well worth familiarizing! For example, it provides the world's most rapidly convergent Ï formulas, and has fascinating properties. Actually, there are infinitely many eccentricities where the circumference EllipticE is expressible in terms radicals and factorials, but they're factorials of fractions!
You may already know that
In[210]:= (1/2)!
Out[210]= âÏ/2
but halves are the only known fractions whose factorials are familiar.
Somewhat amazingly, the 1 à 1/â2 ellipse (bounding box 2Ãâ2) has circumference 9 Ï^(3/2)/(16 (3/4)!^2) + 32 (3/4)!^2/(9âÏ) ~ 5.4025755241907 (compared with the perimeter of the bounding box = 6.82842712474619.)
But this is perhaps the nicest case.
The circumference of a 1 by (1/ð + 1/âð)/â2 ellipse is (where ð := (1+â5)/2, the Golden Ratio)
ArcLength[Circle[{,}, {(1/ð + 1/âð)/â2, 1}]] == 9 Ï^(3/2)/(10 â2 5^(7/8) ð^(1/4) (1/20)! (9/20)!) + 2 â2 5^(3/8) (4 â5 + 10 âð) (1/20)! (9/20)!/(9 ð^(1/4) âÏ) ~ 6.26092807313208, 2Ï-ish because
In[255]:= N[(1/GoldenRatio + 1/âGoldenRatio)/â2]
Out[255]= 0.992908994700242
That's rounder than an Indiana circle.
Ârwg
Oops! I think what I meant to say was a closed form *position* and/or *angle* (central or about the focus) around an elliptical orbit w.r.t. the time. At 11:20 AM 10/17/2018, Henry Baker wrote:
Please stick with ASCII, as it's impossible to read these UTF8 characters (see below).
What about time around an elliptical orbit? Any closed form solutions for -- e.g., approximations to #days from solstice?
At 11:09 AM 10/17/2018, Bill Gosper wrote:
(Mail to some kids.)
Beginners may find it discouraging that the circumference of an ellipse requires this unfamiliar EllipticE function, but it is actually well worth familiarizing! For example, it provides the world's most rapidly convergent Ï formulas, and has fascinating properties. Actually, there are infinitely many eccentricities where the circumference EllipticE is expressible in terms radicals and factorials, but they're factorials of fractions!
You may already know that
In[210]:= (1/2)!
Out[210]= âÏ/2
but halves are the only known fractions whose factorials are familiar.
Somewhat amazingly, the 1 à 1/â2 ellipse (bounding box 2Ãâ2) has circumference 9 Ï^(3/2)/(16 (3/4)!^2) + 32 (3/4)!^2/(9âÏ) ~ 5.4025755241907 (compared with the perimeter of the bounding box = 6.82842712474619.)
But this is perhaps the nicest case.
The circumference of a 1 by (1/ð + 1/âð)/â2 ellipse is (where ð := (1+â5)/2, the Golden Ratio)
ArcLength[Circle[{,}, {(1/ð + 1/âð)/â2, 1}]] == 9 Ï^(3/2)/(10 â2 5^(7/8) ð^(1/4) (1/20)! (9/20)!) + 2 â2 5^(3/8) (4 â5 + 10 âð) (1/20)! (9/20)!/(9 ð^(1/4) âÏ) ~ 6.26092807313208, 2Ï-ish because
In[255]:= N[(1/GoldenRatio + 1/âGoldenRatio)/â2]
Out[255]= 0.992908994700242
That's rounder than an Indiana circle.
Ârwg
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Henry Baker