Science > Physics > A short, exact equation for the perimeter of ellipse, without pi, angles, or eliptic modulus
| Topic: |
Science > Physics |
| User: |
"Golden Boar" |
| Date: |
10 Oct 2005 06:33:02 AM |
| Object: |
A short, exact equation for the perimeter of ellipse, without pi, angles, or eliptic modulus |
An ellipse can be split into an infinite number of evenly spaced lines
of varying length(height). These lines pass through the x-axis at 90
degrees. The first and last lines have a length of 0 and the middle
line has a length equal to the height of the ellipse. The distance
along the x-axis between each line is l/a. The x-axis starts from the
left at 0, there are no negative x values.
To convert these x values into normal Cartesian values: x-l/2.
The distance between a specific line and the first line is:
x = l*b/a
The height of a specific line is:
y = 2*h*sqrt(a*b-b^2)/a
The area of the ellipse is:
A = (l/a)*SUMOF(2*h*sqrt(a*b-b^2)/a)
where b = 0 to a.
a
SUMOF()
b=0
The circumference of the ellipse is:
C =
2*SUMOF(sqrt((l/a)^2+((h/a)*(sqrt(a*(b+1)-(b+1)^2)-sqrt(a*b-b^2)))^2))
where b = 0 to a-1.
a-1
SUMOF()
b=0
l is the length of the ellipse.
h is the height of the ellipse.
a is the number of points equally distributed along l.
b is a specific point along l.
0 = b = a, in steps of 1.
x is the distance between the start of the ellipse and point b.
y is the perpendicular expansion of b, centred on l (the height of a
line going through the x axis at 90 degrees).
A is the area of the ellipse.
C is the circumference of the ellipse.
For an ellipse of length=2 and height=1, the exact infinite series
gives a value of 4.844224110291 (62 terms).
My equation gives a value of 4.84422411023477 (a=1*10^9 points).
I have posted these equations on wiki, you can see them more clearly
at:
http://en.wikipedia.org/wiki/Talk:Ellipse
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| User: "Uncle Al" |
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| Title: Re: A short, exact equation for the perimeter of ellipse, without pi,angles, or eliptic modulus |
10 Oct 2005 12:01:52 PM |
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Golden Boar wrote:
An ellipse can be split into an infinite number of evenly spaced lines
of varying length(height). These lines pass through the x-axis at 90
degrees. The first and last lines have a length of 0 and the middle
line has a length equal to the height of the ellipse. The distance
along the x-axis between each line is l/a. The x-axis starts from the
left at 0, there are no negative x values.
To convert these x values into normal Cartesian values: x-l/2.
[snip]
Ah, fella - calculus.
http://www.csgnetwork.com/circumellipse.html
There is no avoiding the formula. Euler's equation unites analytic
geometry with algebra.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz.pdf
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| User: "Golden Boar" |
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| Title: Re: A short, exact equation for the perimeter of ellipse, without pi, angles, or eliptic modulus |
11 Oct 2005 11:09:51 AM |
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Just in case you are interested, my equation only becomes less accurate
than the one on the above link when using less than 7 points.
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| User: "Golden Boar" |
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| Title: Re: A short, exact equation for the perimeter of ellipse, without pi, angles, or eliptic modulus |
11 Oct 2005 11:14:22 AM |
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I meant 6 points, not 7.
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| User: "Golden Boar" |
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| Title: Re: A short, exact equation for the perimeter of ellipse, without pi, angles, or eliptic modulus |
10 Oct 2005 12:31:40 PM |
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Uncle Al wrote:
Golden Boar wrote:
An ellipse can be split into an infinite number of evenly spaced lines
of varying length(height). These lines pass through the x-axis at 90
degrees. The first and last lines have a length of 0 and the middle
line has a length equal to the height of the ellipse. The distance
along the x-axis between each line is l/a. The x-axis starts from the
left at 0, there are no negative x values.
To convert these x values into normal Cartesian values: x-l/2.
[snip]
Ah, fella - calculus.
http://www.csgnetwork.com/circumellipse.html
There is no avoiding the formula. Euler's equation unites analytic
geometry with algebra.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz.pdf
My equation is a hell of alot more accurate than the one on the above
link.
As I said before, when using a billion(10^9) points, the equation is
near enough exactly the same as the exact infinite series using 62
terms.
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| User: "" |
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| Title: Re: A short, exact equation for the perimeter of ellipse, without pi, angles, or eliptic modulus |
10 Oct 2005 06:39:24 AM |
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I've discovered that if you try to pick a VW up by its bumper, it'll
come right off in your hands. So there.
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| User: "Golden Boar" |
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| Title: Re: A short, exact equation for the perimeter of ellipse, without pi, angles, or eliptic modulus |
10 Oct 2005 06:42:53 AM |
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Good for you, ya loon. Do you want a medal or something?
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