Here some ellipse perimeter calculation methods through history.
These methods are based on approximation mostly.
I have also added integral method and infinite series method.
Some methods are more accurate when ellipse ratio is bigger and some are not.
Yet I have to tell, Ramanujan is not mere man. That man is a beast.

Regards.

ellipse.sm (80,09 KiB) scaricato 1537 volta(e).

ellipse.pdf (147,33 KiB) scaricato 1511 volta(e).

ellipse Faster.sm (16,47 KiB) scaricato 1519 volta(e).

Well your file is a different file which has relation only on integral calculation with my sample.
It has only ellipse perimeter calculation with integral functions, nothing more.

If you want a faster version of my file download it below.
Creating 10 different graphic data for 40 different ratio is time consuming.
Yet original file is slow only due to graphic of infinite serie.
If you disable it calculation will be much faster.

ellipse_without infinite.sm (79,13 KiB) scaricato 1507 volta(e).

Wrote

...
Yet original file is slow only due to graphic of infinite serie.
...

Hi overlord. You can speed up the series calculation using only one eval() in the entire document as pointed. I guess that without rounding errors because the result it's an integer. With that, you can increase the summation limit to 200 or 300 increasing the time only by one or two seconds, given a better approximation to the true percentage of Ramanujan's alien DNA.




Best regards.
Alvaro.

Wrote

Hi overlord. You can speed up the series calculation using only one eval() in the entire document as pointed. I guess that without rounding errors because the result it's an integer. With that, you can increase the summation limit to 200 or 300 increasing the time only by one or two seconds, given a better approximation to the true percentage of Ramanujan's alien DNA.

Best regards.
Alvaro.

Your suggestion really make it faster. Thanks.
I am gonna save the file respect to your correction.
Though after j>133 the eval() hits the border of max number allowed.
Or without using eval(), product gives the error above. Doesn't matter.
By setting the upper limit of product to 133 of infinite serie, error percentage is 0.08.
My brain can't comprehend how the hell Ramanujan found that formula.

Regards

Wrote

...
Though after j>133 the eval() hits the border of max number allowed.
Or without using eval(), product gives the error above. Doesn't matter.
...

Not in my SMath ( Win 32 bits, appVersion(4)="0.99.7822.147" ). I can put 500 as upper limit without problems. Here the plot with a limit of 200:



Best regards.
Alvaro.

Wrote

Not in my SMath ( Win 32 bits, appVersion(4)="0.99.7822.147" ). I can put 500 as upper limit without problems. Here the plot with a limit of 200:
Best regards.
Alvaro.

The result of P8 shall be same with P12 after j>133.
K couldn't be calculated for P8 and K of P12 shall be assigned to P8.
If you add an additional Clear(K) after P8 calculation you will see what I meant.

Regards.

Wrote

Wrote

Not in my SMath ( Win 32 bits, appVersion(4)="0.99.7822.147" ). I can put 500 as upper limit without problems. Here the plot with a limit of 200:
Best regards.
Alvaro.

The result of P8 shall be same with P12 after j>133.
K couldn't be calculated for P8 and K of P12 shall be assigned to P8.
If you add an additional Clear(K) after P8 calculation you will see what I meant.

Regards.

Hi. Thanks, you're right, calculation continues even though they have reached an error, and then they are replaced by the values of the other calculation. Funny that there is no error message when the overflow occurs.

Best regards.
Alvaro.

Ramanujan is a poet! No one understands how he found his beautifully simple and accurate formulas!

Wrote

Here some ellipse perimeter calculation methods through history.
These methods are based on approximation mostly.
I have also added integral method and infinite series method.
Some methods are more accurate when ellipse ratio is bigger and some are not.
Yet I have to tell, Ramanujan is not mere man. That man is a beast.

Regards.

ellipse.sm (80,09 KiB) scaricato 1537 volta(e).

ellipse.pdf (147,33 KiB) scaricato 1511 volta(e).

Wrote

Ramanujan is a poet !

... a Mathematician as well !!!
Here is the exact arc length of the ellipse in Smath.
Cheers ... Jean.

Circle_Ellipse_Parabola [Ellipse ArcLength].sm (20,97 KiB) scaricato 1292 volta(e).