Elliptic integrals are well known as

and since 1959 it is also well known that

 

L=a*(1+(b/a)^s)^(1/s)

 

This is the solvable equivalent of unsolvable integrals !!!!

 

In fact:

 

Thales theorem-N2 says:

at point P1(x,y)

                           a/b=(a^r-x^r)^(1/r)/y

                            A/B=(A^t-x^t)^(1/t)/y

at point P2(a,b)

                            A/B=(A^s-a^s)^(1/s)/b

 

 

 

With an algebraic operation it is proofed that

s=r*t/(r-t)

see: http://local.wasp.uwa.edu.au/~pbourke/

 

When A=B=K, we write [a^s+b^s=K^s]

say: b/a=TAN                 (1+TAN^s)^(1/s)=K is written.

 

What is “K” ?

K is the total arc length on the positive Cartesian,

K is the total area on the positive Cartesian.

K is any unsolved elliptic value on positive Cartesian.

 

What is “s”

s is a variable power s=s(r,TAN).

s is known graphically.For example in the case of an ellipse’s perimeter evaluation sGraph looks like in Fig.I

Fig.I

 

This graph looks like an astroid.We write an (sMod), astroidal math-model, under our control .The overlapping of the graphs are shown on Fig.II

Fig.II

 

Finally we obtain an accurate estimation of the perimeter of an ellipse.

This accuracy reached to error %=-0,000002432…on 2009 March.

You may go for better results,if interested with this trendy problem.
Trendy since Kepler (1609) and the last Master of the world in this concern,

Ramanujan (1920).Read article Ramanujan on Triond.