@salim-barzani 

Q : "but why you go to end and not finish the work i wish i can do like you and i wish always finish the work"

A : I ended my work for four reasons:

1. The expressions of eq0, .., eq10 I got are not the same of those you write: Where do yours come from?
2. In what I did equations eq0, .., eq9 ( you discard eq10) depend upon 14 parameters

   {k, lambda, mu, w, B[1], B[2], a[1], a[2], a[3], a[4], a[5], alpha[0], alpha[1], beta[1]}

   So there is likely an infinity of 14-uples that verify these 10 equations.
3. On your side you want to solve your 10 equation system {eq0, .., eq9} with respect to 7 unknowns:

   {w, a[1], a[2], a[3], alpha[0], alpha[1], beta[1]}

   So it is very likely that you have no solutions at all.
4. Whatever the true expressions of these 10 equations are, they are highly non linear, which means that even if the number of unknowns was "correct", solve will never give you the formal expressions of these unknown (they would be so complex anyway that you would probably not even be able to display them).

So I didn't see the need to go any further until you had checked my expressions of the equations, correctly defined the system to be solved (why do you throw out eq10?) and its unknowns, and understood that its formal resolution is a dead end.

@vv 

The biggest circle which tangents the outer shape at a single point has these characteristics

P3 = NULL
P2 = [1.268415320, 1.896811508]
P1 = [0.6984958110, 0.5060156134]
P0 = [1.134857966, 1.138248648]
r  = 0.7692659625

A_little_help.mw

@vv 

I vote up.

I hadn't been able to pose the problem correctly so that NLPSolve could deal with it (I didn't think to use 8 parameters).

Bravo again

A question: assuming you do "blind" research in the sense you do not know how many contact points there are between the circle and the outer shape (so there are 6 or 8 parameters): how to modify your modelling to handle this?

@C_R 

Limiting the search to  the first quadrant and without using the x2=x1 (or x2=-x1) assumption I got

 [r = .7698626516, [x__ellipse = .699728, y__ellipse = .5051636992]]
 [r = .7698626516, [x__ellipse = .699729, y__ellipse = .5051630067]]

The last one giving a circle center with co-ordinates

[1.138056073, 1.138059427]

Of course these results can be enhanced by adapting the convergence criterion

The optimization algorithm I wrote needs to be enhanced before being delivered.
A first approximation (I did not pushed the iterations far enough) of the maximum radius is 0.7698669432 (contact point with the ellipse [obvious symmetries aside]: x = .6997, y = .5051830906).

Tomorrow will be a rainy day here so I will have some time to refine the algorithm and send you a first draft.


Night

@mmcdara 

I've been waiting six days for your reaction to my last comment.
I consider your silence as a sign of rudeness, to put it mildly.
I've already spent several hours on your problem, some of them to provide a solution, but quite a few also to try and drag from you the information that would have enabled me to consolidate or correct the work I did.
This same work you've always said you weren't satisfied with for a reason you haven't been able to formulate properly... which leads me to doubt you really understand your own problem.
With this in mind, I've decided to destroy all my comments or replies containing worksheets.

@salim-barzani 

All this becomes rather tedious because you're distilling the information drop by drop.
If you knew the procedure for obtaining the expressions on page 5/22, why didn't you make it clear?
I really feel like I'm wasting my time for every time I give you an answer you pull something new out of your hat.

@salim-barzani 

This is my reply to this comment of yours

Please check what I did in the attached file and answer the red highlighted question inside.
Maybe I did some typo somewhere.

Check_and_Clarify.mw

@janhardo 

Stop nit-picking saying "Indeed eq8 is derived, but it is still not the same as in the paper "
Do you mean @salim-barzani is not clever enough to see that eq8 in the paper and in my file are the same?
Would you have preferred me to write something like this
 

Download For_dumb.mw

@salim-barzani 

It took me hours to figure out how to get eq8, but here it is.

Still, I think it's all a bit of art for art's sake, and a lot of wasted time. What do you want to do, check the contents of an old book equation by equation?

 

Download transformations_(complete).mw

Proceed the same way for case lambda > 0.

@minhthien2016 

and read carefully the definition in the reference you give: the dihedral angle is in the range 0..Pi/2 ... unless yoiu have another definition (see also here Wolfram)

The content of this comment has been deleted by the author: an updated and complete version can be found here

Reminder: The paper contains several typos mentioned in the attached new file.

@salim-barzani 
 

1. Open the mw file I sent you within your Maple version.
2. Execute it and save it.
3. Sent a reply including YOUR mw file.
   (I can't do anything with your screen capture)
    

@salim-barzani 

My answer to this reply of yours


Add this at the end of my file
 

for neq in [9, 11] do
  eq := eq_||neq:
  EQ_||neq := lhs(eq) = add(op(1..-2, rhs(eq))) 
              + 
              ``(expand(eval(op(-1, rhs(eq)), G(eta) = 1))) 
              *
              ``(1/indets(rhs(eq), function)[])^2
end do:

neq   := 13:
eq    := eq_||neq:
EQ_||neq := lhs(eq) = op(1, rhs(eq))
            + 
            ``(expand(eval(op(-1, rhs(eq)), G(eta) = 1))) 
            *
            ``(1/indets(rhs(eq), function)[])^2:

print~([EQ_9, EQ_11, EQ_13]):

result:


Where is the problem ???