@Markiyan Hirnyk In the uploaded worksheet SolveEquations is changed to include AllSolutions=true, solutions=1. Option initialpoint which is included in SolveEquations in the original worksheet is not allowed.

When the worksheet is executed, each iteration of the for..do loop takes considerable time.

The DEBUG command following SolveEquations shows that the solved value A is very close to 2.5 for the first 5 iterations, then for the sixth iteration it becomes negative 2.5.

The values of A produced by SolveEquations in the original worksheet, for the first 3 iterations are 2.4696, 2.3305, and 2.2001 through 4 decimal places. These values produce the correct first 3 animation frames.

Archimedean_spiral_cam_follower_2.mw

@vv I will try to restate my worksheet with my Archimedean spirals converted to polar coordinates. 

@Markiyan Hirnyk I have just sent an email to Mr. Moiseev with a link to this Maple forum webpage. I asked for any suggestions which would help me make better use of DirectSearch.

@Markiyan Hirnyk Thank you for your suggestons.

AllSolutions by itself produced 99 solutions. AllSolutions, solutions=1 produced the first of these which gave incorrect values.

@tomleslie I thank you enormously for grasping my problem and putting so much obvious effort into it!

It's going to take me some time to absorb all you have done, so I wanted to reply to you and "boost" your answer now, to show my gratitude. 

@tomleslie Thank you a heap for the lesson on by-pass reference.

I did not know that IntSect[i]  = xxxx automatically implements IntSect as a table if it is not previously otherwise defined as a Matrix or Array.

@Rouben Rostamian  Your reply has taught me a new technique, namely how to delimit two parametrically defined surfaces with respect to each other.

Even after 12 years working with math in Maple it seems I have barely tapped its potential.

I am most grateful to you for this and past assistance.

@rlopez Dr. Lopez, I hope the attached worksheet permits you to see what I want to do and explains my comment about awkward coding.

Desired_display.mw

@vv It seems that plot3d wants a separate procedure for each of the three elements in a surface's parametric definition. This is awkward coding. Perhaps MapleSoft could modify plot3d with an option permitting a single procedure in place of three.

@vv Based on your mention of the role of the intersection, here is another way to determine phi1

phi1 := solve(add(`~`[rhs](el)[i]^2, i = 1 .. 3) = add(`~`[rhs](sp)[i]^2, i = 1 .. 3), phi)[1];
          
 

@vv Please explain the derivation of angle phi1 such that it limits the display of the ellipsoid to the portion of its surface which lies outside of the sphere.

Can you also show the derivation of a comparable angle which will limit the display of the sphere to the portion of its surface which lies inside the ellipsoid? 

@MapleMathMatt Thanks to you I now understand the meaning of map's index. BTW this also works:

`~`[P]([A1, A2, A3, A4], [B1, B2, B3, B4])

@MapleMathMatt I have a lot to learn about map. How does its action change when map is followed by a tilde?

@Joe Riel There is a lot to learn about how map performs. The advice below by MapleMathMatt 55 is informative

@Joe Riel Interesting that the A's must be a list of lists but the B's are a simple list.