https://vk.com/doc242471809_320829803
https://vk.com/doc242471809_321212555 

        Initial curve on the surface:

x3-0.1e-1*exp(x1)/(0.1e-1+x1^4+x2^4)=0,

obtained from the intersection of the sphere:

x1^2+(x2-.5)^2+x3^2-2=0.

      Building a geodesic between selected points. Testing shows that the normal to the surface in any point of the our curve lies on the osculating plane. (Show a picture of all the points together with the planes is not possible, so one pointis shown in two positions and two corresponding osculating planes. In the initial positionof the osculating planeis painted in blue, in the end position – green.)
      Not every geodesic curve is the shortest distance between the points, but we can construct a set of geodetic and select from them the shortest. If an initial curve will be on the other side "bulge",  the first geodesic would be the shortest geodesic.

The same algorithm can be used to construct  parallel curves  on surfaces.
For example, three  parallel curves  on the surface:

(x1-sin((x1^2+x2^2+x3^2)^.5)^2)^2+(x2-sin(x1)^2)^2+(x3-sin(x1)^2)^2-9=0;

.gif file,    4-bar linkage spherical mechanism animation

@Carl Love 

Many examples on my page. Examples constructing geodesic repeated from the beginning (intermittently) to reduce file size and mechanisms try to finish smoothly animation to look smooth and continuous.
I understand the question? I just really do not know much English

@Carl Love 

Yes, it is. GIF. Usually inserted - through the green arrow ... More do not know anything. I have all the files .Gif animation as shown.

http://www.maplesoft.com/applications/view.aspx?SID=34940

and:

https://vk.com/doc242471809_332342637

https://vk.com/doc242471809_332344020

restart:
f:=(x2^20*x4^2)/(x1^10*x3^5*x5*x6^5);
n:=nops(f); 
x := seq(eval(cat('x', i)), i = 1 .. n);
for i to n do
k:= degree(f,x[i]); if k<0 then print( 1/x[i],k); fi ;od:

restart:

f := x1^2/(x2^3*x3^2*x4*x5^10);

n := 5;

x := seq(eval(cat('x', i)), i = 1 .. n);

for i from 2 to n do

op(1, op(i, f))^sign(op(2, op(i, f)))

                                                      end do;

@Hypnos 

restart: with(RootFinding):

f1 := c1+2*c2-3:

f2 := c1+9*c2-5:

f3 := c1^2+c2^2-1:

solve({f1, f2, f3}, {c1, c2});

Isolate([f1, f2, f3], [c1, c2]);

“Solve” is nothing  and  “Isolate”  is  empty... Maple15

“nothing return after solve({y1 = c1*x1 + c2*x2, y2 = c1*x3 + c2*x4, c1^2 + c2^2 = 1}, {c1,c2});”

How many variables (n) and how many equations (m)? It is necessary that n = m  

restart;

f := ln(x);

(diff(f, x))/(diff(ln(x), x));

1;

@Markiyan Hirnyk  ?

http://hostingkartinok.com/show-image.php?id=de847a09264bc43e9ad728995cfeeb59

http://hostingkartinok.com/show-image.php?id=09dcc30c0406a759d0b4e9a9d314e6a4

That was my mistake.

Draghilev.

Small remark: Diaghilev. Draghilev A.V. requested the so to write his name in English.