We consider the following configurations :

Let $X_{1}=(0\ 0\ 0)$, $X_{2}=(\frac{1}{2}\ \frac{-1}{2}\ 0)$, $X_{3}=(\frac{3}{2}\ 1\ 0.3)$, $X_{4}=(\frac{-3}{2}\ \frac{1}{2}\ 0)$, $X_{5}=(\frac{1}{2}\ \frac{1}{3}\ 0.1)$, $X_{6}=(0\ 2\ 0)$,
be fixed points and $Y_{1}=(0\ 0\ 0)$, $Y_{2}=(1\ 1\ 0)$, $Y_{3}=(0\ 1\ 0)$, $Y_{4}=(\frac{3}{2}\ 1\ 0)$, $Y_{5}=(\frac{-3}{2}\ \frac{1}{2}\ 0)$, Y₆=Y₁,
attached to the moving platform.

We choose the square of lengths :
l₁=2, $l_{2}= \frac{15}{2}$, $l_{3}=\frac{17}{4}$, $l_{4}= \frac{49}{4}$, l₅=14, l₆=2.

Let $\alpha_{i}=l_{i}-l_{1}$, we obtain the following system to solve :
 
 

$$-\alpha_{2}a^{2}+3xa^{2}+ya^{2}+4zab-\alpha_{2}b^{2}+3xb^{2}-......ya+2.5a^{2}+4zb+4.5b^{2}+4zc+0.5c^{2}-\alpha_{2}-x-3y-4a+2.5=0$$ (1)

 

$$-\alpha_{3}a^{2}-3ya^{2}-\alpha_{3}b^{2}-3yb^{2}-4zac+4ybc-\a......25a^{2}+4zb+2.25b^{2}-6bc+6.25c^{2}-\alpha_{3}-4x-3y-6a+6.25=0$$ (2)

 

$$-\alpha_{4}a^{2}+xa^{2}+6ya^{2}+6zab-\alpha_{4}b^{2}+xb^{2}-4......a^{2}+4zb+0.25b^{2}+6zc+3bc+11.25c^{2}-\alpha_{4}-3x+9a+2.25=0$$ (3)

 

$$-\alpha_{5}a^{2}+4xa^{2}-4ya^{2}-6zab-\alpha_{5}b^{2}+4xb^{2}......+2ya+8a^{2}+2zb+5b^{2}-6zc-10bc+5c^{2}-\alpha_{5}+2x+2y+8a+2=0$$ (4)

 

$$-\alpha_{6}a^{2}-4xa^{2}-\alpha_{6}b^{2}-4xb^{2}-\alpha_{6}c^{2}-4xc^{2}+4a^{2}+4b^{2}+4c^{2}-\alpha_{6}-4x+4=0$$ (5)

Approximations of the solutions are :
  

t	x	y	z
-0.8559741511414334	1.000000	0.1542734032036417	-0.9880281964923707
0.8559741511414365	1.000000	0.1542734032036381	0.9880281964923713
-0.5208622044322742	1.000000	0.5731959676433147	-0.8194183197106615
0.5208622044322728	1.000000	0.5731959676433166	0.8194183197106604
0.999999980272572	1.000000	1.972742819160095e-08	0.9999999999999998
1.000000019727433	1.000000	-1.972743263130614e-08	0.9999999999999998

 
 

a	b	c	d
-0.30022899732068	-0.2524706689758425	-0.008478921420018205	1.000000
0.3002289973206608	0.2524706689758229	-0.00847892142001521	1.000000
0.7526500320258817	-0.5609206400435618	-0.1171358376325	1.000000
-0.7526500320258833	0.5609206400435761	-0.1171358376324918	1.000000
2.219335519681477e-08	7.397784242297935e-09	4.931857996019877e-09	1.000000
-2.219336279616145e-08	-7.397788962834889e-09	-4.931857226361814e-09	1.000000

We can see than the two last solutions can be considered as one double solution.

1999-10-01